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Andrius Kulikauskas

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Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software

Book.CategoryTheoryGlossary istorija

Paslėpti nežymius pakeitimus - Rodyti galutinio teksto pakeitimus

2019 balandžio 05 d., 10:25 atliko AndriusKulikauskas -
Pridėta 10 eilutė:
* Think through in what sense Product means "and" and Coproduct means "or".
2019 kovo 14 d., 00:03 atliko AndriusKulikauskas -
Pakeista 106 eilutė iš:
* A category is wellpowered if for each object there is only a set of pairwise non-isomorphic '''subobject'''s.
į:
* A category is''' wellpowered''' if for each object there is only a set of pairwise non-isomorphic '''subobject'''s.
2019 kovo 06 d., 09:09 atliko AndriusKulikauskas -
Pakeistos 146-151 eilutės iš
* A category is '''complete''' if all small limits exist
į:
* A category is '''complete''' if all small limits exist.
* A '''finitely complete category'' is a category C which admits all finite limits.
* A '''[[https://ncatlab.org/nlab/show/regular+category | regular category]]''' is a finitely complete category which admits a good notion of image factorization. A primary raison d’être behind regular categories C is to have a decently behaved calculus of relations in C.
* A '''coherent category''' (also called a '''pre-logos''') is a regular category in which the subobject posets Sub(X) all have finite unions which are preserved by the base change functors {$f^*:Sub(Y)\to Sub(X)$}.
* A '''[[https://ncatlab.org/nlab/show/geometric+category | geometric category]]''' is a regular category in which the subobject posets Sub(X) have all small unions which are stable under pullback.
* Well-powered.
2019 kovo 03 d., 22:19 atliko AndriusKulikauskas -
Pridėtos 316-320 eilutės:

* The '''[[https://en.wikipedia.org/wiki/Direct_limit | direct limit of algebraic objects]]''' is a colimit.
* The '''[[https://en.wikipedia.org/wiki/Direct_limit | direct limit in an arbitrary category]]''' is a colimit.
* The '''[[https://en.wikipedia.org/wiki/Inverse_limit | inverse limit of algebraic objects]]''' is a limit.
* The '''[[https://en.wikipedia.org/wiki/Inverse_limit | inverse limit in an arbitrary category]]''' is a limit.
2019 vasario 15 d., 10:07 atliko AndriusKulikauskas -
Pakeista 137 eilutė iš:
* A category is ''monoidal closed''' if it is both monoidal and closed in a compatible way.
į:
* A category is ''closed monoidal''' if it is both monoidal and closed in a compatible way. It is the most general framework which allows currying and uncurrying.
2019 vasario 13 d., 14:52 atliko AndriusKulikauskas -
Pakeistos 15-16 eilutės iš
This is a glossary of properties and concepts in '''category theory''' in '''mathematics'''.
į:
This is a glossary of properties and concepts in '''category theory''' in '''mathematics'''. It is based especially on the following sources:
* [[https://en.wikipedia.org/wiki/Glossary_of_category_theory | Wikipedia: Glossary of category theory]]
2019 vasario 13 d., 14:51 atliko AndriusKulikauskas -
Pridėta 6 eilutė:
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Ištrinta 7 eilutė:
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2019 vasario 13 d., 14:51 atliko AndriusKulikauskas -
Pridėta 6 eilutė:
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Ištrintos 7-8 eilutės:
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2019 vasario 13 d., 14:49 atliko AndriusKulikauskas -
Pridėtos 7-12 eilutės:
>>bgcolor=#FFFFC0<<

* Look for the structure that is needed for each of Grothendieck's six operations.
* Organize the glossary in terms of canonical examples. Provide additional examples.

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2019 vasario 13 d., 14:46 atliko AndriusKulikauskas -
Pridėtos 1-2 eilutės:
>>bgcolor=#E9F5FC<<
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Pridėtos 4-8 eilutės:

'''Investigation: Organize the concepts in category theory to reveal underlying themes'''
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2019 vasario 13 d., 14:09 atliko AndriusKulikauskas -
Pridėta 122 eilutė:
* A category is '''closed''' if it has an internal Hom functor.
Pridėtos 124-125 eilutės:
* A category is ''monoidal closed''' if it is both monoidal and closed in a compatible way.
* A category is '''compact closed''' if it is a monoidal closed category that supports dual objects, as in the case of a finite dimensional vector space.
2019 vasario 12 d., 11:26 atliko AndriusKulikauskas -
Pakeista 251 eilutė iš:
* Given a category ''C'', the left '''Kan extension''' functor along a functor {$f: I \to J$} is the left adjoint (if it exists) to {$f^* = - \circ f: \operatorname{Fct}(J, C) \to \operatorname{Fct}(I, C)$} and is denoted by {$f_!$}. For any {$\alpha: I \to C$}, the functor {$f_! \alpha: J \to C$} is called the left Kan extension of α along ''f''.<ref>http://www.math.harvard.edu/~lurie/282ynotes/LectureXI-Homological.pdf</ref> One can show: {$(f_! \alpha)(j) = \varinjlim_{f(i) \to j} \alpha(i)$} where the colimit runs over all objects {$f(i) \to j$} in the comma category. The right Kan extension functor is the right adjoint (if it exists) to {$f^*$}.
į:
* Given a category ''C'', the left '''Kan extension''' functor along a functor {$f: I \to J$} is the left adjoint (if it exists) to {$f^* = - \circ f: \operatorname{Fct}(J, C) \to \operatorname{Fct}(I, C)$} and is denoted by {$f_!$}. For any {$\alpha: I \to C$}, the functor {$f_! \alpha: J \to C$} is called the left Kan extension of α along ''f''.[[http://www.math.harvard.edu/~lurie/282ynotes/LectureXI-Homological.pdf | reference]]. One can show: {$(f_! \alpha)(j) = \varinjlim_{f(i) \to j} \alpha(i)$} where the colimit runs over all objects {$f(i) \to j$} in the comma category. The right Kan extension functor is the right adjoint (if it exists) to {$f^*$}.
2019 vasario 12 d., 11:26 atliko AndriusKulikauskas -
Pakeistos 258-259 eilutės iš
* A '''monad''' in a category ''X'' is a '''monoid object''' in the monoidal category of endofunctors of ''X'' with the monoidal structure given by composition. For example, given a group ''G'', define an endofunctor ''T'' on '''Set''' by {$T(X) = G \times X$}. Then define the multiplication ''μ'' on ''T'' as the natural transformation {$\mu: T \circ T \to T$} given by {$\mu_X: G \times (G \times X) \to G \times X, \,\, (g, (h, x)) \mapsto (gh, x)$}
and also define the identity map ''η'' in the analogous fashion. Then (''T'', ''μ'', ''η'') constitutes a monad in '''Set'''. More substantially, an adjunction between functors {$F: X \rightleftarrows A : G$} determines a monad in ''X''; namely, one takes {$T = G \circ F$}, the identity map ''η'' on ''T'' to be a unit of the adjunction and also defines ''μ'' using the adjunction.
į:
* A '''monad''' in a category ''X'' is a '''monoid object''' in the monoidal category of endofunctors of ''X'' with the monoidal structure given by composition. For example, given a group ''G'', define an endofunctor ''T'' on '''Set''' by {$T(X) = G \times X$}. Then define the multiplication ''μ'' on ''T'' as the natural transformation {$\mu: T \circ T \to T$} given by {$\mu_X: G \times (G \times X) \to G \times X, \,\, (g, (h, x)) \mapsto (gh, x)$} and also define the identity map ''η'' in the analogous fashion. Then (''T'', ''μ'', ''η'') constitutes a monad in '''Set'''. More substantially, an adjunction between functors {$F: X \rightleftarrows A : G$} determines a monad in ''X''; namely, one takes {$T = G \circ F$}, the identity map ''η'' on ''T'' to be a unit of the adjunction and also defines ''μ'' using the adjunction.
2019 vasario 12 d., 09:57 atliko AndriusKulikauskas -
Pakeistos 258-259 eilutės iš
* A '''monad''' in a category ''X'' is a '''monoid object''' in the monoidal category of endofunctors of ''X'' with the monoidal structure given by composition. For example, given a group ''G'', define an endofunctor ''T'' on '''Set''' by {$T(X) = G \times X$}. Then define the multiplication ''μ'' on ''T'' as the natural transformation {$\mu: T \circ T \to T$} given by
:{$\mu_X: G \times (G \times X) \to G \times X, \,\, (g, (h, x)) \mapsto (gh, x)$}
į:
* A '''monad''' in a category ''X'' is a '''monoid object''' in the monoidal category of endofunctors of ''X'' with the monoidal structure given by composition. For example, given a group ''G'', define an endofunctor ''T'' on '''Set''' by {$T(X) = G \times X$}. Then define the multiplication ''μ'' on ''T'' as the natural transformation {$\mu: T \circ T \to T$} given by {$\mu_X: G \times (G \times X) \to G \times X, \,\, (g, (h, x)) \mapsto (gh, x)$}
2019 vasario 12 d., 09:41 atliko AndriusKulikauskas -
Pakeistos 201-203 eilutės iš
* Given categories ''C'', ''D'', a '''functor''' ''F'' from ''C'' to ''D'' is a structure-preserving map from ''C'' to ''D''; i.e., it consists of an object ''F''(''x'') in ''D'' for each object ''x'' in ''C'' and a morphism ''F''(''f'') in ''D'' for each morphism ''f'' in ''C'' satisfying the conditions: (1) {$F(f \circ g) = F(f) \circ F(g)$} whenever {$f \circ g$} is defined and (2) {$F(\operatorname{id}_x) = \operatorname{id}_{F(x)}$}. For example,
:{$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}, \, S \mapsto \mathfrak{P}(S)$},
where {$\mathfrak{P}(S)$} is the '''power set''' of ''S'' is a functor if we define: for each function {$f: S \to T$}, {$\mathfrak{P}(f): \mathfrak{P}(S) \to \mathfrak{P}(T)$} by {$\mathfrak{P}(f)(A) = f(A)$}.
į:
* Given categories ''C'', ''D'', a '''functor''' ''F'' from ''C'' to ''D'' is a structure-preserving map from ''C'' to ''D''; i.e., it consists of an object ''F''(''x'') in ''D'' for each object ''x'' in ''C'' and a morphism ''F''(''f'') in ''D'' for each morphism ''f'' in ''C'' satisfying the conditions: (1) {$F(f \circ g) = F(f) \circ F(g)$} whenever {$f \circ g$} is defined and (2) {$F(\operatorname{id}_x) = \operatorname{id}_{F(x)}$}.
*
For example, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}, \, S \mapsto \mathfrak{P}(S)$}, where {$\mathfrak{P}(S)$} is the '''power set''' of ''S'' is a functor if we define: for each function {$f: S \to T$}, {$\mathfrak{P}(f): \mathfrak{P}(S) \to \mathfrak{P}(T)$} by {$\mathfrak{P}(f)(A) = f(A)$}.
2019 vasario 12 d., 09:37 atliko AndriusKulikauskas -
Pridėta 218 eilutė:
Pakeistos 220-221 eilutės iš
į:
* A functor π:''C'' → ''D'' is an '''op-fibration''' if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one π-coCartesian morphism ''f'': ''x'' → ''y'' in ''C'' such that π(''f'') = ''g''. In other words, π is the dual of a '''Grothendieck fibration'''.
Ištrintos 224-225 eilutės:
* A set-valued contravariant functor ''F'' on a category ''C'' is said to be '''representable''' if it belongs to the essential image of the '''Yoneda embedding''' {$C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}; i.e., {$F \simeq \operatorname{Hom}_C(-, Z)$} for some object ''Z''. The object ''Z'' is said to be the representing object of ''F''.
* An [adjective] object in a category ''C'' is a contravariant functor (or presheaf) from some fixed category corresponding to the "adjective" to ''C''. For example, a '''simplicial object''' in ''C'' is a contravariant functor from the simplicial category to ''C'' and a '''Γ-object''' is a pointed contravariant functor from '''Γ''' (roughly the pointed category of pointed finite sets) to ''C'' provided ''C'' is pointed.
Pridėtos 227-241 eilutės:
* A '''simplicial object''' in a category ''C'' is roughly a sequence of objects {$X_0, X_1, X_2, \dots$} in ''C'' that forms a simplicial set. In other words, it is a covariant or contravariant functor Δ → ''C''. For example, a '''simplicial presheaf''' is a simplicial object in the category of presheaves.
* A '''simplicial set''' is a contravariant functor from &Delta; to '''Set''', where &Delta; is the '''simplex category''', a category whose objects are the sets [''n''] = { 0, 1, …, ''n'' } and whose morphisms are order-preserving functions. One writes {$X_n = X([n])$} and an element of the set {$X_n$} is called an ''n''-simplex. For example, {$\Delta^n = \operatorname{Hom}_{\Delta}(-, [n])$} is a simplicial set called the standard ''n''-simplex. By Yoneda's lemma, {$X_n \simeq \operatorname{Nat}(\Delta^n, X)$}.
* The functor {$y: C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(-, X)$} is fully faithful and is called the Yoneda embedding. Technical note: the lemma implicitly involves a choice of '''Set'''; i.e., a choice of universe.
* A set-valued contravariant functor ''F'' on a category ''C'' is said to be '''representable''' if it belongs to the essential image of the '''Yoneda embedding''' {$C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}; i.e., {$F \simeq \operatorname{Hom}_C(-, Z)$} for some object ''Z''. The object ''Z'' is said to be the representing object of ''F''.
* An [adjective] object in a category ''C'' is a contravariant functor (or presheaf) from some fixed category corresponding to the "adjective" to ''C''. For example, a '''simplicial object''' in ''C'' is a contravariant functor from the simplicial category to ''C'' and a '''Γ-object''' is a pointed contravariant functor from '''Γ''' (roughly the pointed category of pointed finite sets) to ''C'' provided ''C'' is pointed.

* Given categories ''C'' and ''D'', a '''profunctor''' (or a distributor) from ''C'' to ''D'' is a functor of the form {$D^{\text{op}} \times C \to \mathbf{Set}$}.
* '''distributor'''. Another term for "profunctor".

* If {$f: C \to D, \, g: D \to E$} are functors, then the '''composition''' {$g \circ f$} or {$gf$} is the functor defined by: for an object ''x'' and a morphism ''u'' in ''C'', {$(g \circ f)(x) = g(f(x)), \, (g \circ f)(u) = g(f(u))$}.
* Given categories ''C'', ''D'' and an object ''A'' in ''C'', the '''evaluation''' at ''A'' is the functor:{$\mathbf{Fct}(C, D) \to D, \,\, F \mapsto F(A).$} For example, the '''Eilenberg–Steenrod axioms''' give an instance when the functor is an equivalence.

* A '''(combinatorial) species''' is an endofunctor on the groupoid of finite sets with bijections. It is categorically equivalent to a '''symmetric sequence'''.
* A functor from the category of finite-dimensional vector spaces to itself is called a '''polynomial functor''' if, for each pair of vector spaces ''V'', ''W'', {{nowrap|''F'': Hom(''V'', ''W'') → Hom(''F''(''V''), ''F''(''W''))}} is a polynomial map between the vector spaces. A '''Schur functor''' is a basic example.
Ištrintos 264-279 eilutės:





* '''distributor'''. Another term for "profunctor".

* Given categories ''C'', ''D'' and an object ''A'' in ''C'', the '''evaluation''' at ''A'' is the functor:{$\mathbf{Fct}(C, D) \to D, \,\, F \mapsto F(A).$} For example, the '''Eilenberg–Steenrod axioms''' give an instance when the functor is an equivalence.


* A functor π:''C'' → ''D'' is an '''op-fibration''' if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one π-coCartesian morphism ''f'': ''x'' → ''y'' in ''C'' such that π(''f'') = ''g''. In other words, π is the dual of a '''Grothendieck fibration'''.
* A functor from the category of finite-dimensional vector spaces to itself is called a '''polynomial functor''' if, for each pair of vector spaces ''V'', ''W'', {{nowrap|''F'': Hom(''V'', ''W'') → Hom(''F''(''V''), ''F''(''W''))}} is a polynomial map between the vector spaces. A '''Schur functor''' is a basic example.

* Given categories ''C'' and ''D'', a '''profunctor''' (or a distributor) from ''C'' to ''D'' is a functor of the form {$D^{\text{op}} \times C \to \mathbf{Set}$}.
Ištrintos 265-271 eilutės:

* A '''simplicial object''' in a category ''C'' is roughly a sequence of objects {$X_0, X_1, X_2, \dots$} in ''C'' that forms a simplicial set. In other words, it is a covariant or contravariant functor Δ → ''C''. For example, a '''simplicial presheaf''' is a simplicial object in the category of presheaves.
* A '''simplicial set''' is a contravariant functor from &Delta; to '''Set''', where &Delta; is the '''simplex category''', a category whose objects are the sets [''n''] = { 0, 1, …, ''n'' } and whose morphisms are order-preserving functions. One writes {$X_n = X([n])$} and an element of the set {$X_n$} is called an ''n''-simplex. For example, {$\Delta^n = \operatorname{Hom}_{\Delta}(-, [n])$} is a simplicial set called the standard ''n''-simplex. By Yoneda's lemma, {$X_n \simeq \operatorname{Nat}(\Delta^n, X)$}.

* A '''(combinatorial) species''' is an endofunctor on the groupoid of finite sets with bijections. It is categorically equivalent to a '''symmetric sequence'''.
* If {$f: C \to D, \, g: D \to E$} are functors, then the '''composition''' {$g \circ f$} or {$gf$} is the functor defined by: for an object ''x'' and a morphism ''u'' in ''C'', {$(g \circ f)(x) = g(f(x)), \, (g \circ f)(u) = g(f(u))$}.
* The functor {$y: C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(-, X)$} is fully faithful and is called the Yoneda embedding. Technical note: the lemma implicitly involves a choice of '''Set'''; i.e., a choice of universe.
2019 vasario 12 d., 09:19 atliko AndriusKulikauskas -
Pridėtos 96-97 eilutės:
* The '''simplex category''' &Delta; is the category where an object is a set [''n''] = { 0, 1, …, ''n'' }, n ≥ 0, totally ordered in the standard way and a morphism is an order-preserving function.
Pakeistos 165-166 eilutės iš
į:
* '''simplicial category''' A category enriched over simplicial sets.
Pridėtos 222-223 eilutės:
* Another term for a contravariant functor: a functor from a category ''C''<sup>op</sup> to '''Set''' is a '''presheaf''' of sets on ''C'' and a functor from ''C''<sup>op</sup> to ''s'''''Set''' is a presheaf of simplicial sets or '''simplicial presheaf''', etc. A '''topology''' on ''C'', if any, tells which presheaf is a sheaf (with respect to that topology).
* A set-valued contravariant functor ''F'' on a category ''C'' is said to be '''representable''' if it belongs to the essential image of the '''Yoneda embedding''' {$C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}; i.e., {$F \simeq \operatorname{Hom}_C(-, Z)$} for some object ''Z''. The object ''Z'' is said to be the representing object of ''F''.
Pridėtos 241-243 eilutės:
* The '''nerve functor''' ''N'' is the functor from '''Cat''' to ''s'''''Set''' given by {$N(C)_n = \operatorname{Hom}_{\mathbf{Cat}}([n], C)$}. For example, if {$\varphi$} is a functor in {$N(C)_2$} (called a 2-simplex), let {$x_i = \varphi(i), \, 0 \le i \le 2$}. Then {$\varphi(0 \to 1)$} is a morphism {$f: x_0 \to x_1$} in ''C'' and also {$\varphi(1 \to 2) = g: x_1 \to x_2$} for some ''g'' in ''C''. Since {$0 \to 2$} is {$0 \to 1$} followed by {$1 \to 2$} and since {$\varphi$} is a functor, {$\varphi(0 \to 2) = g \circ f$}. In other words, {$\varphi$} encodes ''f'', ''g'' and their compositions.
* The '''fundamental category functor''' {$\tau_1: s\mathbf{Set} \to \mathbf{Cat}$} is the left adjoint to the nerve functor ''N''. For every category ''C'', {$\tau_1 NC = C$}.
Pakeistos 259-268 eilutės iš
* The '''fundamental category functor''' {$\tau_1: s\mathbf{Set} \to \mathbf{Cat}$} is the left adjoint to the nerve functor ''N''. For every category ''C'', {$\tau_1 NC = C$}.







* The '''nerve functor''' ''N'' is the functor from '''Cat''' to ''s'''''Set''' given by {$N(C)_n = \operatorname{Hom}_{\mathbf{Cat}}([n], C)$}. For example, if {$\varphi$} is a functor in {$N(C)_2$} (called a 2-simplex), let {$x_i = \varphi(i), \, 0 \le i \le 2$}. Then {$\varphi(0 \to 1)$} is a morphism {$f: x_0 \to x_1$} in ''C'' and also {$\varphi(1 \to 2) = g: x_1 \to x_2$} for some ''g'' in ''C''. Since {$0 \to 2$} is {$0 \to 1$} followed by {$1 \to 2$} and since {$\varphi$} is a functor, {$\varphi(0 \to 2) = g \circ f$}. In other words, {$\varphi$} encodes ''f'', ''g'' and their compositions.
į:
Pakeista 262 eilutė iš:
* Another term for a contravariant functor: a functor from a category ''C''<sup>op</sup> to '''Set''' is a presheaf of sets on ''C'' and a functor from ''C''<sup>op</sup> to ''s'''''Set''' is a presheaf of simplicial sets or '''simplicial presheaf''', etc. A '''topology''' on ''C'', if any, tells which presheaf is a sheaf (with respect to that topology).
į:
Pakeista 265 eilutė iš:
* A set-valued contravariant functor ''F'' on a category ''C'' is said to be '''representable''' if it belongs to the essential image of the '''Yoneda embedding''' {$C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}; i.e., {$F \simeq \operatorname{Hom}_C(-, Z)$} for some object ''Z''. The object ''Z'' is said to be the representing object of ''F''.
į:
Pakeistos 267-268 eilutės iš
* The '''simplex category''' &Delta; is the category where an object is a set [''n''] = { 0, 1, …, ''n'' }, n ≥ 0, totally ordered in the standard way and a morphism is an order-preserving function.
* '''simplicial category''' A category enriched over simplicial sets.
į:
2019 vasario 12 d., 09:05 atliko AndriusKulikauskas -
Pridėta 202 eilutė:
* The '''identity functor''' on a category ''C'' is a functor from ''C'' to ''C'' that sends objects and morphisms to themselves.
Pridėtos 204-205 eilutės:
* Given categories ''I'', ''C'', the '''diagonal functor''' is the functor:{$\Delta: C \to \mathbf{Fct}(I, C), \, A \mapsto \Delta_A$} that sends each object ''A'' to the constant functor with value ''A'' and each morphism {$f: A \to B$} to the natural transformation {$\Delta_{f, i}: \Delta_A(i) = A \to \Delta_B(i) =B$} that is ''f'' at each ''i''.
Pakeistos 215-216 eilutės iš
į:
* Given relative categories {$p: F \to C, q: G \to C$} over the same base category ''C'', a functor {$f: F \to G$} over ''C'' is cartesian if it sends cartesian morphisms to cartesian morphisms.
Pridėta 219 eilutė:
* An [adjective] object in a category ''C'' is a contravariant functor (or presheaf) from some fixed category corresponding to the "adjective" to ''C''. For example, a '''simplicial object''' in ''C'' is a contravariant functor from the simplicial category to ''C'' and a '''Γ-object''' is a pointed contravariant functor from '''Γ''' (roughly the pointed category of pointed finite sets) to ''C'' provided ''C'' is pointed.
Pakeistos 224-236 eilutės iš

* Given relative categories {$p: F \to C, q: G \to C$} over the same base category ''C'', a functor {$f: F \to G$} over ''C'' is cartesian if it sends cartesian morphisms to cartesian morphisms.



* Given categories ''I'', ''C'', the '''diagonal functor''' is the functor:{$\Delta: C \to \mathbf{Fct}(I, C), \, A \mapsto \Delta_A$} that sends each object ''A'' to the constant functor with value ''A'' and each morphism {$f: A \to B$} to the natural transformation {$\Delta_{f, i}: \Delta_A(i) = A \to \Delta_B(i) =B$} that is ''f'' at each ''i''.
* '''distributor'''. Another term for "profunctor".

* Given categories ''C'', ''D'' and an object ''A'' in ''C'', the '''evaluation''' at ''A'' is the functor:{$\mathbf{Fct}(C, D) \to D, \,\, F \mapsto F(A).$} For example, the '''Eilenberg–Steenrod axioms''' give an instance when the functor is an equivalence.

* The '''fundamental category functor''' {$\tau_1: s\mathbf{Set} \to \mathbf{Cat}$} is the left adjoint to the nerve functor ''N''. For every category ''C'', {$\tau_1 NC = C$}.
* A functor π: ''C'' → ''D'' is said to exhibit ''C'' as a '''category fibered over''' ''D'' if, for each morphism ''g'': ''x'' → π(''y'') in ''D'', there exists a π-cartesian morphism ''f'': ''x<nowiki>'</nowiki>'' → ''y'' in ''C'' such that π(''f'') = ''g''. If ''D'' is the category of affine schemes (say of finite type over some field), then π is more commonly called a '''prestack'''. '''Note''': π is often a forgetful functor and in fact the '''Grothendieck construction''' implies that every fibered category can be taken to be that form (up to equivalences in a suitable sense).
į:
* A functor is said to be '''monadic''' if it is a constituent of a monadic adjunction.
Pakeistos 228-229 eilutės iš
* The '''identity functor''' on a category ''C'' is a functor from ''C'' to ''C'' that sends objects and morphisms to themselves.
į:
* A functor π: ''C'' ''D'' is said to exhibit ''C'' as a '''category fibered over''' ''D'' if, for each morphism ''g'': ''x'' → π(''y'') in ''D'', there exists a π-cartesian morphism ''f'': ''x<nowiki>'</nowiki>'' → ''y'' in ''C'' such that π(''f'') = ''g''. If ''D'' is the category of affine schemes (say of finite type over some field), then π is more commonly called a '''prestack'''. '''Note''': π is often a forgetful functor and in fact the '''Grothendieck construction''' implies that every fibered category can be taken to be that form (up to equivalences in a suitable sense).
Pridėta 231 eilutė:
Pridėta 235 eilutė:
Ištrinta 239 eilutė:
Pakeistos 241-258 eilutės iš
* A functor is said to be '''monadic''' if it is a constituent of a monadic adjunction.
į:





* '''distributor'''. Another term for "profunctor".

* Given categories ''C'', ''D'' and an object ''A'' in ''C'', the '''evaluation''' at ''A'' is the functor:{$\mathbf{Fct}(C, D) \to D, \,\, F \mapsto F(A)
.$} For example, the '''Eilenberg–Steenrod axioms''' give an instance when the functor is an equivalence.

* The '''fundamental category functor''' {$\tau_1: s\mathbf{Set} \to \mathbf{Cat}$} is the left adjoint to the nerve functor ''N''. For every category ''C'', {$\tau_1 NC = C$}.





Pakeistos 260-261 eilutės iš
* An [adjective] object in a category ''C'' is a contravariant functor (or presheaf) from some fixed category corresponding to the "adjective" to ''C''. For example, a '''simplicial object''' in ''C'' is a contravariant functor from the simplicial category to ''C'' and a '''Γ-object''' is a pointed contravariant functor from '''Γ''' (roughly the pointed category of pointed finite sets) to ''C'' provided ''C'' is pointed.
* A functor π:''C'' → ''D'' is an '''op-fibration''' if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one π-coCartesian morphism ''f'': ''x'' → ''y<nowiki>'</nowiki>
'' in ''C'' such that π(''f'') = ''g''. In other words, π is the dual of a '''Grothendieck fibration'''.
į:
* A functor π:''C'' → ''D'' is an '''op-fibration''' if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one π-coCartesian morphism ''f'': ''x'' ''y'' in ''C'' such that π(''f'') = ''g''. In other words, π is the dual of a '''Grothendieck fibration'''.
2019 vasario 11 d., 14:34 atliko AndriusKulikauskas -
Pridėtos 75-76 eilutės:
* A morphism ''f'' in a category admitting finite limits and finite colimits is '''strict''' if the natural morphism {$\operatorname{Coim}(f) \to \operatorname{Im}(f)$} is an isomorphism.
Pakeistos 87-88 eilutės iš
#For each object ''X'', an identity morphism {$\operatorname{id}_X \in \operatorname{Hom}(X, X)$} subject to the conditions: for any morphisms {$f: X \to Y$}, {$g: Y \to Z$} and {$h: Z \to W$},
*{$(h \circ g) \circ f = h \circ (g \circ f)$} and {$\operatorname{id}_Y \circ f = f \circ \operatorname{id}_X = f$}.
į:
#For each object ''X'', an identity morphism {$\operatorname{id}_X \in \operatorname{Hom}(X, X)$} subject to the conditions: for any morphisms {$f: X \to Y$}, {$g: Y \to Z$} and {$h: Z \to W$}, {$(h \circ g) \circ f = h \circ (g \circ f)$} and {$\operatorname{id}_Y \circ f = f \circ \operatorname{id}_X = f$}.
Pakeistos 101-102 eilutės iš
į:
* A category is '''equivalent''' to another category if there is an '''equivalence''' between them.
Pridėta 189 eilutė:
* A morphism in an ∞-category ''C'' is an '''equivalence''' if it gives an isomorphism in the homotopy category of ''C''.
Ištrintos 201-206 eilutės:
* An '''adjunction''' (also called an adjoint pair) is a pair of functors ''F'': ''C'' → ''D'', ''G'': ''D'' → ''C'' such that there is a "natural" bijection:{$\operatorname{Hom}_D (F(X), Y) \simeq \operatorname{Hom}_C (X, G(Y))$};''F'' is said to be left adjoint to ''G'' and ''G'' to right adjoint to ''F''. Here, "natural" means there is a natural isomorphism {$\operatorname{Hom}_D (F(-), -) \simeq \operatorname{Hom}_C (-, G(-))$} of bifunctors (which are contravariant in the first variable.)

* A functor is amnestic if it has the property: if ''k'' is an isomorphism and ''F''(''k'') is an identity, then ''k'' is an identity.
* A '''bifunctor''' from a pair of categories ''C'' and ''D'' to a category ''E'' is a functor ''C'' × ''D'' → ''E''. For example, for any category ''C'', {$\operatorname{Hom}(-, -)$} is a bifunctor from ''C''<sup>op</sup> and ''C'' to '''Set'''.
* Given relative categories {$p: F \to C, q: G \to C$} over the same base category ''C'', a functor {$f: F \to G$} over ''C'' is cartesian if it sends cartesian morphisms to cartesian morphisms.
* A '''conservative functor''' is a functor that reflects isomorphisms. Many forgetful functors are conservative, but the forgetful functor from '''Top''' to '''Set''' is not conservative.
Pridėtos 203-213 eilutės:
* A functor is said to '''reflect identities''' if it has the property: if ''F''(''k'') is an identity then ''k'' is an identity as well.
* A functor is said to '''reflect isomorphisms''' if it has the property: ''F''(''k'') is an isomorphism then ''k'' is an isomorphism as well.
* A '''conservative functor''' is a functor that reflects isomorphisms. Many forgetful functors are conservative, but the forgetful functor from '''Top''' to '''Set''' is not conservative.
* A functor is amnestic if it has the property: if ''k'' is an isomorphism and ''F''(''k'') is an identity, then ''k'' is an identity.

* A functor is '''faithful''' if it is injective when restricted to each '''hom-set'''.
* A functor is '''full''' if it is surjective when restricted to each '''hom-set'''.
* A functor ''F'' is called '''essentially surjective''' (or isomorphism-dense) if for every object ''B'' there exists an object ''A'' such that ''F''(''A'') is isomorphic to ''B''.
* A functor is an '''equivalence''' if it is faithful, full and essentially surjective.

* '''endofunctor.''' A functor between the same category.
Pridėtos 215-224 eilutės:
* A '''bifunctor''' from a pair of categories ''C'' and ''D'' to a category ''E'' is a functor ''C'' × ''D'' → ''E''. For example, for any category ''C'', {$\operatorname{Hom}(-, -)$} is a bifunctor from ''C''<sup>op</sup> and ''C'' to '''Set'''.

* An '''adjunction''' (also called an adjoint pair) is a pair of functors ''F'': ''C'' → ''D'', ''G'': ''D'' → ''C'' such that there is a "natural" bijection:{$\operatorname{Hom}_D (F(X), Y) \simeq \operatorname{Hom}_C (X, G(Y))$};''F'' is said to be left adjoint to ''G'' and ''G'' to right adjoint to ''F''. Here, "natural" means there is a natural isomorphism {$\operatorname{Hom}_D (F(-), -) \simeq \operatorname{Hom}_C (-, G(-))$} of bifunctors (which are contravariant in the first variable.)
* An adjunction is said to be '''monadic''' if it comes from the monad that it determines by means of the '''Eilenberg–Moore category''' (the category of algebras for the monad).


* Given relative categories {$p: F \to C, q: G \to C$} over the same base category ''C'', a functor {$f: F \to G$} over ''C'' is cartesian if it sends cartesian morphisms to cartesian morphisms.

Pakeistos 227-230 eilutės iš
* '''endofunctor.''' A functor between the same category.
* A functor is an '''equivalence''' if it is faithful, full and essentially surjective. A morphism in an ∞-category ''C'' is an equivalence if it gives an isomorphism in the homotopy category of ''C''.
* A category is equivalent to another category if there is an '''equivalence''' between them.
* A functor ''F'' is called '''essentially surjective''' (or isomorphism-dense) if for every object ''B'' there exists an object ''A'' such that ''F''(''A'') is isomorphic to ''B''.
į:
Pakeista 229 eilutė iš:
* A functor is '''faithful''' if it is injective when restricted to each '''hom-set'''.
į:
Ištrinta 233 eilutė:
* A functor is '''full''' if it is surjective when restricted to each '''hom-set'''.
Pakeista 243 eilutė iš:
* An adjunction is said to be '''monadic''' if it comes from the monad that it determines by means of the '''Eilenberg–Moore category''' (the category of algebras for the monad).
į:
Pakeistos 252-253 eilutės iš
* A functor is said to '''reflect identities''' if it has the property: if ''F''(''k'') is an identity then ''k'' is an identity as well.
* A functor is said to '''reflect isomorphisms''' if it has the property: ''F''(''k'') is an isomorphism then ''k'' is an isomorphism as well.
į:
Pakeista 259 eilutė iš:
* A morphism ''f'' in a category admitting finite limits and finite colimits is '''strict''' if the natural morphism {$\operatorname{Coim}(f) \to \operatorname{Im}(f)$} is an isomorphism.
į:
2019 vasario 11 d., 14:25 atliko AndriusKulikauskas -
Pakeistos 85-86 eilutės iš
#For each object ''X'', an identity morphism {$\operatorname{id}_X \in \operatorname{Hom}(X, X)$}
subject to the conditions: for any morphisms {$f: X \to Y$}, {$g: Y \to Z$} and {$h: Z \to W$},
į:
#For each object ''X'', an identity morphism {$\operatorname{id}_X \in \operatorname{Hom}(X, X)$} subject to the conditions: for any morphisms {$f: X \to Y$}, {$g: Y \to Z$} and {$h: Z \to W$},
Pakeista 87 eilutė iš:
For example, a '''partially ordered set''' can be viewed as a category: the objects are the elements of the set and for each pair of objects ''x'', ''y'', there is a unique morphism {$x \to y$} if and only if {$x \le y$}; the associativity of composition means transitivity.
į:
* For example, a '''partially ordered set''' can be viewed as a category: the objects are the elements of the set and for each pair of objects ''x'', ''y'', there is a unique morphism {$x \to y$} if and only if {$x \le y$}; the associativity of composition means transitivity.
Pakeista 160 eilutė iš:
For example, a category enriched over sets is an ordinary category.
į:
* For example, a category enriched over sets is an ordinary category.
2019 vasario 11 d., 14:23 atliko AndriusKulikauskas -
Pridėtos 123-126 eilutės:
* Given a category ''C'' and an object ''A'' in it, the '''slice category''' {$C_A$} of ''C'' over ''A'' is the category whose objects are all the morphisms in ''C'' with codomain ''A'', whose morphisms are morphisms in ''C'' such that if ''f'' is a morphism from {$p_X: X \to A$} to {$p_Y: Y \to A$}, then {$p_Y \circ f = p_X$} in ''C'' and whose composition is that of ''C''.
* Given functors {$f: C \to B, g: D \to B$}, the '''comma category''' {$(f \downarrow g)$} is a category where (1) the objects are morphisms {$f(c) \to g(d)$} and (2) a morphism from {$\alpha: f(c) \to g(d)$} to {$\beta: f(c') \to g(d')$} consists of {$c \to c'$} and {$d \to d'$} such that {$f(c) \to f(c') \overset{\beta}\to g(d')$} is {$f(c) \overset{\alpha}\to g(d) \to g(d').$} For example, if ''f'' is the identity functor and ''g'' is the constant functor with a value ''b'', then it is the slice category of ''B'' over an object ''b''.
* A '''filtered category''' (also called a filtrant category) is a nonempty category with the properties (1) given objects ''i'' and ''j'', there are an object ''k'' and morphisms ''i'' → ''k'' and ''j'' → ''k'' and (2) given morphisms ''u'', ''v'': ''i'' → ''j'', there are an object ''k'' and a morphism ''w'': ''j'' → ''k'' such that ''w'' ∘ ''u'' = ''w'' ∘ ''v''. A category ''I'' is filtered if and only if, for each finite category ''J'' and functor ''f'': ''J'' → ''I'', the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''.
Pakeistos 147-151 eilutės iš

* Given functors {$f: C \to B, g: D \to B$}, the '''comma category''' {$(f \downarrow g)$} is a category where (1) the objects are morphisms {$f(c) \to g(d)$} and (2) a morphism from {$\alpha: f(c) \to g(d)$} to {$\beta: f(c') \to g(d')$} consists of {$c \to c'$} and {$d \to d'$} such that {$f(c) \to f(c') \overset{\beta}\to g(d')$} is {$f(c) \overset{\alpha}\to g(d) \to g(d').$} For example, if ''f'' is the identity functor and ''g'' is the constant functor with a value ''b'', then it is the slice category of ''B'' over an object ''b''.
į:
* The '''homological dimension''' of an abelian category with enough injectives is the least non-negative integer ''n'' such that every object in the category admits an injective resolution of length at most ''n''. The dimension is ∞ if no such integer exists. For example, the homological dimension of {$Mod_R$} with a principal ideal domain ''R'' is at most one.
Ištrintos 162-175 eilutės:


* A '''filtered category''' (also called a filtrant category) is a nonempty category with the properties (1) given objects ''i'' and ''j'', there are an object ''k'' and morphisms ''i'' → ''k'' and ''j'' → ''k'' and (2) given morphisms ''u'', ''v'': ''i'' → ''j'', there are an object ''k'' and a morphism ''w'': ''j'' → ''k'' such that ''w'' ∘ ''u'' = ''w'' ∘ ''v''. A category ''I'' is filtered if and only if, for each finite category ''J'' and functor ''f'': ''J'' → ''I'', the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''.


* '''Fukaya category'''.



* '''Grothendieck fibration''' A '''fibered category'''.

* '''homotopy category'''. It is closely related to a '''localization of a category'''.
Pakeistos 165-174 eilutės iš
* '''site''' A category equipped with a '''Grothendieck topology'''.

* Given a category ''C'' and an object
''A'' in it, the '''slice category''' {$C_A$} of ''C'' over ''A'' is the category whose objects are all the morphisms in ''C'' with codomain ''A'', whose morphisms are morphisms in ''C'' such that if ''f'' is a morphism from {$p_X: X \to A$} to {$p_Y: Y \to A$}, then {$p_Y \circ f = p_X$} in ''C'' and whose composition is that of ''C''.


* A '''Waldhausen category''' is, roughly, a category with families of cofibrations and weak equivalences.


* The
'''homological dimension''' of an abelian category with enough injectives is the least non-negative integer ''n'' such that every object in the category admits an injective resolution of length at most ''n''. The dimension is ∞ if no such integer exists. For example, the homological dimension of {$Mod_R$} with a principal ideal domain ''R'' is at most one.
į:
* A '''Fukaya category''' is a certain kind of category of Lagrangian submanifolds of a symplectic manifold.
*
'''Grothendieck fibration''' A '''fibered category''', which is used for a general framework of descent theory, to discuss vector bundles, principal bundles and sheaves over topological spaces.
*
'''site''' A category equipped with a '''Grothendieck topology''', which makes its objects act like open sets of a topological space.
*
A '''Waldhausen category''' is, roughly, a category C with families of cofibrations and weak equivalences, which makes it possible to calculate the K-spectrum of C.
* '''homotopy category'''. It is closely related to a
'''localization of a category'''.
2019 vasario 11 d., 14:08 atliko AndriusKulikauskas -
Pridėta 93 eilutė:
* A category is wellpowered if for each object there is only a set of pairwise non-isomorphic '''subobject'''s.
Pakeistos 123-124 eilutės iš
* A category is '''complete''' if all small limits exist.
į:
* A category is '''complete''' if all small limits exist

* A '''monoidal category''', also called a tensor category, is a category ''C'' equipped with (1) a '''bifunctor''' {$\otimes: C \times C \to C$}, (2) an identity object and (3) natural isomorphisms that make ⊗ associative and the identity object an identity for ⊗, subject to certain coherence conditions
.
Pakeistos 139-141 eilutės iš
* A '''monoidal category''', also called a tensor category, is a category ''C'' equipped with (1) a '''bifunctor''' {$\otimes: C \times C \to C$}, (2) an identity object and (3) natural isomorphisms that make ⊗ associative and the identity object an identity for ⊗, subject to certain coherence conditions.

* A category is '''preadditive''' if it is '''enriched''' over the '''monoidal category''' of '''abelian group'''s. More generally, it is {$R$}'''-linear''' if it is enriched over the monoidal category of {$R$}'''-modules''', for ''R''
a '''commutative ring'''.
į:
* A category is '''preadditive''' if it is '''enriched''' over the '''monoidal category''' of '''abelian group'''s. More generally, it is {$R$}'''-linear''' if it is enriched over the monoidal category of {$R$}'''-modules''', for {$R$} a '''commutative ring'''.
Pakeistos 141-142 eilutės iš
į:
* An '''exact category''' is a particular kind of additive category consisting of "short exact sequences".
* A '''Frobenius category''' is an '''exact category''' that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects.
Pakeistos 164-166 eilutės iš
* Given a cardinal number π, a category is said to be π-filtrant if, for each category ''J'' whose set of morphisms has cardinal number strictly less than π, the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''.

* A '''Frobenius category''' is an '''exact category''' that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects.
į:
Pakeistos 184-185 eilutės iš
* A category is wellpowered if for each object there is only a set of pairwise non-isomorphic '''subobject'''s.
į:
Pridėta 190 eilutė:
* Given a cardinal number π, a category is said to be π-filtrant if, for each category ''J'' whose set of morphisms has cardinal number strictly less than π, the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''.
2019 vasario 11 d., 13:53 atliko AndriusKulikauskas -
Pakeista 139 eilutė iš:
* A category is '''preadditive''' if it is '''enriched''' over the '''monoidal category''' of '''abelian group'''s. More generally, it is '''''R''-linear''' if it is enriched over the monoidal category of '''''R''-modules''', for ''R'' a '''commutative ring'''.
į:
* A category is '''preadditive''' if it is '''enriched''' over the '''monoidal category''' of '''abelian group'''s. More generally, it is {$R$}'''-linear''' if it is enriched over the monoidal category of {$R$}'''-modules''', for ''R'' a '''commutative ring'''.
2019 vasario 11 d., 13:52 atliko AndriusKulikauskas -
2019 vasario 11 d., 13:41 atliko AndriusKulikauskas -
Pridėta 92 eilutė:
* A '''small category''' is a category in which the class of all morphisms is a '''set''' (i.e., not a '''proper class'''); otherwise '''large'''. A category is '''locally small''' if the morphisms between every pair of objects ''A'' and ''B'' form a set. Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a '''quasicategory''' is a category whose objects and morphisms merely form a '''conglomerate'''.
Pridėtos 95-97 eilutės:
* The '''functor category''' '''Fct'''(''C'', ''D'') from a category ''C'' to a category ''D'' is the category where the objects are all the functors from ''C'' to ''D'' and the morphisms are all the natural transformations between the functors.
* A '''strict ''n''-category''' is defined inductively: a strict 0-category is a set and a strict ''n''-category is a category whose Hom sets are strict (''n''-1)-categories. Precisely, a strict ''n''-category is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category.
Pakeistos 105-106 eilutės iš
į:
* A category is '''skeletal''' if isomorphic objects are necessarily identical.
Ištrintos 121-122 eilutės:
* The '''functor category''' '''Fct'''(''C'', ''D'') from a category ''C'' to a category ''D'' is the category where the objects are all the functors from ''C'' to ''D'' and the morphisms are all the natural transformations between the functors.
Pridėta 131 eilutė:
* A full subcategory of an abelian category is '''thick''' if it is closed under extensions.
Pridėta 133 eilutė:
* A '''triangulated category''' is a category where one can talk about distinguished triangles, generalization of exact sequences. An abelian category is a prototypical example of a triangulated category.
Pakeistos 135-136 eilutės iš
į:
* A '''derived category''' is a triangulated category that is not necessary an abelian category.
Pakeistos 176-178 eilutės iš
* '''''n''-category''' A '''strict ''n''-category''' is defined inductively: a strict 0-category is a set and a strict ''n''-category is a category whose Hom sets are strict (''n''-1)-categories. Precisely, a strict ''n''-category is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category.
į:
Pakeista 178 eilutė iš:
* A category is '''skeletal''' if isomorphic objects are necessarily identical.
į:
Pakeistos 180-185 eilutės iš
* A '''small category''' is a category in which the class of all morphisms is a '''set''' (i.e., not a '''proper class'''); otherwise '''large'''. A category is '''locally small''' if the morphisms between every pair of objects ''A'' and ''B'' form a set. Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a '''quasicategory''' is a category whose objects and morphisms merely form a '''conglomerate'''.


* A full subcategory of an abelian category is '''thick''' if it is closed under extensions.

* A '''triangulated category''' is a category where one can talk about distinguished triangles, generalization of exact sequences. An abelian category is a prototypical example of a triangulated category. A '''derived category''' is a triangulated category that is not necessary an abelian category.
į:
2019 vasario 11 d., 13:35 atliko AndriusKulikauskas -
Pakeista 80 eilutė iš:
* A '''category (mathematics)|category''' consists of the following data
į:
* A '''category''' consists of the following data
Pakeista 133 eilutė iš:
* A category is '''preadditive''' if it is '''enriched''' over the '''monoidal category''' of '''abelian group'''s. More generally, it is '''preadditive category#R-linear categories|''R''-linear''' if it is enriched over the monoidal category of '''module (mathematics)|''R''-modules''', for ''R'' a '''commutative ring'''.
į:
* A category is '''preadditive''' if it is '''enriched''' over the '''monoidal category''' of '''abelian group'''s. More generally, it is '''''R''-linear''' if it is enriched over the monoidal category of '''''R''-modules''', for ''R'' a '''commutative ring'''.
Pakeistos 176-178 eilutės iš
* A '''small category''' is a category in which the class of all morphisms is a '''set''' (i.e., not a '''proper class'''); otherwise '''large'''. A category is '''locally small''' if the morphisms between every pair of objects ''A'' and ''B'' form a set. Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a '''quasicategory''' is a category whose objects and morphisms merely form a '''conglomerate (set theory)|conglomerate'''.
į:
* A '''small category''' is a category in which the class of all morphisms is a '''set''' (i.e., not a '''proper class'''); otherwise '''large'''. A category is '''locally small''' if the morphisms between every pair of objects ''A'' and ''B'' form a set. Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a '''quasicategory''' is a category whose objects and morphisms merely form a '''conglomerate'''.
Pakeista 222 eilutė iš:
* A '''contravariant functor''' ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a '''presheaf (category theory)|presheaf''' especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let {$\mathfrak{P}(S)$} be the power set of ''S'' and for each function {$f: S \to T$}, define:{$\mathfrak{P}(f): \mathfrak{P}(T) \to \mathfrak{P}(S)$} by sending a subset ''A'' of ''T'' to the pre-image {$f^{-1}(A)$}. With this, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}$} is a contravariant functor.
į:
* A '''contravariant functor''' ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a '''presheaf''' especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let {$\mathfrak{P}(S)$} be the power set of ''S'' and for each function {$f: S \to T$}, define:{$\mathfrak{P}(f): \mathfrak{P}(T) \to \mathfrak{P}(S)$} by sending a subset ''A'' of ''T'' to the pre-image {$f^{-1}(A)$}. With this, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}$} is a contravariant functor.
Pakeista 234 eilutė iš:
* A '''free functor''' is a left adjoint to a forgetful functor. For example, for a ring ''R'', the functor that sends a set ''X'' to the '''free module|free ''R''-module''' generated by ''X'' is a free functor (whence the name).
į:
* A '''free functor''' is a left adjoint to a forgetful functor. For example, for a ring ''R'', the functor that sends a set ''X'' to the '''free ''R''-module''' generated by ''X'' is a free functor (whence the name).
Pakeista 249 eilutė iš:
* An [adjective] object in a category ''C'' is a contravariant functor (or presheaf) from some fixed category corresponding to the "adjective" to ''C''. For example, a '''simplicial object''' in ''C'' is a contravariant functor from the simplicial category to ''C'' and a '''Γ-object''' is a pointed contravariant functor from '''Γ (category theory)|Γ''' (roughly the pointed category of pointed finite sets) to ''C'' provided ''C'' is pointed.
į:
* An [adjective] object in a category ''C'' is a contravariant functor (or presheaf) from some fixed category corresponding to the "adjective" to ''C''. For example, a '''simplicial object''' in ''C'' is a contravariant functor from the simplicial category to ''C'' and a '''Γ-object''' is a pointed contravariant functor from '''Γ''' (roughly the pointed category of pointed finite sets) to ''C'' provided ''C'' is pointed.
Pakeista 300 eilutė iš:
* The '''coproduct''' of a family of objects {$X_i$} in a category ''C'' indexed by a set ''I'' is the inductive limit {$\varinjlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where ''I'' is viewed as a discrete category. It is the dual of the product of the family. For example, a coproduct in ''''''category of groups|Grp'''''' is a '''free product'''.
į:
* The '''coproduct''' of a family of objects {$X_i$} in a category ''C'' indexed by a set ''I'' is the inductive limit {$\varinjlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where ''I'' is viewed as a discrete category. It is the dual of the product of the family. For example, a coproduct in '''Grp''' is a '''free product'''.
Pakeista 325 eilutė iš:
* The '''calculus of functors''' is a technique of studying functors in the manner similar to the way a '''function (mathematics)|function''' is studied via its '''Taylor series''' expansion; whence, the term "calculus".
į:
* The '''calculus of functors''' is a technique of studying functors in the manner similar to the way a '''function''' is studied via its '''Taylor series''' expansion; whence, the term "calculus".
2019 vasario 11 d., 13:31 atliko AndriusKulikauskas -
Pakeistos 63-65 eilutės iš
į:
* A monomorphism is normal if it is the kernel of some morphism.
* An epimorphism is conormal if it is the cokernel of some morphism.
Pridėta 106 eilutė:
* A category is '''normal''' if every monomorphism is normal.
Pakeistos 171-173 eilutės iš
* A category is '''normal''' if every monomorphism is normal.
į:
Pridėtos 296-297 eilutės:
* If f : X → Y is an arbitrary morphism in C, then a '''kernel''' of f is an equaliser of f and the zero morphism from X to Y.
* The dual concept to that of kernel is that of '''cokernel'''. The cokernel of a morphism is its kernel in the opposite category.
2019 vasario 11 d., 13:25 atliko AndriusKulikauskas -
Pakeistos 106-107 eilutės iš
į:
* The '''fundamental groupoid''' {$\Pi_1 X$} of a Kan complex ''X'' is the category where an object is a 0-simplex (vertex) {$\Delta^0 \to X$}, a morphism is a homotopy class of a 1-simplex (path) {$\Delta^1 \to X$} and a composition is determined by the Kan property.

* A '''concrete category''' ''C'' is a category such that there is a faithful functor from ''C'' to '''Set'''; e.g., '''Vec''', '''Grp''' and '''Top'''.
Pridėtos 112-115 eilutės:
* The '''product of a family of categories''' {$C_i$}'s indexed by a set ''I'' is the category denoted by {$\prod_i C_i$} whose class of objects is the product of the classes of objects of {$C_i$}'s and whose hom-sets are {$\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)$}; the morphisms are composed component-wise. It is the dual of the disjoint union.

* The '''functor category''' '''Fct'''(''C'', ''D'') from a category ''C'' to a category ''D'' is the category where the objects are all the functors from ''C'' to ''D'' and the morphisms are all the natural transformations between the functors.
Pakeistos 118-121 eilutės iš
* A '''concrete category''' ''C'' is a category such that there is a faithful functor from ''C'' to '''Set'''; e.g., '''Vec''', '''Grp''' and '''Top'''.

* The '''functor category'''
'''Fct'''(''C'', ''D'') from a category ''C'' to a category ''D'' is the category where the objects are all the functors from ''C'' to ''D'' and the morphisms are all the natural transformations between the functors.
į:
* '''tensor category''' Usually synonymous with '''monoidal category''' (though some authors distinguish between the two concepts.)
* A
'''symmetric monoidal category''' is a '''monoidal category''' (i.e., a category with ⊗) that has maximally symmetric braiding.
* A
'''tensor triangulated category''' is a category that carries the structure of a symmetric monoidal category and that of a triangulated category in a compatible way.
* A
'''PROP''' is a symmetric strict monoidal category whose objects are natural numbers and whose tensor product '''addition''' of natural numbers.
Pakeistos 150-151 eilutės iš
* The '''fundamental groupoid''' {$\Pi_1 X$} of a Kan complex ''X'' is the category where an object is a 0-simplex (vertex) {$\Delta^0 \to X$}, a morphism is a homotopy class of a 1-simplex (path) {$\Delta^1 \to X$} and a composition is determined by the Kan property.
į:
Pakeistos 168-175 eilutės iš
* A category is '''normal''' if every monic is normal.




* The '''product of a family of categories''' {$C_i$}'s indexed by a set ''I'' is the category denoted by {$\prod_i C_i$} whose class of objects is the product of the classes of objects of {$C_i$}'s and whose hom-sets are {$\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)$}; the morphisms are composed component-wise. It is the dual of the disjoint union
.
į:
* A category is '''normal''' if every monomorphism is normal.
Ištrintos 174-178 eilutės:

* '''tensor category''' Usually synonymous with '''monoidal category''' (though some authors distinguish between the two concepts.)
* A '''symmetric monoidal category''' is a '''monoidal category''' (i.e., a category with ⊗) that has maximally symmetric braiding.
* A '''tensor triangulated category''' is a category that carries the structure of a symmetric monoidal category and that of a triangulated category in a compatible way.
* A '''PROP''' is a symmetric strict monoidal category whose objects are natural numbers and whose tensor product '''addition''' of natural numbers.
2019 vasario 11 d., 13:14 atliko AndriusKulikauskas -
Pridėta 92 eilutė:
* A category is '''isomorphic''' to another category if there is an isomorphism between them.
Pridėta 95 eilutė:
* A category ''A'' is a '''subcategory''' of a category ''B'' if there is an inclusion functor from ''A'' to ''B''.
Pridėta 100 eilutė:
* A category is '''finite''' if it has only finitely many morphisms.
Pakeista 102 eilutė iš:
* A category is '''finite''' if it has only finitely many morphisms.
į:
* A '''thin''' is a category where there is at most one morphism between any pair of objects.
Pakeistos 105-107 eilutės iš
į:
* The '''core''' of a category is the maximal groupoid contained in the category.
Pridėtos 114-115 eilutės:
* The '''functor category''' '''Fct'''(''C'', ''D'') from a category ''C'' to a category ''D'' is the category where the objects are all the functors from ''C'' to ''D'' and the morphisms are all the natural transformations between the functors.
Pakeistos 119-122 eilutės iš
į:
* The '''heart''' of a '''t-structure''' ({$D^{\ge 0}$}, {$D^{\le 0}$}) on a triangulated category is the intersection {$D^{\ge 0} \cap D^{\le 0}$}. It is an abelian category.

* A '''monoidal category''', also called a tensor category, is a category ''C'' equipped with (1) a '''bifunctor''' {$\otimes: C \times C \to C$}, (2) an identity object and (3) natural isomorphisms that make ⊗ associative and the identity object an identity for ⊗, subject to certain coherence conditions.
Pakeistos 150-151 eilutės iš
* The '''functor category''' '''Fct'''(''C'', ''D'') from a category ''C'' to a category ''D'' is the category where the objects are all the functors from ''C'' to ''D'' and the morphisms are all the natural transformations between the functors.
į:
Pakeista 153 eilutė iš:
* The '''heart''' of a '''t-structure''' ({$D^{\ge 0}$}, {$D^{\le 0}$}) on a triangulated category is the intersection {$D^{\ge 0} \cap D^{\le 0}$}. It is an abelian category.
į:
Pakeista 156 eilutė iš:
* A category is '''isomorphic''' to another category if there is an isomorphism between them.
į:
Pakeista 158 eilutė iš:
* A '''monoidal category''', also called a tensor category, is a category ''C'' equipped with (1) a '''bifunctor''' {$\otimes: C \times C \to C$}, (2) an identity object and (3) natural isomorphisms that make ⊗ associative and the identity object an identity for ⊗, subject to certain coherence conditions.
į:
Pakeista 164 eilutė iš:
* Given a '''regular cardinal''' κ, a category is '''κ-presentable''' if it admits all small colimits and is '''κ-accessible'''. A category is presentable if it is κ-presentable for some regular cardinal κ (hence presentable for any larger cardinal). '''Note''': Some authors call a presentable category a '''locally presentable category'''.
į:
Pakeistos 166-167 eilutės iš
* A '''PROP''' is a symmetric strict monoidal category whose objects are natural numbers and whose tensor product '''addition''' of natural numbers.
į:
Pakeistos 172-173 eilutės iš
* A category ''A'' is a '''subcategory''' of a category ''B'' if there is an inclusion functor from ''A'' to ''B''.
į:
* '''tensor category''' Usually synonymous with '''monoidal category''' (though some authors distinguish between the two concepts.)
Ištrinta 174 eilutė:
* '''tensor category''' Usually synonymous with '''monoidal category''' (though some authors distinguish between the two concepts.)
Pridėtos 176-178 eilutės:
* A '''PROP''' is a symmetric strict monoidal category whose objects are natural numbers and whose tensor product '''addition''' of natural numbers.
Pakeista 180 eilutė iš:
* A '''thin''' is a category where there is at most one morphism between any pair of objects.
į:
Pakeista 184 eilutė iš:
* The '''core''' of a category is the maximal groupoid contained in the category.
į:
Pridėta 188 eilutė:
* Given a '''regular cardinal''' κ, a category is '''κ-presentable''' if it admits all small colimits and is '''κ-accessible'''. A category is presentable if it is κ-presentable for some regular cardinal κ (hence presentable for any larger cardinal). '''Note''': Some authors call a presentable category a '''locally presentable category'''.
2019 vasario 11 d., 13:03 atliko AndriusKulikauskas -
Pakeistos 89-94 eilutės iš
* A category is '''abelian''' if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.

* A category is '''additive''' if it is preadditive (to be precise, has some pre-additive structure) and admits all finite '''coproduct'''s. Although "preadditive" is an additional structure, one can show "additive" is a ''property'' of a category; i.e., one can ask whether a given category is additive or not.
* An '''adjunction''' (also called an adjoint pair) is a pair of functors ''F'': ''C'' → ''D'', ''G'': ''D'' → ''C'' such that there is a "natural" bijection:{$\operatorname{Hom}_D (F(X), Y) \simeq \operatorname{Hom}_C (X, G(Y))$};''F'' is said to be left adjoint to ''G'' and ''G'' to right adjoint to ''F''. Here, "natural" means there is a natural isomorphism {$\operatorname{Hom}_D (F(-), -) \simeq \operatorname{Hom}_C (-, G(-))$} of bifunctors (which are contravariant in the first variable.)
* A category is '''balanced''' if every bimorphism is an isomorphism.
* A category is '''cartesian closed''' if it has a terminal object and that any two objects have a product and exponential
.
į:
* The '''empty category''' is a category with no object. It is the same thing as the '''empty set''' when the empty set is viewed as a discrete category.
Pakeistos 91-95 eilutės iš
* Given functors {$f: C \to B, g: D \to B$}, the '''comma category''' {$(f \downarrow g)$} is a category where (1) the objects are morphisms {$f(c) \to g(d)$} and (2) a morphism from {$\alpha: f(c) \to g(d)$} to {$\beta: f(c') \to g(d')$} consists of {$c \to c'$} and {$d \to d'$} such that {$f(c) \to f(c') \overset{\beta}\to g(d')$} is {$f(c) \overset{\alpha}\to g(d) \to g(d').$} For example, if ''f'' is the identity functor and ''g'' is the constant functor with a value ''b'', then it is the slice category of ''B'' over an object ''b''.
* A category is '''complete''' if all small limits exist.
* A '''concrete category''' ''C'' is a category such that there is a faithful functor from ''C'' to ''''''Category of sets|Set''''''; e.g., ''''''category of vector spaces|Vec'''''', ''''''category of groups|Grp'''''' and ''''''category of topological spaces|Top''''''.
* A category is '''connected''' if, for each pair of objects ''x'', ''y'', there exists a finite sequence of objects {$z_i$} such that {$z_0 = x, z_n = y$} and either {$\operatorname{Hom}(z_i, z_{i+1})$} or {$\operatorname{Hom}(z_{i+1}, z_i)$} is nonempty for any ''i''.
* A '''differential graded category''' is a category whose Hom sets are equipped with structures of '''differential graded module'''s. In particular, if the category has only one object, it is the same as a differential graded module.
į:
* The '''opposite category''' of a category is obtained by reversing the arrows. For example, if a partially ordered set is viewed as a category, taking its opposite amounts to reversing the ordering.

* A category ''A'' is a '''full subcategory
''' of a category ''B'' if the inclusion functor from ''A'' to ''B'' is full.

* A category
(or ∞-category) is called '''pointed''' if it has a zero object.
Ištrintos 98-110 eilutės:
* Eilenberg–Moore category. Another name for the category of '''algebras for a given monad'''.
* '''enriched category''' Given a monoidal category (''C'', ⊗, 1), a '''category enriched''' over ''C'' is, informally, a category whose Hom sets are in ''C''. More precisely, a category ''D'' enriched over ''C'' is a data consisting of
# A class of objects,
# For each pair of objects ''X'', ''Y'' in ''D'', an object {$\operatorname{Map}_D(X, Y)$} in ''C'', called the '''mapping object''' from ''X'' to ''Y'',
# For each triple of objects ''X'', ''Y'', ''Z'' in ''D'', a morphism in ''C'',
#:{$\circ: \operatorname{Map}_D(Y, Z) \otimes \operatorname{Map}_D(X, Y) \to \operatorname{Map}_D(X, Z)$},
#:called the composition,
#For each object ''X'' in ''D'', a morphism {$1_X: 1 \to \operatorname{Map}_D(X, X)$} in ''C'', called the unit morphism of ''X'' subject to the conditions that (roughly) the compositions are associative and the unit morphisms act as the multiplicative identity.
For example, a category enriched over sets is an ordinary category.
* The '''empty category''' is a category with no object. It is the same thing as the '''empty set''' when the empty set is viewed as a discrete category.
* The '''fundamental groupoid''' {$\Pi_1 X$} of a Kan complex ''X'' is the category where an object is a 0-simplex (vertex) {$\Delta^0 \to X$}, a morphism is a homotopy class of a 1-simplex (path) {$\Delta^1 \to X$} and a composition is determined by the Kan property.
* A '''filtered category''' (also called a filtrant category) is a nonempty category with the properties (1) given objects ''i'' and ''j'', there are an object ''k'' and morphisms ''i'' → ''k'' and ''j'' → ''k'' and (2) given morphisms ''u'', ''v'': ''i'' → ''j'', there are an object ''k'' and a morphism ''w'': ''j'' → ''k'' such that ''w'' ∘ ''u'' = ''w'' ∘ ''v''. A category ''I'' is filtered if and only if, for each finite category ''J'' and functor ''f'': ''J'' → ''I'', the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''.
* Given a cardinal number π, a category is said to be π-filtrant if, for each category ''J'' whose set of morphisms has cardinal number strictly less than π, the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''.
Pakeistos 100-103 eilutės iš
* A '''Frobenius category''' is an '''exact category''' that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects.
*
'''Fukaya category'''.
* A category ''A'' is
a '''full subcategory''' of a category ''B'' if the inclusion functor from ''A'' to ''B'' is full.
* The '''functor category
''' '''Fct'''(''C'', ''D'') from a category ''C'' to a category ''D'' is the category where the objects are all the functors from ''C'' to ''D'' and the morphisms are all the natural transformations between the functors.
į:
* A category is '''balanced''' if every bimorphism is an isomorphism.
* A category is called a '''groupoid''' if every morphism in it is an isomorphism.

* A category is
'''cartesian closed''' if it has a terminal object and that any two objects have a product and exponential.

*
A category is '''complete''' if all small limits exist.

* A
'''concrete category''' ''C'' is a category such that there is a faithful functor from ''C'' to '''Set'''; e.g., '''Vec''', '''Grp''' and '''Top'''.

* A category is '''abelian''' if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.
* An abelian category is '''semisimple''' if every short exact sequence splits. For example, a ring is '''semisimple''' if and only if the category of modules over it is semisimple
.
Pridėtos 112-141 eilutės:

* A category is '''preadditive''' if it is '''enriched''' over the '''monoidal category''' of '''abelian group'''s. More generally, it is '''preadditive category#R-linear categories|''R''-linear''' if it is enriched over the monoidal category of '''module (mathematics)|''R''-modules''', for ''R'' a '''commutative ring'''.
* A category is '''additive''' if it is preadditive (to be precise, has some pre-additive structure) and admits all finite '''coproduct'''s. Although "preadditive" is an additional structure, one can show "additive" is a ''property'' of a category; i.e., one can ask whether a given category is additive or not.


* Given functors {$f: C \to B, g: D \to B$}, the '''comma category''' {$(f \downarrow g)$} is a category where (1) the objects are morphisms {$f(c) \to g(d)$} and (2) a morphism from {$\alpha: f(c) \to g(d)$} to {$\beta: f(c') \to g(d')$} consists of {$c \to c'$} and {$d \to d'$} such that {$f(c) \to f(c') \overset{\beta}\to g(d')$} is {$f(c) \overset{\alpha}\to g(d) \to g(d').$} For example, if ''f'' is the identity functor and ''g'' is the constant functor with a value ''b'', then it is the slice category of ''B'' over an object ''b''.


* A category is '''connected''' if, for each pair of objects ''x'', ''y'', there exists a finite sequence of objects {$z_i$} such that {$z_0 = x, z_n = y$} and either {$\operatorname{Hom}(z_i, z_{i+1})$} or {$\operatorname{Hom}(z_{i+1}, z_i)$} is nonempty for any ''i''.
* A '''differential graded category''' is a category whose Hom sets are equipped with structures of '''differential graded module'''s. In particular, if the category has only one object, it is the same as a differential graded module.

* Eilenberg–Moore category. Another name for the category of '''algebras for a given monad'''.
* '''enriched category''' Given a monoidal category (''C'', ⊗, 1), a '''category enriched''' over ''C'' is, informally, a category whose Hom sets are in ''C''. More precisely, a category ''D'' enriched over ''C'' is a data consisting of
# A class of objects,
# For each pair of objects ''X'', ''Y'' in ''D'', an object {$\operatorname{Map}_D(X, Y)$} in ''C'', called the '''mapping object''' from ''X'' to ''Y'',
# For each triple of objects ''X'', ''Y'', ''Z'' in ''D'', a morphism in ''C'',
#:{$\circ: \operatorname{Map}_D(Y, Z) \otimes \operatorname{Map}_D(X, Y) \to \operatorname{Map}_D(X, Z)$},
#:called the composition,
#For each object ''X'' in ''D'', a morphism {$1_X: 1 \to \operatorname{Map}_D(X, X)$} in ''C'', called the unit morphism of ''X'' subject to the conditions that (roughly) the compositions are associative and the unit morphisms act as the multiplicative identity.
For example, a category enriched over sets is an ordinary category.

* The '''fundamental groupoid''' {$\Pi_1 X$} of a Kan complex ''X'' is the category where an object is a 0-simplex (vertex) {$\Delta^0 \to X$}, a morphism is a homotopy class of a 1-simplex (path) {$\Delta^1 \to X$} and a composition is determined by the Kan property.
* A '''filtered category''' (also called a filtrant category) is a nonempty category with the properties (1) given objects ''i'' and ''j'', there are an object ''k'' and morphisms ''i'' → ''k'' and ''j'' → ''k'' and (2) given morphisms ''u'', ''v'': ''i'' → ''j'', there are an object ''k'' and a morphism ''w'': ''j'' → ''k'' such that ''w'' ∘ ''u'' = ''w'' ∘ ''v''. A category ''I'' is filtered if and only if, for each finite category ''J'' and functor ''f'': ''J'' → ''I'', the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''.
* Given a cardinal number π, a category is said to be π-filtrant if, for each category ''J'' whose set of morphisms has cardinal number strictly less than π, the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''.

* A '''Frobenius category''' is an '''exact category''' that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects.
* '''Fukaya category'''.

* The '''functor category''' '''Fct'''(''C'', ''D'') from a category ''C'' to a category ''D'' is the category where the objects are all the functors from ''C'' to ''D'' and the morphisms are all the natural transformations between the functors.
Ištrinta 142 eilutė:
* A category is called a '''groupoid''' if every morphism in it is an isomorphism. An ∞-category is called an '''∞-groupoid''' if every morphism in it is an equivalence (or equivalently if it is a '''Kan complex'''.
Pakeista 145 eilutė iš:
* '''1=∞-category''' An '''∞-category''' ''C'' is a '''simplicial set''' satisfying the following condition: for each 0 < ''i'' < ''n'', every map of simplicial sets {$f: \Lambda^n_i \to C$} extends to an ''n''-simplex {$f: \Delta^n \to C$}, where &Delta;<sup>''n''</sup> is the standard ''n''-simplex and {$\Lambda^n_i$} is obtained from &Delta;<sup>''n''</sup> by removing the ''i''-th face and the interior (see '''Kan fibration#Definition'''). For example, the '''nerve of a category''' satisfies the condition and thus can be considered as an ∞-category.
į:
Pakeista 149 eilutė iš:
* '''''n''-category''' A '''strict n-category|strict ''n''-category''' is defined inductively: a strict 0-category is a set and a strict ''n''-category is a category whose Hom sets are strict (''n''-1)-categories. Precisely, a strict ''n''-category is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category.
į:
* '''''n''-category''' A '''strict ''n''-category''' is defined inductively: a strict 0-category is a set and a strict ''n''-category is a category whose Hom sets are strict (''n''-1)-categories. Precisely, a strict ''n''-category is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category.
Pakeistos 151-153 eilutės iš
* The '''opposite category''' of a category is obtained by reversing the arrows. For example, if a partially ordered set is viewed as a category, taking its opposite amounts to reversing the ordering.
* A category (or ∞-category) is called '''pointed''' if it has a zero object.
* A category is '''preadditive''' if it is '''enriched category|enriched''' over the '''monoidal category''' of '''abelian group'''s. More generally, it is '''preadditive category#R-linear categories|''R''-linear''' if it is enriched over the monoidal category of '''module (mathematics)|''R''-modules''', for ''R'' a '''commutative ring'''.
į:
Pakeista 157 eilutė iš:
* An abelian category is '''semisimple''' if every short exact sequence splits. For example, a ring is '''semisimple''' if and only if the category of modules over it is semisimple.
į:
Pridėtos 187-188 eilutės:
* An ∞-category is called an '''∞-groupoid''' if every morphism in it is an equivalence (or equivalently if it is a '''Kan complex'''.
* '''1=∞-category''' An '''∞-category''' ''C'' is a '''simplicial set''' satisfying the following condition: for each 0 < ''i'' < ''n'', every map of simplicial sets {$f: \Lambda^n_i \to C$} extends to an ''n''-simplex {$f: \Delta^n \to C$}, where &Delta;<sup>''n''</sup> is the standard ''n''-simplex and {$\Lambda^n_i$} is obtained from &Delta;<sup>''n''</sup> by removing the ''i''-th face and the interior (see '''Kan fibration#Definition'''). For example, the '''nerve of a category''' satisfies the condition and thus can be considered as an ∞-category.
Pridėtos 200-201 eilutės:

* An '''adjunction''' (also called an adjoint pair) is a pair of functors ''F'': ''C'' → ''D'', ''G'': ''D'' → ''C'' such that there is a "natural" bijection:{$\operatorname{Hom}_D (F(X), Y) \simeq \operatorname{Hom}_C (X, G(Y))$};''F'' is said to be left adjoint to ''G'' and ''G'' to right adjoint to ''F''. Here, "natural" means there is a natural isomorphism {$\operatorname{Hom}_D (F(-), -) \simeq \operatorname{Hom}_C (-, G(-))$} of bifunctors (which are contravariant in the first variable.)
2019 vasario 11 d., 12:46 atliko AndriusKulikauskas -
Pakeista 135 eilutė iš:
* The '''product of a family of categories''' ''C''<sub>''i''</sub>'s indexed by a set ''I'' is the category denoted by {$\prod_i C_i$} whose class of objects is the product of the classes of objects of ''C''<sub>''i''</sub>'s and whose hom-sets are {$\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)$}; the morphisms are composed component-wise. It is the dual of the disjoint union.
į:
* The '''product of a family of categories''' {$C_i$}'s indexed by a set ''I'' is the category denoted by {$\prod_i C_i$} whose class of objects is the product of the classes of objects of {$C_i$}'s and whose hom-sets are {$\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)$}; the morphisms are composed component-wise. It is the dual of the disjoint union.
Pakeista 140 eilutė iš:
* Given a category ''C'' and an object ''A'' in it, the '''slice category''' ''C''<sub>/''A''</sub> of ''C'' over ''A'' is the category whose objects are all the morphisms in ''C'' with codomain ''A'', whose morphisms are morphisms in ''C'' such that if ''f'' is a morphism from {$p_X: X \to A$} to {$p_Y: Y \to A$}, then {$p_Y \circ f = p_X$} in ''C'' and whose composition is that of ''C''.
į:
* Given a category ''C'' and an object ''A'' in it, the '''slice category''' {$C_A$} of ''C'' over ''A'' is the category whose objects are all the morphisms in ''C'' with codomain ''A'', whose morphisms are morphisms in ''C'' such that if ''f'' is a morphism from {$p_X: X \to A$} to {$p_Y: Y \to A$}, then {$p_Y \circ f = p_X$} in ''C'' and whose composition is that of ''C''.
Pakeistos 259-260 eilutės iš
* The '''product''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the projective limit {$\varprojlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where ''I'' is viewed as a discrete category. It is denoted by {$\prod_i X_i$} and is the dual of the coproduct of the family.
* The '''coproduct''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the inductive limit {$\varinjlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where ''I'' is viewed as a discrete category. It is the dual of the product of the family. For example, a coproduct in ''''''category of groups|Grp'''''' is a '''free product'''.
į:
* The '''product''' of a family of objects {$X_i$} in a category ''C'' indexed by a set ''I'' is the projective limit {$\varprojlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where ''I'' is viewed as a discrete category. It is denoted by {$\prod_i X_i$} and is the dual of the coproduct of the family.
* The '''coproduct''' of a family of objects {$X_i$} in a category ''C'' indexed by a set ''I'' is the inductive limit {$\varinjlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where ''I'' is viewed as a discrete category. It is the dual of the product of the family. For example, a coproduct in ''''''category of groups|Grp'''''' is a '''free product'''.
Pakeistos 262-263 eilutės iš
* Given a category ''C'' and a set ''I'', the '''fiber product''' over an object ''S'' of a family of objects ''X''<sub>''i''</sub> in ''C'' indexed by ''I'' is the product of the family in the '''slice category''' {$C_{/S}$} of ''C'' over ''S'' (provided there are {$X_i \to S$}). The fiber product of two objects ''X'' and ''Y'' over an object ''S'' is denoted by {$X \times_S Y$} and is also called a '''Cartesian square'''.
į:
* Given a category ''C'' and a set ''I'', the '''fiber product''' over an object ''S'' of a family of objects {$X_i$} in ''C'' indexed by ''I'' is the product of the family in the '''slice category''' {$C_{/S}$} of ''C'' over ''S'' (provided there are {$X_i \to S$}). The fiber product of two objects ''X'' and ''Y'' over an object ''S'' is denoted by {$X \times_S Y$} and is also called a '''Cartesian square'''.
Pakeista 287 eilutė iš:
* '''Grothendieck construction''' Given a functor {$U: C \to \mathbf{Cat}$}, let ''D''<sub>''U''</sub> be the category where the objects are pairs (''x'', ''u'') consisting of an object ''x'' in ''C'' and an object ''u'' in the category ''U''(''x'') and a morphism from (''x'', ''u'') to (''y'', ''v'') is a pair consisting of a morphism ''f'': ''x'' → ''y'' in ''C'' and a morphism ''U''(''f'')(''u'') → ''v'' in ''U''(''y''). The passage from ''U'' to ''D''<sub>''U''</sub> is then called the '''Grothendieck construction'''.
į:
* '''Grothendieck construction''' Given a functor {$U: C \to \mathbf{Cat}$}, let {$D_U$} be the category where the objects are pairs (''x'', ''u'') consisting of an object ''x'' in ''C'' and an object ''u'' in the category ''U''(''x'') and a morphism from (''x'', ''u'') to (''y'', ''v'') is a pair consisting of a morphism ''f'': ''x'' → ''y'' in ''C'' and a morphism ''U''(''f'')(''u'') → ''v'' in ''U''(''y''). The passage from ''U'' to {$D_U$} is then called the '''Grothendieck construction'''.
2019 vasario 11 d., 12:44 atliko AndriusKulikauskas -
Pakeistos 74-75 eilutės iš
* Stated more explicitly, given ''f'' as above, a morphism {$X \to f(u_X)$} in ''D'' is '''universal''' if and only if the natural map {$\operatorname{Hom}_C(u_X, c) \to \operatorname{Hom}_D(X, f(c)), \, \alpha \mapsto (X \to f(u_x) \overset{f(\alpha)}\to f(c))$} is bijective. In particular, if {$\operatorname{Hom}_C(u_X, -) \simeq \operatorname{Hom}_D(X, f(-))$}, then taking ''c'' to be ''u''<sub>''X''</sub> one gets a universal morphism by sending the identity morphism. In other words, having a universal morphism is equivalent to the representability of the functor {$\operatorname{Hom}_D(X, f(-))$}.
į:
* Stated more explicitly, given ''f'' as above, a morphism {$X \to f(u_X)$} in ''D'' is '''universal''' if and only if the natural map {$\operatorname{Hom}_C(u_X, c) \to \operatorname{Hom}_D(X, f(c)), \, \alpha \mapsto (X \to f(u_x) \overset{f(\alpha)}\to f(c))$} is bijective. In particular, if {$\operatorname{Hom}_C(u_X, -) \simeq \operatorname{Hom}_D(X, f(-))$}, then taking ''c'' to be {$u_X$} one gets a universal morphism by sending the identity morphism. In other words, having a universal morphism is equivalent to the representability of the functor {$\operatorname{Hom}_D(X, f(-))$}.
Pakeista 90 eilutė iš:
* Given a '''regular cardinal''' κ, a category is '''κ-accessible''' if it has κ-filtered colimits and there exists a small set ''S'' of κ-compact objects that generates the category under colimits, meaning every object can be written as a colimit of diagrams of objects in ''S''.
į:
Pakeista 99 eilutė iš:
* A category is '''connected''' if, for each pair of objects ''x'', ''y'', there exists a finite sequence of objects ''z''<sub>''i''</sub> such that {$z_0 = x, z_n = y$} and either {$\operatorname{Hom}(z_i, z_{i+1})$} or {$\operatorname{Hom}(z_{i+1}, z_i)$} is nonempty for any ''i''.
į:
* A category is '''connected''' if, for each pair of objects ''x'', ''y'', there exists a finite sequence of objects {$z_i$} such that {$z_0 = x, z_n = y$} and either {$\operatorname{Hom}(z_i, z_{i+1})$} or {$\operatorname{Hom}(z_{i+1}, z_i)$} is nonempty for any ''i''.
Pakeistos 152-154 eilutės iš
* The '''homological dimension''' of an abelian category with enough injectives is the least non-negative integer ''n'' such that every object in the category admits an injective resolution of length at most ''n''. The dimension is ∞ if no such integer exists. For example, the homological dimension of '''Mod<sub>''R''</sub>''' with a principal ideal domain ''R'' is at most one.
į:
* The '''homological dimension''' of an abelian category with enough injectives is the least non-negative integer ''n'' such that every object in the category admits an injective resolution of length at most ''n''. The dimension is ∞ if no such integer exists. For example, the homological dimension of {$Mod_R$} with a principal ideal domain ''R'' is at most one.

* Given a
'''regular cardinal''' κ, a category is '''κ-accessible''' if it has κ-filtered colimits and there exists a small set ''S'' of κ-compact objects that generates the category under colimits, meaning every object can be written as a colimit of diagrams of objects in ''S''.
2019 vasario 11 d., 12:42 atliko AndriusKulikauskas -
Pakeista 51 eilutė iš:
* An object in an abelian category is said to have <span id="finite length"></span>finite length if it has a '''composition series'''. The maximum number of proper subobjects in any such composition series is called the '''length''' of ''A''.
į:
* An object in an abelian category is said to have finite length if it has a '''composition series'''. The maximum number of proper subobjects in any such composition series is called the '''length''' of ''A''.
2019 vasario 11 d., 12:41 atliko AndriusKulikauskas -
Pakeista 73 eilutė iš:
* Given a functor {$f: C \to D$} and an object ''X'' in ''D'', a '''universal morphism''' from ''X'' to ''f'' is an initial object in the '''comma category''' {$(X \downarrow f)$}. (Its dual is also called a universal morphism.) For example, take ''f'' to be the forgetful functor {$\mathbf{Vec}_k \to \mathbf{Set}$} and ''X'' a set. An initial object of {$(X \downarrow f)$} is a function {$j: X \to f(V_X)$}. That it is initial means that if {$k: X \to f(W)$} is another morphism, then there is a unique morphism from ''j'' to ''k'', which consists of a linear map {$V_X \to W$} that extends ''k'' via ''j''; that is to say, {$V_X$} is the '''free vector space''' generated by ''X''.}}
į:
* Given a functor {$f: C \to D$} and an object ''X'' in ''D'', a '''universal morphism''' from ''X'' to ''f'' is an initial object in the '''comma category''' {$(X \downarrow f)$}. (Its dual is also called a universal morphism.) For example, take ''f'' to be the forgetful functor {$\mathbf{Vec}_k \to \mathbf{Set}$} and ''X'' a set. An initial object of {$(X \downarrow f)$} is a function {$j: X \to f(V_X)$}. That it is initial means that if {$k: X \to f(W)$} is another morphism, then there is a unique morphism from ''j'' to ''k'', which consists of a linear map {$V_X \to W$} that extends ''k'' via ''j''; that is to say, {$V_X$} is the '''free vector space''' generated by ''X''.
Pakeista 86 eilutė iš:
For example, a '''partially ordered set''' can be viewed as a category: the objects are the elements of the set and for each pair of objects ''x'', ''y'', there is a unique morphism {$x \to y$} if and only if {$x \le y$}; the associativity of composition means transitivity.}}
į:
For example, a '''partially ordered set''' can be viewed as a category: the objects are the elements of the set and for each pair of objects ''x'', ''y'', there is a unique morphism {$x \to y$} if and only if {$x \le y$}; the associativity of composition means transitivity.
Pakeista 92 eilutė iš:
* An '''adjunction''' (also called an adjoint pair) is a pair of functors ''F'': ''C'' → ''D'', ''G'': ''D'' → ''C'' such that there is a "natural" bijection:{$\operatorname{Hom}_D (F(X), Y) \simeq \operatorname{Hom}_C (X, G(Y))$};''F'' is said to be left adjoint to ''G'' and ''G'' to right adjoint to ''F''. Here, "natural" means there is a natural isomorphism {$\operatorname{Hom}_D (F(-), -) \simeq \operatorname{Hom}_C (-, G(-))$} of bifunctors (which are contravariant in the first variable.)}}
į:
* An '''adjunction''' (also called an adjoint pair) is a pair of functors ''F'': ''C'' → ''D'', ''G'': ''D'' → ''C'' such that there is a "natural" bijection:{$\operatorname{Hom}_D (F(X), Y) \simeq \operatorname{Hom}_C (X, G(Y))$};''F'' is said to be left adjoint to ''G'' and ''G'' to right adjoint to ''F''. Here, "natural" means there is a natural isomorphism {$\operatorname{Hom}_D (F(-), -) \simeq \operatorname{Hom}_C (-, G(-))$} of bifunctors (which are contravariant in the first variable.)
Pakeista 140 eilutė iš:
* Given a category ''C'' and an object ''A'' in it, the '''slice category''' ''C''<sub>/''A''</sub> of ''C'' over ''A'' is the category whose objects are all the morphisms in ''C'' with codomain ''A'', whose morphisms are morphisms in ''C'' such that if ''f'' is a morphism from {$p_X: X \to A$} to {$p_Y: Y \to A$}, then {$p_Y \circ f = p_X$} in ''C'' and whose composition is that of ''C''.}}
į:
* Given a category ''C'' and an object ''A'' in it, the '''slice category''' ''C''<sub>/''A''</sub> of ''C'' over ''A'' is the category whose objects are all the morphisms in ''C'' with codomain ''A'', whose morphisms are morphisms in ''C'' such that if ''f'' is a morphism from {$p_X: X \to A$} to {$p_Y: Y \to A$}, then {$p_Y \circ f = p_X$} in ''C'' and whose composition is that of ''C''.
Pakeista 182 eilutė iš:
* A '''contravariant functor''' ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a '''presheaf (category theory)|presheaf''' especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let {$\mathfrak{P}(S)$} be the power set of ''S'' and for each function {$f: S \to T$}, define:{$\mathfrak{P}(f): \mathfrak{P}(T) \to \mathfrak{P}(S)$} by sending a subset ''A'' of ''T'' to the pre-image {$f^{-1}(A)$}. With this, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}$} is a contravariant functor.}}
į:
* A '''contravariant functor''' ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a '''presheaf (category theory)|presheaf''' especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let {$\mathfrak{P}(S)$} be the power set of ''S'' and for each function {$f: S \to T$}, define:{$\mathfrak{P}(f): \mathfrak{P}(T) \to \mathfrak{P}(S)$} by sending a subset ''A'' of ''T'' to the pre-image {$f^{-1}(A)$}. With this, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}$} is a contravariant functor.
Pakeista 198 eilutė iš:
* Given a category ''C'', the left '''Kan extension''' functor along a functor {$f: I \to J$} is the left adjoint (if it exists) to {$f^* = - \circ f: \operatorname{Fct}(J, C) \to \operatorname{Fct}(I, C)$} and is denoted by {$f_!$}. For any {$\alpha: I \to C$}, the functor {$f_! \alpha: J \to C$} is called the left Kan extension of α along ''f''.<ref>http://www.math.harvard.edu/~lurie/282ynotes/LectureXI-Homological.pdf</ref> One can show: {$(f_! \alpha)(j) = \varinjlim_{f(i) \to j} \alpha(i)$} where the colimit runs over all objects {$f(i) \to j$} in the comma category.}} The right Kan extension functor is the right adjoint (if it exists) to {$f^*$}.
į:
* Given a category ''C'', the left '''Kan extension''' functor along a functor {$f: I \to J$} is the left adjoint (if it exists) to {$f^* = - \circ f: \operatorname{Fct}(J, C) \to \operatorname{Fct}(I, C)$} and is denoted by {$f_!$}. For any {$\alpha: I \to C$}, the functor {$f_! \alpha: J \to C$} is called the left Kan extension of α along ''f''.<ref>http://www.math.harvard.edu/~lurie/282ynotes/LectureXI-Homological.pdf</ref> One can show: {$(f_! \alpha)(j) = \varinjlim_{f(i) \to j} \alpha(i)$} where the colimit runs over all objects {$f(i) \to j$} in the comma category. The right Kan extension functor is the right adjoint (if it exists) to {$f^*$}.
Pakeista 217 eilutė iš:
* Given a ''k''-linear category ''C'' over a field ''k'', a '''Serre functor''' {$f: C \to C$} is an auto-equivalence such that {$\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*$} for any objects ''A'', ''B''.}}
į:
* Given a ''k''-linear category ''C'' over a field ''k'', a '''Serre functor''' {$f: C \to C$} is an auto-equivalence such that {$\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*$} for any objects ''A'', ''B''.
Pakeistos 225-227 eilutės iš
* The functor {$y: C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(-, X)$} is fully faithful and is called the Yoneda embedding.<ref>Technical note: the lemma implicitly involves a choice of '''Set'''; i.e., a choice of universe.</ref> }}
į:
* The functor {$y: C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(-, X)$} is fully faithful and is called the Yoneda embedding. Technical note: the lemma implicitly involves a choice of '''Set'''; i.e., a choice of universe.
Pakeista 247 eilutė iš:
* The '''limit''' (or '''projective limit''') of a functor {$f: I^{\text{op}} \to \mathbf{Set}$} is {$\varprojlim_{i \in I} f(i) = \{ (x_i|i) \in \prod_{i} f(i) | f(s)(x_j) = x_i \text{ for any } s: i \to j \}.$}}}
į:
* The '''limit''' (or '''projective limit''') of a functor {$f: I^{\text{op}} \to \mathbf{Set}$} is {$\varprojlim_{i \in I} f(i) = \{ (x_i|i) \in \prod_{i} f(i) | f(s)(x_j) = x_i \text{ for any } s: i \to j \}.$}
Pakeista 275 eilutė iš:
* The '''homotopy hypothesis''' states an '''∞-groupoid''' is a space (less equivocally, an ''n''-groupoid can be used as a homotopy ''n''-type.)}}
į:
* The '''homotopy hypothesis''' states an '''∞-groupoid''' is a space (less equivocally, an ''n''-groupoid can be used as a homotopy ''n''-type.)
2019 vasario 11 d., 12:38 atliko AndriusKulikauskas -
Pakeistos 43-44 eilutės iš
* An object in a category is said to be '''small''' if it is κ-compact for some regular cardinal κ. The notion prominently appears in Quiilen's '''small object argument''' (cf. https://ncatlab.org/nlab/show/small+object+argument)
į:
* An object in a category is said to be '''small''' if it is κ-compact for some regular cardinal κ. The notion prominently appears in Quiilen's '''small object argument''' [[https://ncatlab.org/nlab/show/small+object+argument | n-lab: small object argument]].
Ištrintos 71-72 eilutės:
2019 vasario 11 d., 12:36 atliko AndriusKulikauskas -
Pridėtos 58-61 eilutės:
* The '''identity morphism''' ''f'' of an object ''A'' is a morphism from ''A'' to ''A'' such that for any morphisms ''g'' with domain ''A'' and ''h'' with codomain ''A'', {$g\circ f=g$} and {$f\circ h=h$}.

* A morphism ''f'' is an '''epimorphism''' if {$g=h$} whenever {$g\circ f=h\circ f$}. In other words, ''f'' is the dual of a monomorphism.
* A morphism ''f'' is a '''monomorphism''' (also called monic) if {$g=h$} whenever {$f\circ g=f\circ h$}; e.g., an '''injection''' in '''Set'''. In other words, ''f'' is the dual of an epimorphism.
Pridėtos 63-69 eilutės:

* A morphism ''f'' is an '''inverse''' to a morphism ''g'' if {$g\circ f$} is defined and is equal to the identity morphism on the codomain of ''g'', and {$f\circ g$} is defined and equal to the identity morphism on the domain of ''g''. The inverse of ''g'' is unique and is denoted by ''{$g^{-1}$}''.
* A morphism ''f'' is an '''isomorphism''' if there exists an ''inverse'' of ''f''.
* ''f'' is a left inverse to ''g'' if {$f\circ g$} is defined and is equal to the identity morphism on the domain of ''g'', and similarly for a right inverse.
* A morphism is a '''section''' if it has a left inverse. For example, the '''axiom of choice''' says that any surjective function admits a section.
* A morphism is a '''retraction''' if it has a right inverse.
Pakeistos 72-78 eilutės iš
* A morphism ''f'' is an '''epimorphism''' if {$g=h$} whenever {$g\circ f=h\circ f$}. In other words, ''f'' is the dual of a monomorphism.
* The '''identity morphism''' ''f'' of an object ''A'' is a morphism from ''A'' to ''A'' such that for any morphisms ''g'' with domain ''A'' and ''h'' with codomain ''A'', {$g\circ f=g$} and {$f\circ h=h$}.
* A morphism ''f'' is an '''inverse''' to a morphism ''g'' if {$g\circ f$} is defined and is equal to the identity morphism on the codomain of ''g'', and {$f\circ g$} is defined and equal to the identity morphism on the domain of ''g''. The inverse of ''g'' is unique and is denoted by ''g''<sup>−1</sup>. ''f'' is a left inverse to ''g'' if {$f\circ g$} is defined and is equal to the identity morphism on the domain of ''g'', and similarly for a right inverse.
* A morphism ''f'' is an '''isomorphism''' if there exists an ''inverse'' of ''f''.
* A morphism ''f'' is a '''monomorphism''' (also called monic) if {$g=h$} whenever {$f\circ g=f\circ h$}; e.g., an '''Injective function|injection''' in ''''''Category of sets|Set''''''. In other words, ''f'' is the dual of an epimorphism.
* A morphism is a '''retraction''' if it has a right inverse.
* A morphism is a '''section''' if it has a left inverse. For example, the '''axiom of choice''' says that any surjective function admits a section.
į:
2019 vasario 11 d., 12:32 atliko AndriusKulikauskas -
Ištrintos 31-34 eilutės:

* A '''Kan complex''' is a '''fibrant object''' in the category of simplicial sets.
* A '''monoid object''' in a monoidal category is an object together with the multiplication map and the identity map that satisfy the expected conditions like associativity. For example, a monoid object in '''Set''' is a usual monoid (unital semigroup) and a monoid object in '''''R''-mod''' is an '''associative algebra''' over a commutative ring ''R''.
Pridėtos 34-35 eilutės:
* A '''monoid object''' in a monoidal category is an object together with the multiplication map and the identity map that satisfy the expected conditions like associativity. For example, a monoid object in '''Set''' is a usual monoid (unital semigroup) and a monoid object in '''''R''-mod''' is an '''associative algebra''' over a commutative ring ''R''.
Pridėtos 37-38 eilutės:

* A '''Kan complex''' is a '''fibrant object''' in the category of simplicial sets.
2019 vasario 11 d., 12:28 atliko AndriusKulikauskas -
Ištrintos 20-26 eilutės:
* Given a '''cardinal number''' κ, an object ''X'' in a category is '''κ-accessible''' (or κ-compact or κ-presentable) if {$\operatorname{Hom}(X, -)$} commutes with κ-filtered colimits.
* '''compact''' Probably synonymous with '''#accessible'''.
* '''perfect''' Sometimes synonymous with "compact". See '''perfect complex'''.
* In a category ''C'', a family of objects {$G_i, i \in I$} is a '''system of generators''' of ''C'' if the functor {$X \mapsto \prod_{i \in I} \operatorname{Hom}(G_i, X)$} is conservative. Its dual is called a system of cogenerators.
* An object ''A'' is '''initial''' if there is exactly one morphism from ''A'' to each object; e.g., '''empty set''' in ''''''Category of sets|Set''''''.
* An object ''A'' in an ∞-category ''C'' is initial if {$\operatorname{Map}_C(A, B)$} is '''contractible''' for each object ''B'' in ''C''.
* An object ''A'' in an abelian category is '''injective''' if the functor {$\operatorname{Hom}(-, A)$} is exact. It is the dual of a projective object.
Pakeistos 22-27 eilutės iš
* A '''Kan complex''' is a '''fibrant object''' in the category of simplicial sets.
* A
'''monoid object''' in a monoidal category is an object together with the multiplication map and the identity map that satisfy the expected conditions like associativity. For example, a monoid object in '''Set''' is a usual monoid (unital semigroup) and a monoid object in '''''R''-mod''' is an '''associative algebra''' over a commutative ring ''R''.
* An object ''A'' in an abelian category is '''projective''' if the functor {$\operatorname{Hom}(A, -)$} is exact. It is the dual of an injective object.
* A simple object in an abelian category is <span id="simple object"></span>an object ''A'' that is not isomorphic to the zero object and whose every '''subobject''' is isomorphic to zero or to ''A''. For example, a '''simple module''' is precisely a simple object in the category of (say left) modules.
* An object in a category is said to be '''small''' if it is κ-compact for some regular cardinal κ. The notion prominently appears in Quiilen's '''small object argument''' (cf. https://ncatlab.org/nlab/show/small+object+argument)
* A '''subterminal object''' is an object ''X'' such that every object has at most one morphism into ''X
''.
į:
* An object ''A'' is '''initial''' if there is exactly one morphism from ''A'' to each object; e.g., '''empty set''' in '''Set'''.
Pakeistos 25-26 eilutės iš
* An object ''A'' in an ∞-category ''C'' is terminal if {$\operatorname{Map}_C(B, A)$} is '''contractible space|contractible''' for every object ''B'' in ''C''.
* A '''zero object''' is an object that is both initial and terminal, such as a '''trivial group''' in ''''''Category of groups|Grp''''''.
į:
* A '''zero object''' is an object that is both initial and terminal, such as a '''trivial group''' in '''Grp'''.
* A '''subterminal object''' is an object ''X'' such that every object has at most one morphism into ''X''.

* In a category
''C'', a family of objects {$G_i, i \in I$} is a '''system of generators''' of ''C'' if the functor {$X \mapsto \prod_{i \in I} \operatorname{Hom}(G_i, X)$} is conservative. Its dual is called a system of cogenerators.

* An object ''A'' in an abelian category is '''injective''' if the functor {$\operatorname{Hom}(-, A)$} is exact. It is the dual of a projective object.
* An object ''A'' in an abelian category is '''projective''' if the functor {$\operatorname{Hom}(A, -)$} is exact. It is the dual of an injective object.

* A '''Kan complex''' is a '''fibrant object''' in the category of simplicial sets.
* A '''monoid object''' in a monoidal category is an object together with the multiplication map and the identity map that satisfy the expected conditions like associativity. For example, a monoid object in '''Set''' is a usual monoid (unital semigroup) and a monoid object in '''''R''-mod''' is an '''associative algebra''' over a commutative ring ''R''.

* A '''simple object''' in an abelian category is an object ''A'' that is not isomorphic to the zero object and whose every '''subobject''' is isomorphic to zero or to ''A''. For example, a '''simple module''' is precisely a simple object in the category of (say left) modules.
Pridėtos 40-44 eilutės:
* Given a '''cardinal number''' κ, an object ''X'' in a category is '''κ-accessible''' (or κ-compact or κ-presentable) if {$\operatorname{Hom}(X, -)$} commutes with κ-filtered colimits.
* '''compact''' Probably synonymous with '''#accessible'''.
* '''perfect''' Sometimes synonymous with "compact". See '''perfect complex'''.
* An object in a category is said to be '''small''' if it is κ-compact for some regular cardinal κ. The notion prominently appears in Quiilen's '''small object argument''' (cf. https://ncatlab.org/nlab/show/small+object+argument)
Pridėtos 47-48 eilutės:
* An object ''A'' in an ∞-category ''C'' is initial if {$\operatorname{Map}_C(A, B)$} is '''contractible''' for each object ''B'' in ''C''.
* An object ''A'' in an ∞-category ''C'' is terminal if {$\operatorname{Map}_C(B, A)$} is '''contractible''' for every object ''B'' in ''C''.
2019 vasario 11 d., 11:36 atliko AndriusKulikauskas -
Pakeista 222 eilutė iš:
* '''A natural transformation is, roughly, a map between functors. Precisely, given a pair of functors ''F'', ''G'' from a category ''C'' to category ''D'', a '''natural transformation''' φ from ''F'' to ''G'' is a set of morphisms in ''D'':{$\{ \phi_x: F(x) \to G(x) \mid x \in \operatorname{Ob}(C) \}$} satisfying the condition: for each morphism ''f'': ''x'' → ''y'' in ''C'', {$\phi_y \circ F(f) = G(f) \circ \phi_x$}. For example, writing {$GL_n(R)$} for the group of invertible ''n''-by-''n'' matrices with coefficients in a commutative ring ''R'', we can view {$GL_n$} as a functor from the category '''CRing''' of commutative rings to the category '''Grp''' of groups. Similarly, {$R \mapsto R^*$} is a functor from '''CRing''' to '''Grp'''. Then the '''determinant''' det is a natural transformation from {$GL_n$} to -<sup>*</sup>.
į:
* A '''natural transformation''' is, roughly, a map between functors. Precisely, given a pair of functors ''F'', ''G'' from a category ''C'' to category ''D'', a '''natural transformation''' φ from ''F'' to ''G'' is a set of morphisms in ''D'':{$\{ \phi_x: F(x) \to G(x) \mid x \in \operatorname{Ob}(C) \}$} satisfying the condition: for each morphism ''f'': ''x'' → ''y'' in ''C'', {$\phi_y \circ F(f) = G(f) \circ \phi_x$}. For example, writing {$GL_n(R)$} for the group of invertible ''n''-by-''n'' matrices with coefficients in a commutative ring ''R'', we can view {$GL_n$} as a functor from the category '''CRing''' of commutative rings to the category '''Grp''' of groups. Similarly, {$R \mapsto R^*$} is a functor from '''CRing''' to '''Grp'''. Then the '''determinant''' det is a natural transformation from {$GL_n$} to -<sup>*</sup>.
Pakeista 251 eilutė iš:
* The coend of a functor {$F: C^{\text{op}} \times C \to X$} is the dual of the '''end (category theory)|end''' of ''F'' and is denoted by {$\int^{c \in C} F(c, c)$}. For example, if ''R'' is a ring, ''M'' a right ''R''-module and ''N'' a left ''R''-module, then the '''tensor product''' of ''M'' and ''N'' is {$M \otimes_R N = \int^{R} M \otimes_{\mathbb{Z}} N$} where ''R'' is viewed as a category with one object whose morphisms are the elements of ''R''.
į:
* The '''coend''' of a functor {$F: C^{\text{op}} \times C \to X$} is the dual of the '''end''' of ''F'' and is denoted by {$\int^{c \in C} F(c, c)$}. For example, if ''R'' is a ring, ''M'' a right ''R''-module and ''N'' a left ''R''-module, then the '''tensor product''' of ''M'' and ''N'' is {$M \otimes_R N = \int^{R} M \otimes_{\mathbb{Z}} N$} where ''R'' is viewed as a category with one object whose morphisms are the elements of ''R''.
2019 vasario 11 d., 11:34 atliko AndriusKulikauskas -
Pakeistos 233-234 eilutės iš
* A '''cone''' is a way to express the '''universal property''' of a colimit (or dually a limit). One can show<ref>{{harvnb|Kashiwara|Schapira|2006|loc=Ch. 2, Exercise 2.8.}}</ref> that the colimit {$\varinjlim$} is the left adjoint to the diagonal functor {$\Delta: C \to \operatorname{Fct}(I, C)$}, which sends an object ''X'' to the constant functor with value ''X''; that is, for any ''X'' and any functor {$f: I \to C$}, {$\operatorname{Hom}(\varinjlim f, X) \simeq \operatorname{Hom}(f, \Delta_X),$} provided the colimit in question exists. The right-hand side is then the set of cones with vertex ''X''.
į:
* A '''cone''' is a way to express the '''universal property''' of a colimit (or dually a limit). One can show that the colimit {$\varinjlim$} is the left adjoint to the diagonal functor {$\Delta: C \to \operatorname{Fct}(I, C)$}, which sends an object ''X'' to the constant functor with value ''X''; that is, for any ''X'' and any functor {$f: I \to C$}, {$\operatorname{Hom}(\varinjlim f, X) \simeq \operatorname{Hom}(f, \Delta_X),$} provided the colimit in question exists. The right-hand side is then the set of cones with vertex ''X''.
Pakeista 250 eilutė iš:
* The '''end''' of a functor {$F: C^{\text{op}} \times C \to X$} is the limit {$\int_{c \in C} F(c, c) = \varprojlim (F^{\#}: C^{\#} \to X)$} where {$C^{\#}$} is the category (called the '''subdivision category''' of ''C'') whose objects are symbols {$c^{\#}, u^{\#}$} for all objects ''c'' and all morphisms ''u'' in ''C'' and whose morphisms are {$b^{\#} \to u^{\#}$} and {$u^{\#} \to c^{\#}$} if {$u: b \to c$} and where {$F^{\#}$} is induced by ''F'' so that {$c^{\#}$} would go to {$F(c, c)$} and {$u^{\#}, u: b \to c$} would go to {$F(b, c)$}. For example, for functors {$F, G: C \to X$}, {$\int_{c \in C} \operatorname{Hom}(F(c), G(c))$} is the set of natural transformations from ''F'' to ''G''. For more examples, see [http://mathoverflow.net/questions/78471/intuition-for-coends this mathoverflow thread]. The dual of an end is a coend.
į:
* The '''end''' of a functor {$F: C^{\text{op}} \times C \to X$} is the limit {$\int_{c \in C} F(c, c) = \varprojlim (F^{\#}: C^{\#} \to X)$} where {$C^{\#}$} is the category (called the '''subdivision category''' of ''C'') whose objects are symbols {$c^{\#}, u^{\#}$} for all objects ''c'' and all morphisms ''u'' in ''C'' and whose morphisms are {$b^{\#} \to u^{\#}$} and {$u^{\#} \to c^{\#}$} if {$u: b \to c$} and where {$F^{\#}$} is induced by ''F'' so that {$c^{\#}$} would go to {$F(c, c)$} and {$u^{\#}, u: b \to c$} would go to {$F(b, c)$}. For example, for functors {$F, G: C \to X$}, {$\int_{c \in C} \operatorname{Hom}(F(c), G(c))$} is the set of natural transformations from ''F'' to ''G''. For more examples, see [[http://mathoverflow.net/questions/78471/intuition-for-coends | this mathoverflow thread]]. The dual of an end is a coend.
2019 vasario 11 d., 11:33 atliko AndriusKulikauskas -
Pridėta 230 eilutė:
* Given a category ''C'', a '''diagram''' in ''C'' is a functor {$f: I \to C$} from a small category ''I''.
Pridėtos 232-240 eilutės:

* A '''cone''' is a way to express the '''universal property''' of a colimit (or dually a limit). One can show<ref>{{harvnb|Kashiwara|Schapira|2006|loc=Ch. 2, Exercise 2.8.}}</ref> that the colimit {$\varinjlim$} is the left adjoint to the diagonal functor {$\Delta: C \to \operatorname{Fct}(I, C)$}, which sends an object ''X'' to the constant functor with value ''X''; that is, for any ''X'' and any functor {$f: I \to C$}, {$\operatorname{Hom}(\varinjlim f, X) \simeq \operatorname{Hom}(f, \Delta_X),$} provided the colimit in question exists. The right-hand side is then the set of cones with vertex ''X''.

* The '''limit''' (or '''projective limit''') of a functor {$f: I^{\text{op}} \to \mathbf{Set}$} is {$\varprojlim_{i \in I} f(i) = \{ (x_i|i) \in \prod_{i} f(i) | f(s)(x_j) = x_i \text{ for any } s: i \to j \}.$}}}
* The limit {$\varprojlim_{i \in I} f(i)$} of a functor {$f: I^{\text{op}} \to C$} is an object, if any, in ''C'' that satisfies: for any object ''X'' in ''C'', {$\operatorname{Hom}(X, \varprojlim_{i \in I} f(i)) = \varprojlim_{i \in I} \operatorname{Hom}(X, f(i))$}; i.e., it is an object representing the functor {$X \mapsto \varprojlim_i \operatorname{Hom}(X, f(i)).$}
* The '''colimit''' (or '''inductive limit''') {$\varinjlim_{i \in I} f(i)$} is the dual of a limit; i.e., given a functor {$f: I \to C$}, it satisfies: for any ''X'', {$\operatorname{Hom}(\varinjlim f(i), X) = \varprojlim \operatorname{Hom}(f(i), X)$}. Explicitly, to give {$\varinjlim f(i) \to X$} is to give a family of morphisms {$f(i) \to X$} such that for any {$i \to j$}, {$f(i) \to X$} is {$f(i) \to f(j) \to X$}. Perhaps the simplest example of a colimit is a '''coequalizer'''. For another example, take ''f'' to be the identity functor on ''C'' and suppose {$L = \varinjlim_{X \in C} f(X)$} exists; then the identity morphism on ''L'' corresponds to a compatible family of morphisms {$\alpha_X: X \to L$} such that {$\alpha_L$} is the identity. If {$f: X \to L$} is any morphism, then {$f = \alpha_L \circ f = \alpha_X$}; i.e., ''L'' is a final object of ''C''.
* '''ind-limit''' A colimit (or inductive limit) in {$\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}.

* The '''equalizer''' of a pair of morphisms {$f, g: A \to B$} is the limit of the pair. It is the dual of a coequalizer.
Pridėta 242 eilutė:
* The '''image''' of a morphism ''f'': ''X'' → ''Y'' is the equalizer of {$Y \rightrightarrows Y \sqcup_X Y$}.
Pakeistos 244-246 eilutės iš
* A '''cone (category theory)|cone''' is a way to express the '''universal property''' of a colimit (or dually a limit). One can show<ref>{{harvnb|Kashiwara|Schapira|2006|loc=Ch. 2, Exercise 2.8.}}</ref> that the colimit {$\varinjlim$} is the left adjoint to the diagonal functor {$\Delta: C \to \operatorname{Fct}(I, C)$}, which sends an object ''X'' to the constant functor with value ''X''; that is, for any ''X'' and any functor {$f: I \to C$},
:{$\operatorname{Hom}(\varinjlim f, X) \simeq \operatorname{Hom}(f, \Delta_X),$}
provided the colimit in question exists. The right-hand side is then the set of cones with vertex ''X''
.
į:
* The '''product''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the projective limit {$\varprojlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where ''I'' is viewed as a discrete category. It is denoted by {$\prod_i X_i$} and is the dual of the coproduct of the family.
Pakeistos 247-253 eilutės iš
* Given a category ''C'', a '''diagram''' in ''C'' is a functor {$f: I \to C$} from a small category ''I''.
* The '''end''' of a functor
{$F: C^{\text{op}} \times C \to X$} is the limit
:{$\int_{c \in C} F(c, c) = \varprojlim (F^{\#}: C^{\#} \to X)$}
where {$C^{\#}$} is the category (called the '''subdivision category''' of ''C'') whose objects are symbols {$c^{\#}, u^{\#}$} for all objects ''c'' and all morphisms ''u'' in ''C'' and whose morphisms are {$b^{\#} \to u^{\#}$} and {$u^{\#} \to c^{\#}$} if {$u: b \to c$} and where {$F^{\#}$} is induced by ''F'' so that {$c^{\#}$} would go to {$F(c, c)$} and {$u^{\#}, u: b \to c$} would go to {$F(b, c)$}. For example, for functors {$F, G: C \to X$},
:{$\int_{c \in C} \operatorname{Hom}(F(c), G(c))$}
is the set of natural transformations from ''F'' to ''G''. For more examples, see [http://mathoverflow.net/questions/78471/intuition-for-coends this mathoverflow thread]. The dual of an end is a coend.
* The '''equalizer (mathematics)|equalizer''' of a pair of morphisms {$f, g: A \to B$} is the limit of the pair. It is the dual of a coequalizer.
į:
* Given a monoidal category ''B'', the '''tensor product of functors''' {$F: C^{\text{op}} \to B$} and {$G: C \to B$} is the coend: {$F \otimes_C G = \int^{c \in C} F(c) \otimes G(c).$}
Pakeistos 249-255 eilutės iš
* The '''image''' of a morphism ''f'': ''X'' → ''Y'' is the equalizer of {$Y \rightrightarrows Y \sqcup_X Y$}.
* '''ind-limit''' A colimit
(or inductive limit) in {$\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}.
* The
'''limit''' (or '''projective limit''') of a functor {$f: I^{\text{op}} \to \mathbf{Set}$} is: {$\varprojlim_{i \in I} f(i) = \{ (x_i|i) \in \prod_{i} f(i) | f(s)(x_j) = x_i \text{ for any } s: i \to j \}.$}}}
* The limit
{$\varprojlim_{i \in I} f(i)$} of a functor {$f: I^{\text{op}} \to C$} is an object, if any, in ''C'' that satisfies: for any object ''X'' in ''C'', {$\operatorname{Hom}(X, \varprojlim_{i \in I} f(i)) = \varprojlim_{i \in I} \operatorname{Hom}(X, f(i))$}; i.e., it is an object representing the functor {$X \mapsto \varprojlim_i \operatorname{Hom}(X, f(i)).$}
* The '''colimit''' (or '''inductive limit''') {$\varinjlim_{i \in I} f(i)$} is the dual of a limit; i.e., given a functor {$f: I \to C$}, it satisfies: for any ''X'', {$\operatorname{Hom}(\varinjlim f(i), X) = \varprojlim \operatorname{Hom}(f(i), X)$}. Explicitly, to give {$\varinjlim f(i) \to X$} is to give a family of morphisms {$f(i) \to X$} such that for any {$i \to j$}, {$f(i) \to X$} is {$f(i) \to f(j) \to X$}. Perhaps the simplest example of a colimit is a '''coequalizer'''. For another example, take ''f'' to be the identity functor on ''C'' and suppose {$L = \varinjlim_{X \in C} f(X)$} exists; then the identity morphism on ''L'' corresponds to a compatible family of morphisms {$\alpha_X: X \to L$} such that {$\alpha_L$} is the identity. If {$f: X \to L$} is any morphism, then {$f = \alpha_L \circ f = \alpha_X$}; i.e., ''L'' is a final object of ''C''.
* The '''product''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the projective limit {$\varprojlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where ''I'' is viewed as a discrete category. It is denoted by {$\prod_i X_i$} and is the dual of the coproduct of the family.
* Given a monoidal category ''B'', the '''tensor product of functors''' {$F: C^{\text{op}} \to B$} and {$G: C \to B$} is the coend: {$F \otimes_C G = \int^{c \in C} F(c) \otimes G(c).$}
į:
* The '''end''' of a functor {$F: C^{\text{op}} \times C \to X$} is the limit {$\int_{c \in C} F(c, c) = \varprojlim (F^{\#}: C^{\#} \to X)$} where {$C^{\#}$} is the category (called the '''subdivision category''' of ''C'') whose objects are symbols {$c^{\#}, u^{\#}$} for all objects ''c'' and all morphisms ''u'' in ''C'' and whose morphisms are {$b^{\#} \to u^{\#}$} and {$u^{\#} \to c^{\#}$} if {$u: b \to c$} and where {$F^{\#}$} is induced by ''F'' so that {$c^{\#}$} would go to {$F(c, c)$} and {$u^{\#}, u: b \to c$} would go to {$F(b, c)$}. For example, for functors {$F, G: C \to X$}, {$\int_{c \in C} \operatorname{Hom}(F(c), G(c))$} is the set of natural transformations from ''F'' to ''G''. For more examples, see [http://mathoverflow.net/questions/78471/intuition-for-coends this mathoverflow thread]. The dual of an end is a coend.
Pridėta 252 eilutė:
2019 vasario 11 d., 00:02 atliko AndriusKulikauskas -
Pridėta 188 eilutė:
* If {$F: C \to D$} is a functor and ''y'' is the Yoneda embedding of ''C'', then the '''Yoneda extension''' of ''F'' is the left Kan extension of ''F'' along ''y''.
Pakeista 214 eilutė iš:
{{defn|no=2|1=If {$F: C \to D$} is a functor and ''y'' is the Yoneda embedding of ''C'', then the '''Yoneda extension''' of ''F'' is the left Kan extension of ''F'' along ''y''.
į:
2019 vasario 11 d., 00:00 atliko AndriusKulikauskas -
Pakeistos 38-48 eilutės iš
į:
* Given a monad ''T'' in a category ''X'', an '''algebra for a algebra for ''T''''' or a ''T''-algebra is an object in ''X'' with a '''monoid action''' of ''T'' ("algebra" is misleading and "''T''-object" is perhaps a better term.) For example, given a group ''G'' that determines a monad ''T'' in '''Set''' in the standard way, a ''T''-algebra is a set with an '''action''' of ''G''.

[+Variations of objects+]

* In a category, a '''sieve''' is a set ''S'' of objects having the property: if ''f'' is a morphism with the codomain in ''S'', then the domain of ''f'' is in ''S''.
* Given an object ''A'' in a category, a '''subobject''' of ''A'' is an equivalence class of monomorphisms to ''A''; two monomorphisms ''f'', ''g'' are considered equivalent if ''f'' factors through ''g'' and ''g'' factors through ''f''.
* An object in an abelian category is said to have <span id="finite length"></span>finite length if it has a '''composition series'''. The maximum number of proper subobjects in any such composition series is called the '''length''' of ''A''.
* A '''subquotient''' is a quotient of a subobject.
* A '''symmetric sequence''' is a sequence of objects with actions of '''symmetric group'''s. It is categorically equivalent to a '''(combinatorial) species'''.
* '''Hall algebra of a category'''. See '''Ringel–Hall algebra'''. (based on equivalence classes of objects)
Pakeistos 140-158 eilutės iš
į:
* The '''homological dimension''' of an abelian category with enough injectives is the least non-negative integer ''n'' such that every object in the category admits an injective resolution of length at most ''n''. The dimension is ∞ if no such integer exists. For example, the homological dimension of '''Mod<sub>''R''</sub>''' with a principal ideal domain ''R'' is at most one.

[+Variation of categories+]

* A topology on a category is '''subcanonical''' if every representable contravariant functor on ''C'' is a sheaf with respect to that topology. Generally speaking, some '''flat topology''' may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical.
* A '''t-structure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞-category''') that axiomatizes the notions of complexes whose cohomology concentrated in non-negative degrees or non-positive degrees.

[+Generalizations of a category+]

* A '''bicategory''' is a model of a weak '''2-category'''.
* '''colored operad''' Another term for '''multicategory''', a generalized category where a morphism can have several domains. The notion of "colored operad" is more primitive than that of operad: in fact, an operad can be defined as a colored operad with a single object.
* A '''multicategory''' is a generalization of a category in which a morphism is allowed to have more than one domain. It is the same thing as a '''colored operad'''.
* An ∞-category is '''stable''' if (1) it has a zero object, (2) every morphism in it admits a fiber and a cofiber and (3) a triangle in it is a fiber sequence if and only if it is a cofiber sequence.
* A strict 0-category is a set and for any integer ''n'' > 0, a '''strict ''n''-category''' is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category. '''Note''': the term "''n''-category" typically refers to "'''weak ''n''-category'''"; not strict one.
* One can define an ∞-category as a kind of a colim of ''n''-categories. Conversely, if one has the notion of a (weak) ∞-category (say a '''quasi-category''') in the beginning, then a weak ''n''-category can be defined as a type of a truncated ∞-category.
* The notion of a '''weak ''n''-category''' is obtained from the strict one by weakening the conditions like associativity of composition to hold only up to '''coherent isomorphism'''s in the weak sense.
* A '''Dwyer–Kan equivalence''' is a generalization of an equivalence of categories to the simplicial context.
Pridėta 187 eilutė:
* Given a group or monoid ''M'', the '''Day convolution''' is the tensor product in {$\mathbf{Fct}(M, \mathbf{Set})$}. Day convolution is equivalently a left Kan extension.
Pridėta 192 eilutė:
* An adjunction is said to be '''monadic''' if it comes from the monad that it determines by means of the '''Eilenberg–Moore category''' (the category of algebras for the monad).
Pridėtos 215-226 eilutės:
[+Generalizations of a functor+]

* The term "'''lax functor'''" is essentially synonymous with "'''pseudo-functor'''".

[+Natural transformations+]

* '''A natural transformation is, roughly, a map between functors. Precisely, given a pair of functors ''F'', ''G'' from a category ''C'' to category ''D'', a '''natural transformation''' φ from ''F'' to ''G'' is a set of morphisms in ''D'':{$\{ \phi_x: F(x) \to G(x) \mid x \in \operatorname{Ob}(C) \}$} satisfying the condition: for each morphism ''f'': ''x'' → ''y'' in ''C'', {$\phi_y \circ F(f) = G(f) \circ \phi_x$}. For example, writing {$GL_n(R)$} for the group of invertible ''n''-by-''n'' matrices with coefficients in a commutative ring ''R'', we can view {$GL_n$} as a functor from the category '''CRing''' of commutative rings to the category '''Grp''' of groups. Similarly, {$R \mapsto R^*$} is a functor from '''CRing''' to '''Grp'''. Then the '''determinant''' det is a natural transformation from {$GL_n$} to -<sup>*</sup>.
* Natural transformations are composed pointwise: if {$\varphi: f \to g, \, \psi: g \to h$} are natural transformations, then {$\psi \circ \varphi$} is the natural transformation given by {$(\psi \circ \varphi)_x = \psi_x \circ \varphi_x$}.
* A '''natural isomorphism''' is a natural transformation that is an isomorphism (i.e., admits the inverse).
* Given a functor ''F'': ''C'' → ''D'', the '''identity natural transformation''' from ''F'' to ''F'' is a natural transformation consisting of the identity morphisms of ''F''(''X'') in ''D'' for the objects ''X'' in ''C''.
Pakeistos 253-266 eilutės iš
[+Generalizations of a category+]

* A '''bicategory''' is a model of a weak '''2-category'''.
* '''colored operad''' Another term for
'''multicategory''', a generalized category where a morphism can have several domains. The notion of "colored operad" is more primitive than that of operad: in fact, an operad can be defined as a colored operad with a single object.
* A '''multicategory''' is a generalization of a category in which a morphism is allowed to have more than one domain. It is the same thing as a '''colored operad'''.
* An ∞-category is '''stable''' if (1) it has a zero object, (2) every morphism in it admits a fiber and a cofiber and (3) a triangle in it is a fiber sequence if and only if it is a cofiber sequence.
* A strict 0-category is a set and for any integer ''n'' > 0, a '''strict ''n''-category''' is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category. '''Note''': the term "''n''-category" typically refers to "'''weak ''n''-category'''"; not strict one.
* One can define an ∞-category as a kind of a colim of ''n''-categories. Conversely, if one has the notion of a (weak) ∞-category (say a '''quasi-category''') in the beginning, then a weak ''n''-category can be defined as a type of a truncated ∞-category.
* The notion of a '''weak ''n''-category''' is obtained from the strict one by weakening the conditions like associativity of composition to hold only up to '''coherent isomorphism'''s in the weak sense.

[+Generalizations of a functor+]

* The term "'''lax functor'''" is essentially synonymous with "'''pseudo-functor'''"
.
į:
[+Spaces+]

* The '''classifying space of a category''' ''C'' is the geometric realization of the nerve of ''C''.
*
'''Segal space'''s were certain simplicial spaces, introduced as models for '''(∞,&nbsp;1)-categories'''.
Pridėtos 277-281 eilutės:
[+Dualities+]

* '''co-''' Often used synonymous with op-; for example, a '''colimit''' refers to an op-limit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an op-fibration is not the same thing as a '''cofibration'''.
* The '''Tannakian duality''' states that, in an appropriate setup, to give a morphism {$f: X \to Y$} is to give a pullback functor {$f^*$} along it. In other words, the Hom set {$\operatorname{Hom}(X, Y)$} can be identified with the functor category {$\operatorname{Fct}(D(Y), D(X))$}, perhaps in the '''derived sense''', where {$D(X)$} is the category associated to ''X'' (e.g., the derived category).
Pakeistos 286-329 eilutės iš
* Higher category theory is a subfield of category theory that concerns the study of n-categories and ∞-categories.

[+Variations of objects+]

* In a category, a '''sieve''' is a set ''S'' of objects having the property: if ''f'' is a morphism with the codomain in ''S'', then the domain of ''f'' is in ''S''.
* Given an object ''A'' in a category, a '''subobject''' of ''A'' is an equivalence class of monomorphisms to ''A''; two monomorphisms ''f'', ''g'' are considered equivalent if ''f'' factors through ''g'' and ''g'' factors through ''f''.
* An object in an abelian category is said to have <span id="finite length"></span>finite length if it has a '''composition series'''. The maximum number of proper subobjects in any such composition series is called the '''length''' of ''A''.
* A '''subquotient''' is a quotient of a subobject.
* A '''symmetric sequence''' is a sequence of objects with actions of '''symmetric group'''s. It is categorically equivalent to a '''(combinatorial) species'''.
* '''Hall algebra of a category'''. See '''Ringel–Hall algebra'''. (based on equivalence classes of objects)

[+Variation of categories+]

* A topology on a category is '''subcanonical''' if every representable contravariant functor on ''C'' is a sheaf with respect to that topology. Generally speaking, some '''flat topology''' may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical.
* A '''t-structure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞-category''') that axiomatizes the notions of complexes whose cohomology concentrated in non-negative degrees or non-positive degrees.

[+Dualities+]

* '''co-''' Often used synonymous with op-; for example, a '''colimit''' refers to an op-limit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an op-fibration is not the same thing as a '''cofibration'''.
* The '''Tannakian duality''' states that, in an appropriate setup, to give a morphism {$f: X \to Y$} is to give a pullback functor {$f^*$} along it. In other words, the Hom set {$\operatorname{Hom}(X, Y)$} can be identified with the functor category {$\operatorname{Fct}(D(Y), D(X))$}, perhaps in the '''derived sense''', where {$D(X)$} is the category associated to ''X'' (e.g., the derived category).


[+Other+]

* Given a monad ''T'' in a category ''X'', an '''algebra for a algebra for ''T''''' or a ''T''-algebra is an object in ''X'' with a '''monoid action''' of ''T'' ("algebra" is misleading and "''T''-object" is perhaps a better term.) For example, given a group ''G'' that determines a monad ''T'' in '''Set''' in the standard way, a ''T''-algebra is a set with an '''action''' of ''G''.

* The '''classifying space of a category''' ''C'' is the geometric realization of the nerve of ''C''.

* Given a group or monoid ''M'', the '''Day convolution''' is the tensor product in {$\mathbf{Fct}(M, \mathbf{Set})$}.

* A '''Dwyer–Kan equivalence''' is a generalization of an equivalence of categories to the simplicial context.

* The '''homological dimension''' of an abelian category with enough injectives is the least non-negative integer ''n'' such that every object in the category admits an injective resolution of length at most ''n''. The dimension is ∞ if no such integer exists. For example, the homological dimension of '''Mod<sub>''R''</sub>''' with a principal ideal domain ''R'' is at most one.

* An adjunction is said to be '''monadic''' if it comes from the monad that it determines by means of the '''Eilenberg–Moore category''' (the category of algebras for the monad).

* '''A natural transformation is, roughly, a map between functors. Precisely, given a pair of functors ''F'', ''G'' from a category ''C'' to category ''D'', a '''natural transformation''' φ from ''F'' to ''G'' is a set of morphisms in ''D'':{$\{ \phi_x: F(x) \to G(x) \mid x \in \operatorname{Ob}(C) \}$} satisfying the condition: for each morphism ''f'': ''x'' → ''y'' in ''C'', {$\phi_y \circ F(f) = G(f) \circ \phi_x$}. For example, writing {$GL_n(R)$} for the group of invertible ''n''-by-''n'' matrices with coefficients in a commutative ring ''R'', we can view {$GL_n$} as a functor from the category '''CRing''' of commutative rings to the category '''Grp''' of groups. Similarly, {$R \mapsto R^*$} is a functor from '''CRing''' to '''Grp'''. Then the '''determinant''' det is a natural transformation from {$GL_n$} to -<sup>*</sup>.
* Natural transformations are composed pointwise: if {$\varphi: f \to g, \, \psi: g \to h$} are natural transformations, then {$\psi \circ \varphi$} is the natural transformation given by {$(\psi \circ \varphi)_x = \psi_x \circ \varphi_x$}.
* A '''natural isomorphism''' is a natural transformation that is an isomorphism (i.e., admits the inverse).
* Given a functor ''F'': ''C'' → ''D'', the '''identity natural transformation''' from ''F'' to ''F'' is a natural transformation consisting of the identity morphisms of ''F''(''X'') in ''D'' for the objects ''X'' in ''C''.

* '''Segal space'''s were certain simplicial spaces, introduced as models for '''(∞,&nbsp;1)-categories'''.
į:
* Higher category theory is a subfield of category theory that concerns the study of n-categories and ∞-categories.
2019 vasario 10 d., 22:03 atliko AndriusKulikauskas -
Pridėtos 162-163 eilutės:
* A '''finitary monad''' or an algebraic monad is a monad on '''Set''' whose underlying endofunctor commutes with filtered colimits.
* A '''comonad''' in a category ''X'' is a '''comonid''' in the monoidal category of endofunctors of ''X''.
Pakeistos 257-258 eilutės iš
į:
* '''Hall algebra of a category'''. See '''Ringel–Hall algebra'''. (based on equivalence classes of objects)
Pakeistos 261-262 eilutės iš
į:
* A topology on a category is '''subcanonical''' if every representable contravariant functor on ''C'' is a sheaf with respect to that topology. Generally speaking, some '''flat topology''' may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical.
* A '''t-structure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞-category''') that axiomatizes the notions of complexes whose cohomology concentrated in non-negative degrees or non-positive degrees.
Ištrintos 275-277 eilutės:

* A '''comonad''' in a category ''X'' is a '''comonid''' in the monoidal category of endofunctors of ''X''.
Pakeistos 279-282 eilutės iš
* A '''finitary monad''' or an algebraic monad is a monad on '''Set''' whose underlying endofunctor commutes with filtered colimits.

* '''Hall algebra of a category'''. See '''Ringel–Hall algebra'''.
į:
Ištrintos 291-293 eilutės:
* A topology on a category is '''subcanonical''' if every representable contravariant functor on ''C'' is a sheaf with respect to that topology. Generally speaking, some '''flat topology''' may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical.

* A '''t-structure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞-category''') that axiomatizes the notions of complexes whose cohomology concentrated in non-negative degrees or non-positive degrees.
2019 vasario 10 d., 21:51 atliko AndriusKulikauskas -
Pakeistos 129-130 eilutės iš
į:
* The '''core''' of a category is the maximal groupoid contained in the category.
Ištrinta 243 eilutė:
Pakeistos 246-265 eilutės iš
į:
* Higher category theory is a subfield of category theory that concerns the study of n-categories and ∞-categories.

[+Variations of objects+]

* In a category, a '''sieve''' is a set ''S'' of objects having the property: if ''f'' is a morphism with the codomain in ''S'', then the domain of ''f'' is in ''S''.
* Given an object ''A'' in a category, a '''subobject''' of ''A'' is an equivalence class of monomorphisms to ''A''; two monomorphisms ''f'', ''g'' are considered equivalent if ''f'' factors through ''g'' and ''g'' factors through ''f''.
* An object in an abelian category is said to have <span id="finite length"></span>finite length if it has a '''composition series'''. The maximum number of proper subobjects in any such composition series is called the '''length''' of ''A''.
* A '''subquotient''' is a quotient of a subobject.
* A '''symmetric sequence''' is a sequence of objects with actions of '''symmetric group'''s. It is categorically equivalent to a '''(combinatorial) species'''.

[+Variation of categories+]



[+Dualities+]

* '''co-''' Often used synonymous with op-; for example, a '''colimit''' refers to an op-limit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an op-fibration is not the same thing as a '''cofibration'''.
* The '''Tannakian duality''' states that, in an appropriate setup, to give a morphism {$f: X \to Y$} is to give a pullback functor {$f^*$} along it. In other words, the Hom set {$\operatorname{Hom}(X, Y)$} can be identified with the functor category {$\operatorname{Fct}(D(Y), D(X))$}, perhaps in the '''derived sense''', where {$D(X)$} is the category associated to ''X'' (e.g., the derived category).
Pakeistos 271-272 eilutės iš
* '''co-''' Often used synonymous with op-; for example, a '''colimit''' refers to an op-limit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an op-fibration is not the same thing as a '''cofibration'''.}}
į:
Ištrinta 274 eilutė:
* The '''core''' of a category is the maximal groupoid contained in the category.
Ištrinta 279 eilutė:
Ištrintos 283-286 eilutės:
* Given a functor ''F'': ''C'' → ''D'', the '''identity natural transformation''' from ''F'' to ''F'' is a natural transformation consisting of the identity morphisms of ''F''(''X'') in ''D'' for the objects ''X'' in ''C''.

* An object in an abelian category is said to have <span id="finite length"></span>finite length if it has a '''composition series'''. The maximum number of proper subobjects in any such composition series is called the '''length''' of ''A''.
Pridėtos 289-290 eilutės:
* Given a functor ''F'': ''C'' → ''D'', the '''identity natural transformation''' from ''F'' to ''F'' is a natural transformation consisting of the identity morphisms of ''F''(''X'') in ''D'' for the objects ''X'' in ''C''.
Pakeistos 292-294 eilutės iš
* In a category, a '''sieve''' is a set ''S'' of objects having the property: if ''f'' is a morphism with the codomain in ''S'', then the domain of ''f'' is in ''S''.

* Given an object ''A'' in a category, a '''subobject''' of ''A'' is an equivalence class of monomorphisms to ''A''; two monomorphisms ''f'', ''g'' are considered equivalent if ''f'' factors through ''g'' and ''g'' factors through ''f''.
į:
Pakeistos 295-301 eilutės iš
* A '''subquotient''' is a quotient of a subobject.
* A
'''symmetric sequence''' is a sequence of objects with actions of '''symmetric group'''s. It is categorically equivalent to a '''(combinatorial) species'''.
* A '''t-structure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞-category''') that axiomatizes the notions of complexes whose cohomology concentrated in non-negative degrees or non-positive degrees.
* The '''Tannakian duality''' states that, in an appropriate setup, to give a morphism {$f: X \to Y$} is to give a pullback functor {$f^*$} along it. In other words, the Hom set {$\operatorname{Hom}(X, Y)$} can be identified with the functor category {$\operatorname{Fct}(D(Y), D(X))$}, perhaps in the '''derived sense''', where {$D(X)$} is the category associated to ''X'' (e.g., the derived category).

į:
* A '''t-structure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞-category''') that axiomatizes the notions of complexes whose cohomology concentrated in non-negative degrees or non-positive degrees.
2019 vasario 10 d., 21:42 atliko AndriusKulikauskas -
Pakeistos 51-53 eilutės iš
į:
* Given a functor {$f: C \to D$} and an object ''X'' in ''D'', a '''universal morphism''' from ''X'' to ''f'' is an initial object in the '''comma category''' {$(X \downarrow f)$}. (Its dual is also called a universal morphism.) For example, take ''f'' to be the forgetful functor {$\mathbf{Vec}_k \to \mathbf{Set}$} and ''X'' a set. An initial object of {$(X \downarrow f)$} is a function {$j: X \to f(V_X)$}. That it is initial means that if {$k: X \to f(W)$} is another morphism, then there is a unique morphism from ''j'' to ''k'', which consists of a linear map {$V_X \to W$} that extends ''k'' via ''j''; that is to say, {$V_X$} is the '''free vector space''' generated by ''X''.}}
* Stated more explicitly, given ''f'' as above, a morphism {$X \to f(u_X)$} in ''D'' is '''universal''' if and only if the natural map {$\operatorname{Hom}_C(u_X, c) \to \operatorname{Hom}_D(X, f(c)), \, \alpha \mapsto (X \to f(u_x) \overset{f(\alpha)}\to f(c))$} is bijective. In particular, if {$\operatorname{Hom}_C(u_X, -) \simeq \operatorname{Hom}_D(X, f(-))$}, then taking ''c'' to be ''u''<sub>''X''</sub> one gets a universal morphism by sending the identity morphism. In other words, having a universal morphism is equivalent to the representability of the functor {$\operatorname{Hom}_D(X, f(-))$}.
Pakeistos 237-241 eilutės iš
[+Other+]

* Given a monad ''T'' in a category ''X'', an '''algebra for a algebra for ''T''''' or a ''T''-algebra is an object in ''X'' with a '''monoid action''' of ''T'' ("algebra" is misleading and "''T''-object" is perhaps a better term.) For example, given a group ''G'' that determines a monad ''T'' in '''Set''' in the standard way, a ''T''-algebra is a set with an '''action''' of ''G''.
į:
* '''localization of a category'''
Pridėtos 239-243 eilutės:
* '''Simplicial localization''' is a method of localizing a category.

[+Areas of math+]
Pakeistos 245-250 eilutės iš
į:
* '''Grothendieck's Galois theory''' A category-theoretic generalization of '''Galois theory'''.

[+Other+]

* Given a monad ''T'' in a category ''X'', an '''algebra for a algebra for ''T''''' or a ''T''-algebra is an object in ''X'' with a '''monoid action''' of ''T'' ("algebra" is misleading and "''T''-object" is perhaps a better term.) For example, given a group ''G'' that determines a monad ''T'' in '''Set''' in the standard way, a ''T''-algebra is a set with an '''action''' of ''G''.
Ištrinta 255 eilutė:
Pakeistos 261-262 eilutės iš
* '''Grothendieck's Galois theory''' A category-theoretic generalization of '''Galois theory'''; see '''Grothendieck's Galois theory'''.
į:
Pakeista 264 eilutė iš:
* '''Higher category theory''' is a subfield of category theory that concerns the study of '''''n''-categories''' and '''∞-categories'''.
į:
Pakeista 270 eilutė iš:
* '''localization of a category'''
į:
Pakeistos 278-279 eilutės iš
* '''Simplicial localization''' is a method of localizing a category.
į:
Pakeistos 286-287 eilutės iš
* Given a functor {$f: C \to D$} and an object ''X'' in ''D'', a '''universal morphism''' from ''X'' to ''f'' is an initial object in the '''comma category''' {$(X \downarrow f)$}. (Its dual is also called a universal morphism.) For example, take ''f'' to be the forgetful functor {$\mathbf{Vec}_k \to \mathbf{Set}$} and ''X'' a set. An initial object of {$(X \downarrow f)$} is a function {$j: X \to f(V_X)$}. That it is initial means that if {$k: X \to f(W)$} is another morphism, then there is a unique morphism from ''j'' to ''k'', which consists of a linear map {$V_X \to W$} that extends ''k'' via ''j''; that is to say, {$V_X$} is the '''free vector space''' generated by ''X''.}}
* Stated more explicitly, given ''f'' as above, a morphism {$X \to f(u_X)$} in ''D'' is '''universal''' if and only if the natural map {$\operatorname{Hom}_C(u_X, c) \to \operatorname{Hom}_D(X, f(c)), \, \alpha \mapsto (X \to f(u_x) \overset{f(\alpha)}\to f(c))$} is bijective. In particular, if {$\operatorname{Hom}_C(u_X, -) \simeq \operatorname{Hom}_D(X, f(-))$}, then taking ''c'' to be ''u''<sub>''X''</sub> one gets a universal morphism by sending the identity morphism. In other words, having a universal morphism is equivalent to the representability of the functor {$\operatorname{Hom}_D(X, f(-))$}.
į:
2019 vasario 10 d., 21:36 atliko AndriusKulikauskas -
Pridėtos 22-23 eilutės:
* '''compact''' Probably synonymous with '''#accessible'''.
* '''perfect''' Sometimes synonymous with "compact". See '''perfect complex'''.
Pakeistos 177-179 eilutės iš
į:
* The functor {$y: C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(-, X)$} is fully faithful and is called the Yoneda embedding.<ref>Technical note: the lemma implicitly involves a choice of '''Set'''; i.e., a choice of universe.</ref> }}
{{defn|no=2|1=If {$F: C \to D$} is a functor and ''y'' is the Yoneda embedding of ''C'', then the '''Yoneda extension''' of ''F'' is the left Kan extension of ''F'' along ''y''.
Pakeistos 213-219 eilutės iš
į:
* One can define an ∞-category as a kind of a colim of ''n''-categories. Conversely, if one has the notion of a (weak) ∞-category (say a '''quasi-category''') in the beginning, then a weak ''n''-category can be defined as a type of a truncated ∞-category.
* The notion of a '''weak ''n''-category''' is obtained from the strict one by weakening the conditions like associativity of composition to hold only up to '''coherent isomorphism'''s in the weak sense.

[+Generalizations of a functor+]

* The term "'''lax functor'''" is essentially synonymous with "'''pseudo-functor'''".
Pakeistos 227-229 eilutės iš
į:
* Yoneda’s Lemma asserts ... in more evocative terms, a mathematical object X is best thought of in the context of a category surrounding it, and is determined by the network of relations it enjoys with all the objects of that category. Moreover, to understand X it might be more germane to deal directly with the functor representing it. This is reminiscent of Wittgenstein’s ’language game’; i.e., that the meaning of a word is—in essence—determined by, in fact is nothing more than, its relations to all the utterances in a language.
* The '''Yoneda lemma''' says: for each set-valued contravariant functor ''F'' on ''C'' and an object ''X'' in ''C'', there is a natural bijection {$F(X) \simeq \operatorname{Nat}(\operatorname{Hom}_C(-, X), F)$} where Nat means the set of natural transformations.
Pakeistos 247-248 eilutės iš
* '''compact''' Probably synonymous with '''#accessible'''.
į:
Pakeista 261 eilutė iš:
* The term "'''lax functor'''" is essentially synonymous with "'''pseudo-functor'''".
į:
Ištrintos 265-266 eilutės:
* The notion of a '''weak ''n''-category''' is obtained from the strict one by weakening the conditions like associativity of composition to hold only up to '''coherent isomorphism'''s in the weak sense.
* One can define an ∞-category as a kind of a colim of ''n''-categories. Conversely, if one has the notion of a (weak) ∞-category (say a '''quasi-category''') in the beginning, then a weak ''n''-category can be defined as a type of a truncated ∞-category.
Ištrinta 268 eilutė:
* '''perfect''' Sometimes synonymous with "compact". See '''perfect complex'''.
Pakeista 270 eilutė iš:
* In a category, a '''sieve''' is a set ''S'' of objects having the property: if ''f'' is a morphism with the codomain in ''S'', then the domain of ''f'' is in ''S''.}} conflict with the literature?
į:
* In a category, a '''sieve''' is a set ''S'' of objects having the property: if ''f'' is a morphism with the codomain in ''S'', then the domain of ''f'' is in ''S''.
Pridėta 279 eilutė:
Pakeistos 281-285 eilutės iš
{{defn|no=2|Stated more explicitly, given ''f'' as above, a morphism {$X \to f(u_X)$} in ''D'' is universal if and only if the natural map {$\operatorname{Hom}_C(u_X, c) \to \operatorname{Hom}_D(X, f(c)), \, \alpha \mapsto (X \to f(u_x) \overset{f(\alpha)}\to f(c))$} is bijective. In particular, if {$\operatorname{Hom}_C(u_X, -) \simeq \operatorname{Hom}_D(X, f(-))$}, then taking ''c'' to be ''u''<sub>''X''</sub> one gets a universal morphism by sending the identity morphism. In other words, having a universal morphism is equivalent to the representability of the functor {$\operatorname{Hom}_D(X, f(-))$}.

* Yoneda’s Lemma asserts ... in more evocative terms, a mathematical object X is best thought of in the context of a category surrounding it, and is determined by the network of relations it enjoys with all the objects of that category. Moreover, to understand X it might be more germane to deal directly with the functor representing it. This is reminiscent of Wittgenstein’s ’language game’; i.e., that the meaning of a word is—in essence—determined by, in fact is nothing more than, its relations to all the utterances in a language.
* The '''Yoneda lemma''' says: for each set-valued contravariant functor ''F'' on ''C'' and an object ''X'' in ''C'', there is a natural bijection {$F(X) \simeq \operatorname{Nat}(\operatorname{Hom}_C(-, X), F)$} where Nat means the set of natural transformations. In particular, the functor {$y: C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(-, X)$} is fully faithful and is called the Yoneda embedding.<ref>Technical note: the lemma implicitly involves a choice of '''Set'''; i.e., a choice of universe.</ref> }}
{{defn|no=2|1=If {$F: C \to D$} is a functor and ''y'' is the Yoneda embedding of ''C'', then the '''Yoneda extension''' of ''F'' is the left Kan extension of ''F'' along ''y''.
į:
* Stated more explicitly, given ''f'' as above, a morphism {$X \to f(u_X)$} in ''D'' is '''universal''' if and only if the natural map {$\operatorname{Hom}_C(u_X, c) \to \operatorname{Hom}_D(X, f(c)), \, \alpha \mapsto (X \to f(u_x) \overset{f(\alpha)}\to f(c))$} is bijective. In particular, if {$\operatorname{Hom}_C(u_X, -) \simeq \operatorname{Hom}_D(X, f(-))$}, then taking ''c'' to be ''u''<sub>''X''</sub> one gets a universal morphism by sending the identity morphism. In other words, having a universal morphism is equivalent to the representability of the functor {$\operatorname{Hom}_D(X, f(-))$}.
2019 vasario 10 d., 21:00 atliko AndriusKulikauskas -
Pakeistos 61-62 eilutės iš
į:
* '''composition''' A composition of morphisms in a category is part of the datum defining the category.
Pakeistos 174-175 eilutės iš
į:
* If {$f: C \to D, \, g: D \to E$} are functors, then the '''composition''' {$g \circ f$} or {$gf$} is the functor defined by: for an object ''x'' and a morphism ''u'' in ''C'', {$(g \circ f)(x) = g(f(x)), \, (g \circ f)(u) = g(f(u))$}.
Pakeistos 202-205 eilutės iš
[+Other+]

* Given a monad ''T'' in a category ''X'', an '''algebra for a algebra for ''T''''' or a ''T''-algebra is an object in ''X'' with a '''monoid action''' of ''T'' ("algebra" is misleading and "''T''-object" is perhaps a better term.) For example, given a group ''G'' that determines a monad ''T'' in '''Set''' in the standard way, a ''T''-algebra is a set with an '''action''' of ''G''.
* '''Beck's theorem''' characterizes the category of '''algebras for a given monad'''.
į:
[+Generalizations of a category+]
Pakeistos 205-219 eilutės iš
* '''Bousfield localization'''
į:
* '''colored operad''' Another term for '''multicategory''', a generalized category where a morphism can have several domains. The notion of "colored operad" is more primitive than that of operad: in fact, an operad can be defined as a colored operad with a single object.
* A '''multicategory''' is a generalization of a category in which a morphism is allowed to have more than one domain. It is the same thing as a '''colored operad'''.
* An ∞-category is '''stable''' if (1) it has a zero object, (2) every morphism in it admits a fiber and a cofiber and (3) a triangle in it is a fiber sequence if and only if it is a cofiber sequence.
* A strict 0-category is a set and for any integer ''n'' > 0, a '''strict ''n''-category''' is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category. '''Note''': the term "''n''-category" typically refers to "'''weak ''n''-category'''"; not strict one.

[+Theorems+]

* '''Beck's theorem''' characterizes the category of '''algebras for a given monad'''.
* The '''density theorem''' states that every presheaf (a set-valued contravariant functor) is a colimit of representable presheaves. Yoneda's lemma embeds a category ''C'' into the category of presheaves on ''C''. The density theorem then says the image is "dense", so to say. The name "density" is because of the analogy with the '''Jacobson density theorem''' (or other variants) in abstract algebra.
* The '''homotopy hypothesis''' states an '''∞-groupoid''' is a space (less equivocally, an ''n''-groupoid can be used as a homotopy ''n''-type.)}}
* '''Quillen’s theorem A''' provides a criterion for a functor to be a weak equivalence.
* The '''Gabriel–Popescu theorem''' says an abelian category is a '''quotient''' of the category of modules.

[+Techniques+]
Ištrinta 220 eilutė:
* '''Categorical logic''' is an approach to '''mathematical logic''' that uses category theory.
Pridėtos 222-230 eilutės:
* '''Grothendieck construction''' Given a functor {$U: C \to \mathbf{Cat}$}, let ''D''<sub>''U''</sub> be the category where the objects are pairs (''x'', ''u'') consisting of an object ''x'' in ''C'' and an object ''u'' in the category ''U''(''x'') and a morphism from (''x'', ''u'') to (''y'', ''v'') is a pair consisting of a morphism ''f'': ''x'' → ''y'' in ''C'' and a morphism ''U''(''f'')(''u'') → ''v'' in ''U''(''y''). The passage from ''U'' to ''D''<sub>''U''</sub> is then called the '''Grothendieck construction'''.

[+Other+]

* Given a monad ''T'' in a category ''X'', an '''algebra for a algebra for ''T''''' or a ''T''-algebra is an object in ''X'' with a '''monoid action''' of ''T'' ("algebra" is misleading and "''T''-object" is perhaps a better term.) For example, given a group ''G'' that determines a monad ''T'' in '''Set''' in the standard way, a ''T''-algebra is a set with an '''action''' of ''G''.

* '''Bousfield localization'''
* '''Categorical logic''' is an approach to '''mathematical logic''' that uses category theory.
Pakeista 233 eilutė iš:
* '''colored operad''' Another term for '''multicategory''', a generalized category where a morphism can have several domains. The notion of "colored operad" is more primitive than that of operad: in fact, an operad can be defined as a colored operad with a single object.}}
į:
Ištrintos 236-238 eilutės:
* '''composition''' A composition of morphisms in a category is part of the datum defining the category.
# If {$f: C \to D, \, g: D \to E$} are functors, then the composition {$g \circ f$} or {$gf$} is the functor defined by: for an object ''x'' and a morphism ''u'' in ''C'', {$(g \circ f)(x) = g(f(x)), \, (g \circ f)(u) = g(f(u))$}.
# Natural transformations are composed pointwise: if {$\varphi: f \to g, \, \psi: g \to h$} are natural transformations, then {$\psi \circ \varphi$} is the natural transformation given by {$(\psi \circ \varphi)_x = \psi_x \circ \varphi_x$}.
Pakeista 239 eilutė iš:
* The '''density theorem''' states that every presheaf (a set-valued contravariant functor) is a colimit of representable presheaves. Yoneda's lemma embeds a category ''C'' into the category of presheaves on ''C''. The density theorem then says the image is "dense", so to say. The name "density" is because of the analogy with the '''Jacobson density theorem''' (or other variants) in abstract algebra.
į:
Ištrinta 241 eilutė:
* The '''Gabriel–Popescu theorem''' says an abelian category is a '''quotient''' of the category of modules.
Pakeista 243 eilutė iš:
* '''Grothendieck construction''' Given a functor {$U: C \to \mathbf{Cat}$}, let ''D''<sub>''U''</sub> be the category where the objects are pairs (''x'', ''u'') consisting of an object ''x'' in ''C'' and an object ''u'' in the category ''U''(''x'') and a morphism from (''x'', ''u'') to (''y'', ''v'') is a pair consisting of a morphism ''f'': ''x'' → ''y'' in ''C'' and a morphism ''U''(''f'')(''u'') → ''v'' in ''U''(''y''). The passage from ''U'' to ''D''<sub>''U''</sub> is then called the '''Grothendieck construction'''.
į:
Pakeista 247 eilutė iš:
* The '''homotopy hypothesis''' states an '''∞-groupoid''' is a space (less equivocally, an ''n''-groupoid can be used as a homotopy ''n''-type.)}}
į:
Pakeista 253 eilutė iš:
* A '''multicategory''' is a generalization of a category in which a morphism is allowed to have more than one domain. It is the same thing as a '''colored operad'''.
į:
Pridėta 257 eilutė:
* Natural transformations are composed pointwise: if {$\varphi: f \to g, \, \psi: g \to h$} are natural transformations, then {$\psi \circ \varphi$} is the natural transformation given by {$(\psi \circ \varphi)_x = \psi_x \circ \varphi_x$}.
Ištrinta 259 eilutė:
* '''Quillen’s theorem A''' provides a criterion for a functor to be a weak equivalence.
Pakeistos 263-264 eilutės iš
* An ∞-category is '''stable''' if (1) it has a zero object, (2) every morphism in it admits a fiber and a cofiber and (3) a triangle in it is a fiber sequence if and only if it is a cofiber sequence.
* A strict 0-category is a set and for any integer ''n'' > 0, a '''strict ''n''-category''' is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category. '''Note''': the term "''n''-category" typically refers to "'''weak ''n''-category'''"; not strict one.
į:
2019 vasario 10 d., 20:50 atliko AndriusKulikauskas -
Pakeistos 259-294 eilutės iš
{{defn|no=2|1=If {$F: C \to D$} is a functor and ''y'' is the Yoneda embedding of ''C'', then the '''Yoneda extension''' of ''F'' is the left Kan extension of ''F'' along ''y''.



==References==
*{{cite book
| first = Michael
| last = Artin
| authorlink = Michael Artin
|editor='''Alexandre Grothendieck''' |editor2='''Jean-Louis Verdier'''
title = Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1
}}
*{{Cite book
| last=Kashiwara
| first=Masaki
| last2=Schapira
| first2=Pierre
| title=Categories and sheaves
| year=2006
| ref=harv
|author-link = Masaki Kashiwara|author-link2 = Pierre Schapira (mathematician)}}
*'''André Joyal|A. Joyal''', [http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf The theory of quasi-categories II] (Volume I is missing??)
*'''Jacob Lurie|Lurie, J.''', ''[https://web.archive.org/web/20150206092410/http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf Higher Algebra]''
*Lurie, J., '''''Higher Topos Theory'''''
* {{cite book | last=Mac Lane | first=Saunders | authorlink=Saunders Mac Lane | title='''Categories for the Working Mathematician''' | edition=2nd | series='''Graduate Texts in Mathematics''' | volume=5 | location=New York, NY | publisher='''Springer-Verlag''' | year=1998 | isbn=0-387-98403-8 | zbl=0906.18001 |ref=harv}}
* {{cite book | editor1-last=Pedicchio | editor1-first=Maria Cristina | editor2-last=Tholen | editor2-first=Walter | title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory | series=Encyclopedia of Mathematics and Its Applications | volume=97 | location=Cambridge | publisher='''Cambridge University Press''' | year=2004 | isbn=0-521-83414-7 | zbl=1034.18001 }}
* {{Cite arxiv |title = Notes on Grothendieck topologies, fibered categories and descent theory |eprint = math/0412512|date = 2004-12-28|first = Angelo|last = Vistoli |ref=harv}}

[+Further reading+]
* Groth, M., [http://www.math.uni-bonn.de/~mgroth/InfinityCategories.pdf A Short Course on ∞-categories]
* [http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf Cisinski's notes]
*'''History of topos theory'''
*http://plato.stanford.edu/entries/category-theory/
*{{Cite book |last=Leinster|first=Tom|date=2014|title=Basic Category Theory|series=Cambridge Studies in Advanced Mathematics |publisher=Cambridge University Press|volume=143 |arxiv=1612.09375|bibcode=2016arXiv161209375L}}
*Emily Riehl, [http://www.math.jhu.edu/~eriehl/ssets.pdf A leisurely introduction to simplicial sets]
* [http://www.andrew.cmu.edu/user/awodey/catlog/ Categorical Logic] lecture notes by '''Steve Awodey'''
į:
{{defn|no=2|1=If {$F: C \to D$} is a functor and ''y'' is the Yoneda embedding of ''C'', then the '''Yoneda extension''' of ''F'' is the left Kan extension of ''F'' along ''y''.
2019 vasario 10 d., 20:29 atliko AndriusKulikauskas -
Pakeistos 35-36 eilutės iš
į:
* A '''zero object''' is an object that is both initial and terminal, such as a '''trivial group''' in ''''''Category of groups|Grp''''''.
Pakeistos 257-277 eilutės iš
{{term|1=Yoneda}}
{{defn|no=1|1={{quote box
|quote=
Yoneda’s Lemma asserts ... in more evocative terms, a mathematical object X is best thought of in the context of a category surrounding it, and is determined by the network of relations it enjoys with all the objects of that category. Moreover, to understand X it might be more germane to deal directly with the functor representing it. This is reminiscent of Wittgenstein’s ’language game’; i.e., that the meaning of a word is—in essence—determined by, in fact is nothing more than, its relations to all the utterances in a language.
|author='''Barry Mazur'''
|source=[http://www.math.harvard.edu/~mazur/papers/Thinking.about.Grothendieck%285%29.pdf Thinking about Grothendieck]
|width=33%}}The
'''Yoneda lemma''' says: for each set-valued contravariant functor ''F'' on ''C'' and an object ''X'' in ''C'', there is a natural bijection
:
{$F(X) \simeq \operatorname{Nat}(\operatorname{Hom}_C(-, X), F)$}
where Nat means the set of natural transformations. In particular, the functor
:{$y: C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(-, X)$}
is fully faithful and is called the Yoneda embedding.<ref>Technical note: the lemma implicitly involves a choice of '''Set'''; i.e., a choice of universe.</ref> }}
{{defn|no=2|1=If {$F: C \to D$} is a functor and ''y'' is the Yoneda embedding of ''C'', then the '''Yoneda extension''' of ''F'' is the left Kan extension of ''F'' along ''y''.}}
{{glossary end}}

[+Z+]
{{glossary}}
{{term|1=zero}}
{{defn|1=A '''zero object''' is an object that is both initial and terminal, such as a '''trivial group''' in ''''''Category of groups|Grp''''''.}}
{{glossary end}}

==Notes==
{{reflist}}
į:
* Yoneda’s Lemma asserts ... in more evocative terms, a mathematical object X is best thought of in the context of a category surrounding it, and is determined by the network of relations it enjoys with all the objects of that category. Moreover, to understand X it might be more germane to deal directly with the functor representing it. This is reminiscent of Wittgenstein’s ’language game’; i.e., that the meaning of a word is—in essence—determined by, in fact is nothing more than, its relations to all the utterances in a language.
* The '''Yoneda lemma''' says: for each set-valued contravariant functor ''F'' on ''C'' and an object ''X'' in ''C'', there is a natural bijection {$F(X) \simeq \operatorname{Nat}(\operatorname{Hom}_C(-, X), F)$} where Nat means the set of natural transformations. In particular, the functor {$y: C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(-, X)$} is fully faithful and is called the Yoneda embedding.<ref>Technical note: the lemma implicitly involves a choice of '''Set'''; i.e., a choice of universe.</ref> }}
{{defn|no=2|1=If {$F: C \to D$} is a functor and ''y'' is the Yoneda embedding of ''C'', then the '''Yoneda extension''' of ''F'' is the left Kan extension of ''F'' along ''y''.
2019 vasario 10 d., 20:24 atliko AndriusKulikauskas -
Pakeistos 122-124 eilutės iš
į:
* A '''Waldhausen category''' is, roughly, a category with families of cofibrations and weak equivalences.
* A category is wellpowered if for each object there is only a set of pairwise non-isomorphic '''subobject'''s.
Ištrintos 252-253 eilutės:
Pakeistos 254-269 eilutės iš
{{defn|no=2|Stated more explicitly, given ''f'' as above, a morphism {$X \to f(u_X)$} in ''D'' is universal if and only if the natural map
:{$\operatorname{Hom}_C(u_X, c) \to \operatorname{Hom}_D(X, f(c)), \, \alpha \mapsto (X \to f(u_x) \overset{f(\alpha)}\to f(c))$}
is bijective. In particular, if {$\operatorname{Hom}_C(u_X, -) \simeq \operatorname{Hom}_D(X, f(-))$}, then taking ''c'' to be ''u''<sub>''X''</sub> one gets a universal morphism by sending the identity morphism. In other words, having a universal morphism is equivalent to the representability of the functor {$\operatorname{Hom}_D(X, f(-))$}.}}
{{glossary end}}

==W==
{{glossary}}
{{term|1=Waldhausen category}}
{{defn|1=A '''Waldhausen category''' is, roughly, a category with families of cofibrations and weak equivalences.}}

{{term|1=wellpowered}}
{{defn|1=A category is wellpowered if for each object there is only a set of pairwise non-isomorphic '''subobject'''s.}}
{{glossary end}}

==Y==
{{glossary}}
į:
{{defn|no=2|Stated more explicitly, given ''f'' as above, a morphism {$X \to f(u_X)$} in ''D'' is universal if and only if the natural map {$\operatorname{Hom}_C(u_X, c) \to \operatorname{Hom}_D(X, f(c)), \, \alpha \mapsto (X \to f(u_x) \overset{f(\alpha)}\to f(c))$} is bijective. In particular, if {$\operatorname{Hom}_C(u_X, -) \simeq \operatorname{Hom}_D(X, f(-))$}, then taking ''c'' to be ''u''<sub>''X''</sub> one gets a universal morphism by sending the identity morphism. In other words, having a universal morphism is equivalent to the representability of the functor {$\operatorname{Hom}_D(X, f(-))$}.
2019 vasario 10 d., 20:22 atliko AndriusKulikauskas -
Pakeistos 32-35 eilutės iš
į:
* A '''subterminal object''' is an object ''X'' such that every object has at most one morphism into ''X''.
* An object ''A'' is '''terminal''' (also called final) if there is exactly one morphism from each object to ''A''; e.g., '''singleton'''s in '''Set'''. It is the dual of an '''initial object'''.
* An object ''A'' in an ∞-category ''C'' is terminal if {$\operatorname{Map}_C(B, A)$} is '''contractible space|contractible''' for every object ''B'' in ''C''.
Pakeistos 115-122 eilutės iš
į:
* A category ''A'' is a '''subcategory''' of a category ''B'' if there is an inclusion functor from ''A'' to ''B''.
* A '''symmetric monoidal category''' is a '''monoidal category''' (i.e., a category with ⊗) that has maximally symmetric braiding.
* '''tensor category''' Usually synonymous with '''monoidal category''' (though some authors distinguish between the two concepts.)
* A '''tensor triangulated category''' is a category that carries the structure of a symmetric monoidal category and that of a triangulated category in a compatible way.
* A full subcategory of an abelian category is '''thick''' if it is closed under extensions.
* A '''thin''' is a category where there is at most one morphism between any pair of objects.
* A '''triangulated category''' is a category where one can talk about distinguished triangles, generalization of exact sequences. An abelian category is a prototypical example of a triangulated category. A '''derived category''' is a triangulated category that is not necessary an abelian category.
Pakeistos 170-171 eilutės iš
į:
* A '''(combinatorial) species''' is an endofunctor on the groupoid of finite sets with bijections. It is categorically equivalent to a '''symmetric sequence'''.
Pakeistos 195-197 eilutės iš
į:
* Given a monoidal category ''B'', the '''tensor product of functors''' {$F: C^{\text{op}} \to B$} and {$G: C \to B$} is the coend: {$F \otimes_C G = \int^{c \in C} F(c) \otimes G(c).$}
* The coend of a functor {$F: C^{\text{op}} \times C \to X$} is the dual of the '''end (category theory)|end''' of ''F'' and is denoted by {$\int^{c \in C} F(c, c)$}. For example, if ''R'' is a ring, ''M'' a right ''R''-module and ''N'' a left ''R''-module, then the '''tensor product''' of ''M'' and ''N'' is {$M \otimes_R N = \int^{R} M \otimes_{\mathbb{Z}} N$} where ''R'' is viewed as a category with one object whose morphisms are the elements of ''R''.
Ištrintos 208-212 eilutės:
* The coend of a functor {$F: C^{\text{op}} \times C \to X$} is the dual of the '''end (category theory)|end''' of ''F'' and is denoted by
:{$\int^{c \in C} F(c, c)$}.
For example, if ''R'' is a ring, ''M'' a right ''R''-module and ''N'' a left ''R''-module, then the '''tensor product of modules|tensor product''' of ''M'' and ''N'' is
:{$M \otimes_R N = \int^{R} M \otimes_{\mathbb{Z}} N$}
where ''R'' is viewed as a category with one object whose morphisms are the elements of ''R''.
Pakeistos 242-247 eilutės iš
'''Simplicial localization''' is a method of localizing a category.


* A '''(combinatorial) species''' is an endofunctor on the groupoid of finite sets with bijections. It is categorically equivalent to a '''symmetric sequence'''.
į:
* '''Simplicial localization''' is a method of localizing a category.
Ištrintos 243-244 eilutės:
Pakeistos 245-304 eilutės iš
* A topology on a category is '''subcanonical''' if every representable contravariant functor on ''C'' is a sheaf with respect to that topology.<ref>{{harvnb|Vistoli|2004|loc=Definition 2.57.}}</ref> Generally speaking, some '''flat topology''' may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical.

{{term|1=subcategory}}
{{defn|1=A category ''A'' is
a '''subcategory''' of a category ''B'' if there is an inclusion functor from ''A'' to ''B''.}}

{{term|1=subobject}}
{{defn|1=Given an object ''A'' in a category, a '''subobject''' of ''A'' is an equivalence class of monomorphisms to
''A''; two monomorphisms ''f'', ''g'' are considered equivalent if ''f'' factors through ''g'' and ''g'' factors through ''f''.}}

{{term|1=subquotient}}
{{defn|1=A '''subquotient'''
is a quotient of a subobject.}}

{{term|1=subterminal object}}
{{defn|1=
A '''subterminal object''' is an object ''X'' such that every object has at most one morphism into ''X''.}}

{{term|1=symmetric monoidal category}}
{{defn|1=A '''symmetric monoidal category''' is a '''monoidal category
''' (i.e., a category with ⊗) that has maximally symmetric braiding.}}

{{term|1=symmetric sequence}}
{{defn|1=A '''symmetric sequence''' is
a sequence of objects with actions of '''symmetric group'''s. It is categorically equivalent to a '''combinatorial species|(combinatorial) species'''.}}

{{glossary end
}}

==T==
{{glossary}}
{{term|1=t-structure}}
{{defn|1=A
'''t-structure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞-category''') that axiomatizes the notions of complexes whose cohomology concentrated in non-negative degrees or non-positive degrees.}}

{{term|1=Tannakian duality}}
{{defn|The
'''Tannakian duality''' states that, in an appropriate setup, to give a morphism {$f: X \to Y$} is to give a pullback functor {$f^*$} along it. In other words, the Hom set {$\operatorname{Hom}(X, Y)$} can be identified with the functor category {$\operatorname{Fct}(D(Y), D(X))$}, perhaps in the '''derived algebraic geometry|derived sense''', where {$D(X)$} is the category associated to ''X'' (e.g., the derived category).<ref>Jacob Lurie. Tannaka duality for geometric stacks. http://math.harvard.edu/~lurie/, 2004.</ref><ref>{{cite arxiv|last=Bhatt|first=Bhargav|date=2014-04-29|title=Algebraization and Tannaka duality|eprint=1404.7483|class=math.AG}}</ref>}}

{{term|1=tensor category}}
{{defn|1=Usually synonymous with
'''monoidal category''' (though some authors distinguish between the two concepts.)}}

{{term|1=tensor triangulated category}}
{{defn|1=A
'''tensor triangulated category''' is a category that carries the structure of a symmetric monoidal category and that of a triangulated category in a compatible way.}}

{{term|tensor product}}
{{defn|Given a monoidal category ''B'', the '''tensor product of functors''' {$F: C^{\text{op}} \to B$} and {$G: C \to B$} is the coend:
:
{$F \otimes_C G = \int^{c \in C} F(c) \otimes G(c).$}}}

{{term|1=terminal}}
{{defn|no=1|An object ''A'' is '''terminal object|terminal''' (also called final) if there is exactly one morphism from each object to ''A''; e.g., '''singleton (mathematics)|singleton'''s in ''''''Category of sets|Set''''''. It is the dual of an '''initial object'''.}}
{{defn|no=2|An object ''A'' in an ∞-category ''C'' is terminal if {$\operatorname{Map}_C(B, A)$} is '''contractible space|contractible''' for every object ''B'' in ''C''.}}

{{term|1=thick subcategory}}
{{defn|1=A full subcategory of an abelian category is '''thick subcategory|thick''' if it is closed under extensions.}}

{{term|1=thin}}
{{defn|1=A '''thin category|thin''' is a category where there is at most one morphism between any pair of objects.}}

{{term|1=triangulated category}}
{{defn|1=A '''triangulated category''' is a category where one can talk about distinguished triangles, generalization of exact sequences. An abelian category is a prototypical example of a triangulated category. A '''derived category''' is a triangulated category that is not necessary an abelian category.}}

{{glossary end}}

[+U+]
{{glossary}}
{{term|1=universal}}
{{defn|no=1|Given a functor {$f: C \to D$} and an object ''X'' in ''D'', a '''universal morphism''' from ''X'' to ''f'' is an initial object in the '''comma category''' {$(X \downarrow f)$}. (Its dual is also called a universal morphism.) For example, take ''f'' to be the forgetful functor {$\mathbf{Vec}_k \to \mathbf{Set}$} and ''X'' a set. An initial object of {$(X \downarrow f)$} is a function {$j: X \to f(V_X)$}. That it is initial means that if {$k: X \to f(W)$} is another morphism, then there is a unique morphism from ''j'' to ''k'', which consists of a linear map {$V_X \to W$} that extends ''k'' via ''j''; that is to say, {$V_X$} is the '''free vector space''' generated by ''X''
.}}
į:
* Given an object ''A'' in a category, a '''subobject''' of ''A'' is an equivalence class of monomorphisms to ''A''; two monomorphisms ''f'', ''g'' are considered equivalent if ''f'' factors through ''g'' and ''g'' factors through ''f''.
* A topology on a category is '''subcanonical''' if every representable contravariant functor on ''C'' is a sheaf with respect to that topology. Generally speaking, some '''flat topology''' may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical.
* A
'''subquotient''' is a quotient of a subobject.
* A
'''symmetric sequence''' is a sequence of objects with actions of '''symmetric group'''s. It is categorically equivalent to a '''(combinatorial) species'''.
* A '''t-structure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞-category''') that axiomatizes the notions of complexes whose cohomology concentrated in non-negative degrees or non-positive degrees.
* The
'''Tannakian duality''' states that, in an appropriate setup, to give a morphism {$f: X \to Y$} is to give a pullback functor {$f^*$} along it. In other words, the Hom set {$\operatorname{Hom}(X, Y)$} can be identified with the functor category {$\operatorname{Fct}(D(Y), D(X))$}, perhaps in the '''derived sense''', where {$D(X)$} is the category associated to ''X'' (e.g., the derived category).


* Given
a functor {$f: C \to D$} and an object ''X'' in ''D'', a '''universal morphism''' from ''X'' to ''f'' is an initial object in the '''comma category''' {$(X \downarrow f)$}. (Its dual is also called a universal morphism.) For example, take ''f'' to be the forgetful functor {$\mathbf{Vec}_k \to \mathbf{Set}$} and ''X'' a set. An initial object of {$(X \downarrow f)$} is a function {$j: X \to f(V_X)$}. That it is initial means that if {$k: X \to f(W)$} is another morphism, then there is a unique morphism from ''j'' to ''k'', which consists of a linear map {$V_X \to W$} that extends ''k'' via ''j''; that is to say, {$V_X$} is the '''free vector space''' generated by ''X''.}}
{{defn|no=2|Stated more explicitly, given ''f'' as above,
a morphism {$X \to f(u_X)$} in ''D'' is universal if and only if the natural map
:
{$\operatorname{Hom}_C(u_X, c) \to \operatorname{Hom}_D(X, f(c)), \, \alpha \mapsto (X \to f(u_x) \overset{f(\alpha)}\to f(c))$}
is bijective. In particular, if {$\operatorname{Hom}_C(u_X, -) \simeq \operatorname{Hom}_D(X, f(-))$}, then taking ''c'' to be ''u''<sub>''X''</sub> one gets a universal morphism by sending the identity morphism. In other words, having a universal morphism is equivalent to the representability of the functor {$\operatorname{Hom}_D(X, f(-))$}.}}
2019 vasario 10 d., 19:50 atliko AndriusKulikauskas -
Pakeistos 108-112 eilutės iš
į:
* '''site''' A category equipped with a '''Grothendieck topology'''.
* A category is '''skeletal''' if isomorphic objects are necessarily identical.
* Given a category ''C'' and an object ''A'' in it, the '''slice category''' ''C''<sub>/''A''</sub> of ''C'' over ''A'' is the category whose objects are all the morphisms in ''C'' with codomain ''A'', whose morphisms are morphisms in ''C'' such that if ''f'' is a morphism from {$p_X: X \to A$} to {$p_Y: Y \to A$}, then {$p_Y \circ f = p_X$} in ''C'' and whose composition is that of ''C''.}}
* A '''small category''' is a category in which the class of all morphisms is a '''set''' (i.e., not a '''proper class'''); otherwise '''large'''. A category is '''locally small''' if the morphisms between every pair of objects ''A'' and ''B'' form a set. Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a '''quasicategory''' is a category whose objects and morphisms merely form a '''conglomerate (set theory)|conglomerate'''.
Pakeistos 159-160 eilutės iš
į:
* A morphism ''f'' in a category admitting finite limits and finite colimits is '''strict''' if the natural morphism {$\operatorname{Coim}(f) \to \operatorname{Im}(f)$} is an isomorphism.
Ištrintos 237-242 eilutės:

* '''site''' A category equipped with a '''Grothendieck topology'''.
* A category is '''skeletal''' if isomorphic objects are necessarily identical.
* Given a category ''C'' and an object ''A'' in it, the '''slice category''' ''C''<sub>/''A''</sub> of ''C'' over ''A'' is the category whose objects are all the morphisms in ''C'' with codomain ''A'', whose morphisms are morphisms in ''C'' such that if ''f'' is a morphism from {$p_X: X \to A$} to {$p_Y: Y \to A$}, then {$p_Y \circ f = p_X$} in ''C'' and whose composition is that of ''C''.}}
* A '''small category''' is a category in which the class of all morphisms is a '''set''' (i.e., not a '''proper class'''); otherwise '''large'''. A category is '''locally small''' if the morphisms between every pair of objects ''A'' and ''B'' form a set. Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a '''quasicategory''' is a category whose objects and morphisms merely form a '''conglomerate (set theory)|conglomerate'''.
Pakeistos 242-244 eilutės iš
* A morphism ''f'' in a category admitting finite limits and finite colimits is '''strict''' if the natural morphism {$\operatorname{Coim}(f) \to \operatorname{Im}(f)$} is an isomorphism.

A strict 0-category is a set and for any integer ''n'' > 0, a '''strict ''n''-category''' is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category. '''Note''': the term "''n''-category" typically refers to "'''weak ''n''-category'''"; not strict one.
į:
* A strict 0-category is a set and for any integer ''n'' > 0, a '''strict ''n''-category''' is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category. '''Note''': the term "''n''-category" typically refers to "'''weak ''n''-category'''"; not strict one.
2019 vasario 10 d., 19:48 atliko AndriusKulikauskas -
Pakeistos 31-32 eilutės iš
į:
* An object in a category is said to be '''small''' if it is κ-compact for some regular cardinal κ. The notion prominently appears in Quiilen's '''small object argument''' (cf. https://ncatlab.org/nlab/show/small+object+argument)
Pakeistos 153-155 eilutės iš
į:
* A '''simplicial object''' in a category ''C'' is roughly a sequence of objects {$X_0, X_1, X_2, \dots$} in ''C'' that forms a simplicial set. In other words, it is a covariant or contravariant functor Δ → ''C''. For example, a '''simplicial presheaf''' is a simplicial object in the category of presheaves.
* A '''simplicial set''' is a contravariant functor from &Delta; to '''Set''', where &Delta; is the '''simplex category''', a category whose objects are the sets [''n''] = { 0, 1, …, ''n'' } and whose morphisms are order-preserving functions. One writes {$X_n = X([n])$} and an element of the set {$X_n$} is called an ''n''-simplex. For example, {$\Delta^n = \operatorname{Hom}_{\Delta}(-, [n])$} is a simplicial set called the standard ''n''-simplex. By Yoneda's lemma, {$X_n \simeq \operatorname{Nat}(\Delta^n, X)$}.
Pakeistos 232-264 eilutės iš
{{term|1=simplicial object}}
{{defn|1=A
'''simplicial object''' in a category ''C'' is roughly a sequence of objects {$X_0, X_1, X_2, \dots$} in ''C'' that forms a simplicial set. In other words, it is a covariant or contravariant functor Δ → ''C''. For example, a '''simplicial presheaf''' is a simplicial object in the category of presheaves.}}

{{term|1=simplicial set}}
{{defn|1=A
'''simplicial set''' is a contravariant functor from &Delta; to '''Set''', where &Delta; is the '''simplex category''', a category whose objects are the sets [''n''] = { 0, 1, …, ''n'' } and whose morphisms are order-preserving functions. One writes {$X_n = X([n])$} and an element of the set {$X_n$} is called an ''n''-simplex. For example, {$\Delta^n = \operatorname{Hom}_{\Delta}(-, [n])$} is a simplicial set called the standard ''n''-simplex. By Yoneda's lemma, {$X_n \simeq \operatorname{Nat}(\Delta^n, X)$}.}}

{{term|1=site}}
{{defn|1=A category equipped with a
'''Grothendieck topology'''.}}

{{term|1=skeletal}}
{{defn|1=A category is '''Skeleton (category theory)|skeletal''' if isomorphic objects are necessarily identical.}}

{{term|1=slice}}
{{defn|1=Given a category ''C'' and an object
''A'' in it, the '''slice category''' ''C''<sub>/''A''</sub> of ''C'' over ''A'' is the category whose objects are all the morphisms in ''C'' with codomain ''A'', whose morphisms are morphisms in ''C'' such that if ''f'' is a morphism from {$p_X: X \to A$} to {$p_Y: Y \to A$}, then {$p_Y \circ f = p_X$} in ''C'' and whose composition is that of ''C''.}}

{{term|1=small}}
{{defn|no=1|1=A
'''small category''' is a category in which the class of all morphisms is a '''Set (mathematics)|set''' (i.e., not a '''proper class'''); otherwise '''large'''. A category is '''locally small''' if the morphisms between every pair of objects ''A'' and ''B'' form a set. Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a '''quasicategory''' is a category whose objects and morphisms merely form a '''conglomerate (set theory)|conglomerate'''.<ref>{{cite book |last=Adámek |first=Jiří |author2=Herrlich, Horst |author3=Strecker, George E |title=Abstract and Concrete Categories (The Joy of Cats) |origyear=1990 |url=http://katmat.math.uni-bremen.de/acc/ |format=PDF |year=2004 |publisher= Wiley & Sons |location=New York |isbn=0-471-60922-6 |page=40}}</ref> (NB: some authors use the term "quasicategory" with a different meaning.<ref>{{cite journal|doi=10.1016/S0022-4049(02)00135-4|last=Joyal|first=A.|title=Quasi-categories and Kan complexes|journal=Journal of Pure and Applied Algebra|volume=175|year=2002|issue=1–3|pages=207–222|ref=harv}}</ref>)}}
{{defn|no=2|An object in a category is said to be '''small object|small''' if it is κ-compact for some regular cardinal κ. The notion prominently appears in Quiilen's '''small object argument''' (cf. https://ncatlab.org/nlab/show/small+object+argument)}}

{{term|species}}
{{defn|1=A '''combinatorial species|(combinatorial) species''' is an endofunctor on the groupoid of finite sets with bijections. It is categorically equivalent to a '''symmetric sequence'''.}}

{{term|1=stable}}
{{defn|1=An ∞-category is '''stable ∞-category|stable''' if (1) it has a zero object, (2) every morphism in it admits a fiber and a cofiber and (3) a triangle in it is a fiber sequence if and only if it is a cofiber sequence.}}

{{term|strict}}
{{defn|A morphism ''f'' in a category admitting finite limits and finite colimits is '''strict morphism|strict''' if the natural morphism {$\operatorname{Coim}(f) \to \operatorname{Im}(f)$} is an isomorphism.}}

{{term|strict ''n''-category}}
{{defn|A strict 0-category is a set and for any integer ''n'' > 0, a '''strict n-category|strict ''n''-category''' is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category. '''Note''': the term "''n''-category" typically refers to "'''weak n-category|weak ''n''-category'''"; not strict one.}}

{{term|1=subcanonical}}
{{defn|1=A topology on a category is '''subcanonical''' if every representable contravariant functor on ''C'' is a sheaf with respect to that topology.<ref>{{harvnb|Vistoli|2004|loc=Definition 2.57.}}</ref> Generally speaking, some '''flat topology''' may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical.}}
į:

*
'''site''' A category equipped with a '''Grothendieck topology'''.
* A category is '''skeletal''' if isomorphic objects are necessarily identical.
* Given a category
''C'' and an object ''A'' in it, the '''slice category''' ''C''<sub>/''A''</sub> of ''C'' over ''A'' is the category whose objects are all the morphisms in ''C'' with codomain ''A'', whose morphisms are morphisms in ''C'' such that if ''f'' is a morphism from {$p_X: X \to A$} to {$p_Y: Y \to A$}, then {$p_Y \circ f = p_X$} in ''C'' and whose composition is that of ''C''.}}
* A '''small category''' is a category in which the class of all morphisms is a '''
set''' (i.e., not a '''proper class'''); otherwise '''large'''. A category is '''locally small''' if the morphisms between every pair of objects ''A'' and ''B'' form a set. Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a '''quasicategory''' is a category whose objects and morphisms merely form a '''conglomerate (set theory)|conglomerate'''.

*
A '''(combinatorial) species''' is an endofunctor on the groupoid of finite sets with bijections. It is categorically equivalent to a '''symmetric sequence'''.

* An ∞-category is
'''stable''' if (1) it has a zero object, (2) every morphism in it admits a fiber and a cofiber and (3) a triangle in it is a fiber sequence if and only if it is a cofiber sequence.

* A morphism
''f'' in a category admitting finite limits and finite colimits is '''strict''' if the natural morphism {$\operatorname{Coim}(f) \to \operatorname{Im}(f)$} is an isomorphism.

A strict 0-category is a set and for any integer
''n'' > 0, a '''strict ''n''-category''' is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category. '''Note''': the term "''n''-category" typically refers to "'''weak ''n''-category'''"; not strict one.

* A topology on
a category is '''subcanonical''' if every representable contravariant functor on ''C'' is a sheaf with respect to that topology.<ref>{{harvnb|Vistoli|2004|loc=Definition 2.57.}}</ref> Generally speaking, some '''flat topology''' may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical.
2019 vasario 10 d., 19:31 atliko AndriusKulikauskas -
Pakeistos 30-31 eilutės iš
į:
* A simple object in an abelian category is <span id="simple object"></span>an object ''A'' that is not isomorphic to the zero object and whose every '''subobject''' is isomorphic to zero or to ''A''. For example, a '''simple module''' is precisely a simple object in the category of (say left) modules.
Pakeistos 43-44 eilutės iš
į:
* A morphism is a '''section''' if it has a left inverse. For example, the '''axiom of choice''' says that any surjective function admits a section.
Pakeistos 106-107 eilutės iš
į:
* An abelian category is '''semisimple''' if every short exact sequence splits. For example, a ring is '''semisimple''' if and only if the category of modules over it is semisimple.
Pakeistos 146-152 eilutės iš
į:
* A functor is said to '''reflect identities''' if it has the property: if ''F''(''k'') is an identity then ''k'' is an identity as well.
* A functor is said to '''reflect isomorphisms''' if it has the property: ''F''(''k'') is an isomorphism then ''k'' is an isomorphism as well.
* A set-valued contravariant functor ''F'' on a category ''C'' is said to be '''representable''' if it belongs to the essential image of the '''Yoneda embedding''' {$C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}; i.e., {$F \simeq \operatorname{Hom}_C(-, Z)$} for some object ''Z''. The object ''Z'' is said to be the representing object of ''F''.
* Given a ''k''-linear category ''C'' over a field ''k'', a '''Serre functor''' {$f: C \to C$} is an auto-equivalence such that {$\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*$} for any objects ''A'', ''B''.}}
* The '''simplex category''' &Delta; is the category where an object is a set [''n''] = { 0, 1, …, ''n'' }, n ≥ 0, totally ordered in the standard way and a morphism is an order-preserving function.
* '''simplicial category''' A category enriched over simplicial sets.
Pakeistos 224-257 eilutės iš


* A functor is said to '''reflect identities''' if it has the property: if ''F''(''k'') is an identity then ''k'' is an identity as well.
* A functor is said to
'''reflect isomorphisms''' if it has the property: ''F''(''k'') is an isomorphism then ''k'' is an isomorphism as well.
* A set-valued contravariant functor ''F'' on a category ''C'' is said to be '''representable''' if it belongs to the essential image of the '''Yoneda embedding''' {$C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}; i.e., {$F \simeq \operatorname{Hom}_C(-, Z)$} for some object ''Z''. The object ''Z'' is said to be the representing object of ''F''.


==S==
{{glossary}}
{{term|1=section}}
{{defn|1=A morphism is a '''section (category theory)|section''' if it has a left inverse. For example, the '''axiom of choice''' says that any surjective function admits a section.}}

{{term|1=Segal space}}
{{defn|1='''Segal space'''s were certain simplicial spaces, introduced as models for '''(infinity,1)-category|(∞,&nbsp;1)-categories'''.}}

{{term|1=semisimple}}
{{defn|1=An abelian category is '''semisimple category|semisimple''' if every short exact sequence splits. For example, a ring is '''semisimple ring|semisimple''' if and only if the category of modules over it is semisimple.}}

{{term|1=Serre functor}}
{{defn|1=Given a ''k''-linear category ''C'' over a field ''k'', a '''Serre functor''' {$f: C \to C$} is an auto-equivalence such that {$\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*$} for any objects ''A'', ''B''.}}
<!--{{term|1=sieve}}
{{defn|1=In a category, a '''sieve (category theory)|sieve''' is a set ''S'' of objects having the property: if ''f'' is a morphism with the codomain in ''S'', then the domain of ''f'' is in ''S''.}} conflict with the literature? -->
{{term|1=simple object}}
{{defn|1=A simple object in an abelian category is <span id="simple object"></span>an object ''A'' that is not isomorphic to the zero object and whose every '''subobject''' is isomorphic to zero or to ''A''. For example, a '''simple module''' is precisely a simple object in the category of (say left) modules.}}

{{term|1=simplex category}}
{{defn|1=The '''simplex category''' &Delta; is the category where an object is a set [''n''] = { 0, 1, …, ''n'' }, n ≥ 0, totally ordered in the standard way and a morphism is an order-preserving function.}}

{{term|1=simplicial category}}
{{defn|1=A category enriched over simplicial sets.}}

{{term|1=Simplicial localization}}
{{defn|1='''Simplicial localization''' is a method of localizing a category.}}
į:
* '''Segal space'''s were certain simplicial spaces, introduced as models for '''(∞,&nbsp;1)-categories'''.
* In a category, a
'''sieve''' is a set ''S'' of objects having the property: if ''f'' is a morphism with the codomain in ''S'', then the domain of ''f'' is in ''S''.}} conflict with the literature?

'''Simplicial localization''' is a method of localizing a category.
2019 vasario 10 d., 19:08 atliko AndriusKulikauskas -
Pakeistos 41-42 eilutės iš
į:
* A morphism is a '''retraction''' if it has a right inverse.
Pakeistos 218-233 eilutės iš



==R==
{{glossary}}
{{term|1=
reflect}}
{{defn|no=1|1=A functor is said to reflect identities
if it has the property: if ''F''(''k'') is an identity then ''k'' is an identity as well.}}
{{defn|no=2|1=A
functor is said to reflect isomorphisms if it has the property: ''F''(''k'') is an isomorphism then ''k'' is an isomorphism as well.}}

{{term|1=representable}}
{{defn|1=A
set-valued contravariant functor ''F'' on a category ''C'' is said to be '''representable functor|representable''' if it belongs to the essential image of the '''Yoneda embedding''' {$C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}; i.e., {$F \simeq \operatorname{Hom}_C(-, Z)$} for some object ''Z''. The object ''Z'' is said to be the representing object of ''F''.}}

{{term|1=retraction}}
{{defn|1='''File:Section retract.svg|150px|thumb|''f'' is a retraction of ''g''. ''g'' is a section of ''f''.'''A morphism is a '''section (category theory)|retraction''' if it has a right inverse.}}
{{glossary end}}
į:
* A functor is said to '''reflect identities''' if it has the property: if ''F''(''k'') is an identity then ''k'' is an identity as well.
* A functor is said to '''reflect isomorphisms''' if it has the property: ''F''(''k'') is an isomorphism then ''k'' is an isomorphism as well.
* A set-valued contravariant functor ''F'' on a category ''C'' is said to be '''representable''' if it belongs to the essential image of the '''Yoneda embedding''' {$C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}; i.e., {$F \simeq \operatorname{Hom}_C(-, Z)$} for some object ''Z''. The object ''Z'' is said to be the representing object of ''F''.
2019 vasario 10 d., 19:04 atliko AndriusKulikauskas -
Pakeistos 29-30 eilutės iš
į:
* An object ''A'' in an abelian category is '''projective''' if the functor {$\operatorname{Hom}(A, -)$} is exact. It is the dual of an injective object.
Pakeistos 99-100 eilutės iš
į:
* A category is '''preadditive''' if it is '''enriched category|enriched''' over the '''monoidal category''' of '''abelian group'''s. More generally, it is '''preadditive category#R-linear categories|''R''-linear''' if it is enriched over the monoidal category of '''module (mathematics)|''R''-modules''', for ''R'' a '''commutative ring'''.
* Given a '''regular cardinal''' κ, a category is '''κ-presentable''' if it admits all small colimits and is '''κ-accessible'''. A category is presentable if it is κ-presentable for some regular cardinal κ (hence presentable for any larger cardinal). '''Note''': Some authors call a presentable category a '''locally presentable category'''.
* The '''product of a family of categories''' ''C''<sub>''i''</sub>'s indexed by a set ''I'' is the category denoted by {$\prod_i C_i$} whose class of objects is the product of the classes of objects of ''C''<sub>''i''</sub>'s and whose hom-sets are {$\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)$}; the morphisms are composed component-wise. It is the dual of the disjoint union.
* A '''PROP''' is a symmetric strict monoidal category whose objects are natural numbers and whose tensor product '''addition''' of natural numbers.
Pakeistos 139-140 eilutės iš
į:
* A functor from the category of finite-dimensional vector spaces to itself is called a '''polynomial functor''' if, for each pair of vector spaces ''V'', ''W'', {{nowrap|''F'': Hom(''V'', ''W'') → Hom(''F''(''V''), ''F''(''W''))}} is a polynomial map between the vector spaces. A '''Schur functor''' is a basic example.
* Another term for a contravariant functor: a functor from a category ''C''<sup>op</sup> to '''Set''' is a presheaf of sets on ''C'' and a functor from ''C''<sup>op</sup> to ''s'''''Set''' is a presheaf of simplicial sets or '''simplicial presheaf''', etc. A '''topology''' on ''C'', if any, tells which presheaf is a sheaf (with respect to that topology).
* Given categories ''C'' and ''D'', a '''profunctor''' (or a distributor) from ''C'' to ''D'' is a functor of the form {$D^{\text{op}} \times C \to \mathbf{Set}$}.
Pakeistos 165-166 eilutės iš
į:
* The '''product''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the projective limit {$\varprojlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where ''I'' is viewed as a discrete category. It is denoted by {$\prod_i X_i$} and is the dual of the coproduct of the family.
Pakeistos 213-245 eilutės iš

{{term|1=polynomial}}
{{defn|1=A functor from the category of finite-dimensional vector spaces
to itself is called a '''polynomial functor''' if, for each pair of vector spaces ''V'', ''W'', {{nowrap|''F'': Hom(''V'', ''W'') → Hom(''F''(''V''), ''F''(''W''))}} is a polynomial map between the vector spaces. A '''Schur functor''' is a basic example.}}

{{term|1
=preadditive}}
{{defn|1
=A category is '''preadditive category|preadditive''' if it is '''enriched category|enriched''' over the '''monoidal category''' of '''abelian group'''s. More generally, it is '''preadditive category#R-linear categories|''R''-linear''' if it is enriched over the monoidal category of '''module (mathematics)|''R''-modules''', for ''R'' a '''commutative ring'''.}}

{{term|1
=presentable}}
{{defn|Given a '''regular cardinal''' κ, a category is '''presentabl category|κ-presentable''' if it admits all small colimits and is '''#accessible|κ-accessible'''. A category is presentable if it is κ-presentable for some regular cardinal κ (hence presentable for any larger cardinal). '''Note''': Some authors call a presentable category a '''locally presentable category'''.}}

{{term|1
=presheaf}}
{{defn|1=Another term for a contravariant functor: a functor from a category ''C''<sup>op</sup> to '''Set''' is a presheaf of sets on ''C'' and a functor from ''C''<sup>op</sup> to ''s'''''Set''' is a presheaf of simplicial sets or '''simplicial presheaf''', etc. A '''Grothendieck topology|topology''' on ''C'', if any, tells which presheaf is a sheaf (with respect to that topology).}}

{{term|1=product}}
{{defn|no=1|The '''product (category theory)|product''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the projective limit {$\varprojlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where ''I'' is viewed as a discrete category. It is denoted by {$\prod_i X_i$} and is the dual of the coproduct of the family.}}
{{defn|no=2|The '''product of categories|product of a family of categories''' ''C''<sub>''i''</sub>'s indexed by a set ''I'' is the category denoted by {$\prod_i C_i$} whose class of objects is the product of the classes of objects of ''C''<sub>''i''</sub>'s and whose hom-sets are {$\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)$}; the morphisms are composed component-wise. It is the dual of the disjoint union.}}

{{term|1=profunctor}}
{{defn|1=Given categories ''C'' and ''D'', a '''profunctor''' (or a distributor) from ''C'' to ''D'' is a functor of the form {$D^{\text{op}} \times C \to \mathbf{Set}$}.}}

{{term|1=projective}}
{{defn|1=An object ''A'' in an abelian category is '''projective object|projective''' if the functor {$\operatorname{Hom}(A, -)$} is exact. It is the dual of an injective object.}}

{{term|1=PROP}}
{{defn|1=A '''PROP (category theory)|PROP''' is a symmetric strict monoidal category whose objects are natural numbers and whose tensor product '''addition''' of natural numbers.}}
{{glossary end}}

==Q==
{{glossary}}
{{term|1=Quillen}}
{{defn|1='''Quillen’s theorem A''' provides a criterion for a functor to be a weak equivalence.}}
{{glossary end}}
į:
* '''Quillen’s theorem A''' provides a criterion for a functor to be a weak equivalence.







==R==
2019 vasario 10 d., 12:05 atliko AndriusKulikauskas -
Pridėtos 19-20 eilutės:
* An '''object''' is part of a data defining a category.
Pakeistos 95-99 eilutės iš
į:
* A category is '''normal''' if every monic is normal.
* The '''opposite category''' of a category is obtained by reversing the arrows. For example, if a partially ordered set is viewed as a category, taking its opposite amounts to reversing the ordering.
* A category (or ∞-category) is called '''pointed''' if it has a zero object.
Pakeistos 133-134 eilutės iš
* A category is '''normal''' if every monic is normal.
į:
* An [adjective] object in a category ''C'' is a contravariant functor (or presheaf) from some fixed category corresponding to the "adjective" to ''C''. For example, a '''simplicial object''' in ''C'' is a contravariant functor from the simplicial category to ''C'' and a '''Γ-object''' is a pointed contravariant functor from '''Γ (category theory)|Γ''' (roughly the pointed category of pointed finite sets) to ''C'' provided ''C'' is pointed.
* A functor π:''C'' → ''D'' is an '''op-fibration''' if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one π-coCartesian morphism ''f'': ''x'' → ''y<nowiki>'</nowiki>'' in ''C'' such that π(''f'') = ''g''. In other words, π is the dual of a '''Grothendieck fibration'''.
Pakeistos 203-229 eilutės iš
* '''A natural transformation is, roughly, a map between functors. Precisely, given a pair of functors ''F'', ''G'' from a category ''C'' to category ''D'', a '''natural transformation''' φ from ''F'' to ''G'' is a set of morphisms in ''D'':{$\{ \phi_x: F(x) \to G(x) \mid x \in \operatorname{Ob}(C) \}$}
satisfying the condition: for each morphism ''f'': ''x'' → ''y'' in ''C'', {$\phi_y \circ F(f) = G(f) \circ \phi_x$}. For example, writing {$GL_n(R)$} for the group of invertible ''n''-by-''n'' matrices with coefficients in a commutative ring ''R'', we can view {$GL_n$} as a functor from the category '''CRing''' of commutative rings to the category '''Grp''' of groups. Similarly, {$R \mapsto R^*$} is a functor from '''CRing''' to '''Grp'''. Then the '''determinant''' det is a natural transformation from {$GL_n$} to -<sup>*</sup>.
{{defn|no=2|A '''natural isomorphism''' is a natural transformation that is an isomorphism (i.e., admits the inverse).}}




==O==
{{glossary}}
{{term|1=object}}
{{defn|no=1|1=An object is part of a data defining a category
.}}
{{defn|no=2|1=An [adjective] object in a category ''C'' is a contravariant functor (or presheaf) from some fixed category corresponding to the "adjective" to ''C''. For example, a '''simplicial object''' in ''C'' is a contravariant functor from the simplicial category to ''C'' and a '''Γ-object''' is a pointed contravariant functor from '''Γ (category theory)|Γ''' (roughly the pointed category of pointed finite sets) to ''C'' provided ''C'' is pointed.}}

{{term|1=op-fibration}}
{{defn|1=A functor π:''C'' → ''D'' is an '''op-fibration''' if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one π-coCartesian morphism ''f'': ''x'' → ''y<nowiki>'</nowiki>'' in ''C'' such that π(''f'') = ''g''. In other words, π is the dual of a '''Grothendieck fibration'''.}}

{{term|1=opposite}}
{{defn|1=The '''opposite category''' of a category is obtained by reversing the arrows. For example, if a partially ordered set is viewed as a category, taking its opposite amounts to reversing the ordering.}}
{{glossary end}}

==P==
{{glossary}}
{{term|1=perfect}}
{{defn|Sometimes synonymous with "compact". See '''perfect complex'''.}}

{{term|1=pointed}}
{{defn|1=A category (or ∞-category) is called pointed if it has a zero object.}}
į:
* '''A natural transformation is, roughly, a map between functors. Precisely, given a pair of functors ''F'', ''G'' from a category ''C'' to category ''D'', a '''natural transformation''' φ from ''F'' to ''G'' is a set of morphisms in ''D'':{$\{ \phi_x: F(x) \to G(x) \mid x \in \operatorname{Ob}(C) \}$} satisfying the condition: for each morphism ''f'': ''x'' → ''y'' in ''C'', {$\phi_y \circ F(f) = G(f) \circ \phi_x$}. For example, writing {$GL_n(R)$} for the group of invertible ''n''-by-''n'' matrices with coefficients in a commutative ring ''R'', we can view {$GL_n$} as a functor from the category '''CRing''' of commutative rings to the category '''Grp''' of groups. Similarly, {$R \mapsto R^*$} is a functor from '''CRing''' to '''Grp'''. Then the '''determinant''' det is a natural transformation from {$GL_n$} to -<sup>*</sup>.
* A '''natural isomorphism''' is a natural transformation that is an isomorphism (i.e., admits the inverse).
* '''perfect''' Sometimes synonymous with "compact". See '''perfect complex'''.
2019 vasario 10 d., 11:59 atliko AndriusKulikauskas -
Pakeistos 26-27 eilutės iš
į:
* A '''monoid object''' in a monoidal category is an object together with the multiplication map and the identity map that satisfy the expected conditions like associativity. For example, a monoid object in '''Set''' is a usual monoid (unital semigroup) and a monoid object in '''''R''-mod''' is an '''associative algebra''' over a commutative ring ''R''.
Pakeistos 37-38 eilutės iš
į:
* A morphism ''f'' is a '''monomorphism''' (also called monic) if {$g=h$} whenever {$f\circ g=f\circ h$}; e.g., an '''Injective function|injection''' in ''''''Category of sets|Set''''''. In other words, ''f'' is the dual of an epimorphism.
Pakeistos 91-93 eilutės iš
į:
* A '''monoidal category''', also called a tensor category, is a category ''C'' equipped with (1) a '''bifunctor''' {$\otimes: C \times C \to C$}, (2) an identity object and (3) natural isomorphisms that make ⊗ associative and the identity object an identity for ⊗, subject to certain coherence conditions.
* '''''n''-category''' A '''strict n-category|strict ''n''-category''' is defined inductively: a strict 0-category is a set and a strict ''n''-category is a category whose Hom sets are strict (''n''-1)-categories. Precisely, a strict ''n''-category is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category.
Pakeistos 122-128 eilutės iš
į:
* A '''monad''' in a category ''X'' is a '''monoid object''' in the monoidal category of endofunctors of ''X'' with the monoidal structure given by composition. For example, given a group ''G'', define an endofunctor ''T'' on '''Set''' by {$T(X) = G \times X$}. Then define the multiplication ''μ'' on ''T'' as the natural transformation {$\mu: T \circ T \to T$} given by
:{$\mu_X: G \times (G \times X) \to G \times X, \,\, (g, (h, x)) \mapsto (gh, x)$}
and also define the identity map ''η'' in the analogous fashion. Then (''T'', ''μ'', ''η'') constitutes a monad in '''Set'''. More substantially, an adjunction between functors {$F: X \rightleftarrows A : G$} determines a monad in ''X''; namely, one takes {$T = G \circ F$}, the identity map ''η'' on ''T'' to be a unit of the adjunction and also defines ''μ'' using the adjunction.
* A functor is said to be '''monadic''' if it is a constituent of a monadic adjunction.
* The '''nerve functor''' ''N'' is the functor from '''Cat''' to ''s'''''Set''' given by {$N(C)_n = \operatorname{Hom}_{\mathbf{Cat}}([n], C)$}. For example, if {$\varphi$} is a functor in {$N(C)_2$} (called a 2-simplex), let {$x_i = \varphi(i), \, 0 \le i \le 2$}. Then {$\varphi(0 \to 1)$} is a morphism {$f: x_0 \to x_1$} in ''C'' and also {$\varphi(1 \to 2) = g: x_1 \to x_2$} for some ''g'' in ''C''. Since {$0 \to 2$} is {$0 \to 1$} followed by {$1 \to 2$} and since {$\varphi$} is a functor, {$\varphi(0 \to 2) = g \circ f$}. In other words, {$\varphi$} encodes ''f'', ''g'' and their compositions.
* A category is '''normal''' if every monic is normal.
Pakeistos 190-242 eilutės iš


'''localization of a category'''

* A '''monad''' in a category ''X'' is a '''monoid object''' in the monoidal category of endofunctors of ''X'' with the monoidal structure given by composition. For example, given a group ''G'', define an endofunctor ''T'' on '''Set''' by {$T(X) = G \times X$}. Then define the multiplication ''μ'' on ''T'' as the natural transformation {$\mu: T \circ T \to T$} given by
:{$\mu_X: G \times (G \times X) \to G \times X, \,\, (g, (h, x)) \mapsto (gh, x)$}
and also define the identity map
''η'' in the analogous fashion. Then (''T'', ''μ'', ''η'') constitutes a monad in '''Set'''. More substantially, an adjunction between functors {$F: X \rightleftarrows A : G$} determines a monad in ''X''; namely, one takes {$T = G \circ F$}, the identity map ''η'' on ''T'' to be a unit of the adjunction and also defines ''μ'' using the adjunction.}}

{{term|1=monadic}}
{{defn|no=1|1=An adjunction is said to be
'''monadic adjunction|monadic''' if it comes from the monad that it determines by means of the '''Eilenberg–Moore category''' (the category of algebras for the monad).}}
{{defn|no=2|1=A functor is said to be
'''monadic functor|monadic''' if it is a constituent of a monadic adjunction.}}

{{term|1=monoidal category
}}
{{defn|1=A '''monoidal category
''', also called a tensor category, is a category ''C'' equipped with (1) a '''bifunctor''' {$\otimes: C \times C \to C$}, (2) an identity object and (3) natural isomorphisms that make ⊗ associative and the identity object an identity for ⊗, subject to certain coherence conditions.}}

{{term|1=monoid object}}
{{defn|1=A '''monoid object''' in a monoidal
category is an object together with the multiplication map and the identity map that satisfy the expected conditions like associativity. For example, a monoid object in '''Set''' is a usual monoid (unital semigroup) and a monoid object in ''''''category of modules|''R''-mod'''''' is an '''associative algebra''' over a commutative ring ''R''.}}

{{term|1=monomorphism}}
{{defn|1
=A morphism ''f'' is a '''monomorphism''' (also called monic) if {$g=h$} whenever {$f\circ g=f\circ h$}; e.g., an '''Injective function|injection''' in ''''''Category of sets|Set''''''. In other words, ''f'' is the dual of an epimorphism.}}

{{term|1=multicategory}}
{{defn|1=A '''multicategory''' is a generalization of a category in which a morphism is allowed to have more than one domain. It is the same thing as a '''colored operad'''.<ref>https://ncatlab.org/nlab/show/multicategory</ref>}}
{{glossary end}}

==N==
{{glossary}}

{{term|1=''n''-category}}
{{quote box
|quote=[T]he issue of comparing definitions of weak ''n''-category is a slippery one, as it is hard to say what it even ''means'' for two such definitions to be equivalent. [...] It is widely held that the structure formed by weak ''n''-categories and the functors, transformations, ... between them should be a weak (''n'' + 1)-category; and if this is the case then the question is whether your weak (''n'' + 1)-category of weak ''n-''categories is equivalent to mine—but whose definition of weak (''n'' + 1)-category are we using here... ?
|source=[http://www.tac.mta.ca/tac/volumes/10/1/10-01abs.html A survey of definitions of ''n''-category]
|author=Tom Leinster
|align=right
|width=33%
}}{{defn|no=1|1=A '''strict n-category|strict ''n''-category''' is defined inductively: a strict 0-category is a set and a strict ''n''-category is a category whose Hom sets are strict (''n''-1)-categories. Precisely, a strict ''n''-category is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category.}}
{{defn|no=2|1=The notion of a '''weak n-category|weak ''n''-category''' is obtained from the strict one by weakening the conditions like associativity of composition to hold only up to '''coherent isomorphism'''s in the weak sense.}}
{{defn|no=3|1=One can define an ∞-category as a kind of a colim of ''n''-categories. Conversely, if one has the notion of a (weak) ∞-category (say a '''quasi-category''') in the beginning, then a weak ''n''-category can be defined as a type of a truncated ∞-category.}}

{{term|1=natural}}
{{defn|no=1|A natural transformation is, roughly, a map between functors. Precisely, given a pair of functors ''F'', ''G'' from a category ''C'' to category ''D'', a '''natural transformation''' φ from ''F'' to ''G'' is a set of morphisms in ''D''
:{$\{ \phi_x: F(x) \to G(x) \mid x \in \operatorname{Ob}(C) \}$}
satisfying the condition: for each morphism ''f'': ''x'' → ''y'' in ''C'', {$\phi_y \circ F(f) = G(f) \circ \phi_x$}. For example, writing {$GL_n(R)$} for the group of invertible ''n''-by-''n'' matrices with coefficients in a commutative ring ''R'', we can view {$GL_n$} as a functor from the category '''CRing''' of commutative rings to the category '''Grp''' of groups. Similarly, {$R \mapsto R^*$} is a functor from '''CRing''' to '''Grp'''. Then the '''determinant''' det is a natural transformation from {$GL_n$} to -<sup>*</sup>.}}
{{defn|no=2|A '''natural isomorphism''' is a natural transformation that is an isomorphism (i.e., admits the inverse).}}

'''Image:Nerve-2-simplex.png|thumb|right|The composition is encoded as a 2-simplex.'''
{{term|1=nerve}}
{{defn|1=The '''nerve functor''' ''N'' is the functor from '''Cat''' to ''s'''''Set''' given by {$N(C)_n = \operatorname{Hom}_{\mathbf{Cat}}([n], C)$}. For example, if {$\varphi$} is a functor in {$N(C)_2$} (called a 2-simplex), let {$x_i = \varphi(i), \, 0 \le i \le 2$}. Then {$\varphi(0 \to 1)$} is a morphism {$f: x_0 \to x_1$} in ''C'' and also {$\varphi(1 \to 2) = g: x_1 \to x_2$} for some ''g'' in ''C''. Since {$0 \to 2$} is {$0 \to 1$} followed by {$1 \to 2$} and since {$\varphi$} is a functor, {$\varphi(0 \to 2) = g \circ f$}. In other words, {$\varphi$} encodes ''f'', ''g'' and their compositions.}}

{{term|1=normal}}
{{defn|1=A category is '''normal category|normal''' if every monic is normal.{{citation needed|date=October 2015}}}}
{{glossary end}}
į:
* '''localization of a category'''
* An adjunction is said to be '''monadic''' if it comes from the monad that it determines by means of the '''Eilenberg–Moore category''' (the category of algebras for the monad).
* A
'''multicategory''' is a generalization of a category in which a morphism is allowed to have more than one domain. It is the same thing as a '''colored operad'''.
* The notion of a '''weak ''n''-category''' is obtained from the strict one by weakening the conditions like associativity of composition to hold only up to
'''coherent isomorphism'''s in the weak sense.
* One can define an ∞-category as a kind of a colim of ''n''-categories. Conversely, if one has the notion of a (weak) ∞-category (say a '''quasi-category''') in the beginning, then a weak ''n''-category can be defined as a type of a truncated ∞-category.
* '''A natural transformation is, roughly, a map between functors. Precisely, given a pair of functors
''F'', ''G'' from a category ''C'' to category ''D'', a '''natural transformation''' φ from ''F'' to ''G'' is a set of morphisms in ''D'':{$\{ \phi_x: F(x) \to G(x) \mid x \in \operatorname{Ob}(C) \}$}
satisfying the condition: for each morphism ''f'': ''x'' → ''y'' in ''C'', {$\phi_y \circ F(f) = G(f) \circ \phi_x$}. For example, writing {$GL_n(R)$} for the group of invertible ''n''-by-''n'' matrices with coefficients in a commutative ring ''R'', we can view {$GL_n$} as a functor from the category '''CRing''' of commutative rings to the category '''Grp''' of groups. Similarly, {$R \mapsto R^*$} is a functor from '''CRing''' to '''Grp'''. Then the '''determinant''' det is a natural transformation from {$GL_n$} to -<sup>*</sup>.
{{defn|no=2|A
'''natural isomorphism''' is a natural transformation that is an isomorphism (i.e., admits the inverse).}}




==O==
2019 vasario 10 d., 11:35 atliko AndriusKulikauskas -
Pakeistos 24-26 eilutės iš
į:
* An object is '''isomorphic''' to another object if there is an isomorphism between them.
* A '''Kan complex''' is a '''fibrant object''' in the category of simplicial sets.
Pakeistos 87-89 eilutės iš
į:
* A category is '''isomorphic''' to another category if there is an isomorphism between them.
* Given a monad ''T'', the '''Kleisli category''' of ''T'' is the full subcategory of the category of ''T''-algebras (called Eilenberg–Moore category) that consists of free ''T''-algebras.
Pakeistos 117-118 eilutės iš
į:
* Given a category ''C'', the left '''Kan extension''' functor along a functor {$f: I \to J$} is the left adjoint (if it exists) to {$f^* = - \circ f: \operatorname{Fct}(J, C) \to \operatorname{Fct}(I, C)$} and is denoted by {$f_!$}. For any {$\alpha: I \to C$}, the functor {$f_! \alpha: J \to C$} is called the left Kan extension of α along ''f''.<ref>http://www.math.harvard.edu/~lurie/282ynotes/LectureXI-Homological.pdf</ref> One can show: {$(f_! \alpha)(j) = \varinjlim_{f(i) \to j} \alpha(i)$} where the colimit runs over all objects {$f(i) \to j$} in the comma category.}} The right Kan extension functor is the right adjoint (if it exists) to {$f^*$}.
Pakeistos 137-138 eilutės iš
* '''ind-limit''' A colimit (or inductive limit) in {$\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}.}}
į:
* '''ind-limit''' A colimit (or inductive limit) in {$\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}.
* The '''limit''' (or '''projective limit''') of a functor {$f: I^{\text{op}} \to \mathbf{Set}$} is: {$\varprojlim_{i \in I} f(i) = \{ (x_i|i) \in \prod_{i} f(i) | f(s)(x_j) = x_i \text{ for any } s: i \to j \}.$}}}
* The limit {$\varprojlim_{i \in I} f(i)$} of a functor {$f: I^{\text{op}} \to C$} is an object, if any, in ''C'' that satisfies: for any object ''X'' in ''C'', {$\operatorname{Hom}(X, \varprojlim_{i \in I} f(i)) = \varprojlim_{i \in I} \operatorname{Hom}(X, f(i))$}; i.e., it is an object representing the functor {$X \mapsto \varprojlim_i \operatorname{Hom}(X, f(i)).$}
* The '''colimit''' (or '''inductive limit''') {$\varinjlim_{i \in I} f(i)$} is the dual of a limit; i.e., given a functor {$f: I \to C$}, it satisfies: for any ''X'', {$\operatorname{Hom}(\varinjlim f(i), X) = \varprojlim \operatorname{Hom}(f(i), X)$}. Explicitly, to give {$\varinjlim f(i) \to X$} is to give a family of morphisms {$f(i) \to X$} such that for any {$i \to j$}, {$f(i) \to X$} is {$f(i) \to f(j) \to X$}. Perhaps the simplest example of a colimit is a '''coequalizer'''. For another example, take ''f'' to be the identity functor on ''C'' and suppose {$L = \varinjlim_{X \in C} f(X)$} exists; then the identity morphism on ''L'' corresponds to a compatible family of morphisms {$\alpha_X: X \to L$} such that {$\alpha_L$} is the identity. If {$f: X \to L$} is any morphism, then {$f = \alpha_L \circ f = \alpha_X$}; i.e., ''L'' is a final object of ''C''.
Pakeistos 178-220 eilutės iš


* An object is '''isomorphic''' to another object if there is an isomorphism between them.

* A category is '''isomorphic''' to another category
if there is an isomorphism between them.

{{term|1=Kan complex}}
{{defn|1=A '''Kan complex''' is a '''fibrant object''' in the category of simplicial sets.}}

{{term|1=Kan extension}}
{{defn|no=1|Given a category
''C'', the left '''Kan extension''' functor along a functor {$f: I \to J$} is the left adjoint (if it exists) to {$f^* = - \circ f: \operatorname{Fct}(J, C) \to \operatorname{Fct}(I, C)$} and is denoted by {$f_!$}. For any {$\alpha: I \to C$}, the functor {$f_! \alpha: J \to C$} is called the left Kan extension of α along ''f''.<ref>http://www.math.harvard.edu/~lurie/282ynotes/LectureXI-Homological.pdf</ref> One can show:
:{$(f_! \alpha)(j) = \varinjlim_{f(i) \to j} \alpha(i)$}
where the colimit runs over all objects {$f(i) \to j$} in the comma category.}}
{{defn|no=2|The right Kan extension functor is the right adjoint (if it exists) to {$f^*$}.}}

{{term|1=Kleisli category}}
{{defn|1=Given a monad ''T'', the '''Kleisli category''' of ''T'' is the full subcategory of the category of ''T''-algebras (called Eilenberg–Moore category) that consists of free ''T''-algebras.}}
{{glossary end}}

==L==
{{glossary}}
{{term|1=lax}}
{{defn|1=The term "'''lax functor'''" is essentially synonymous with "'''pseudo-functor'''".}}

{{term|1=length}}
{{defn|1=An object in an abelian category is said to have <span id="finite length"></span>finite length if it has a '''composition series'''. The maximum number of proper subobjects in any such composition series is called the '''length''' of ''A''.<ref>{{harvnb|Kashiwara|Schapira|2006|loc=exercise 8.20}}</ref>}}

{{term|1=limit}}
{{defn|no=1|The '''limit (category theory)|limit''' (or '''projective limit''') of a functor {$f: I^{\text{op}} \to \mathbf{Set}$} is
::{$\varprojlim_{i \in I} f(i) = \{ (x_i|i) \in \prod_{i} f(i) | f(s)(x_j) = x_i \text{ for any } s: i \to j \}.$}}}
{{defn|no=2|The limit {$\varprojlim_{i \in I} f(i)$} of a functor {$f: I^{\text{op}} \to C$} is an object, if any, in ''C'' that satisfies: for any object ''X'' in ''C'', {$\operatorname{Hom}(X, \varprojlim_{i \in I} f(i)) = \varprojlim_{i \in I} \operatorname{Hom}(X, f(i))$}; i.e., it is an object representing the functor {$X \mapsto \varprojlim_i \operatorname{Hom}(X, f(i)).$}}}
{{defn|no=3|The '''colimit''' (or '''inductive limit''') {$\varinjlim_{i \in I} f(i)$} is the dual of a limit; i.e., given a functor {$f: I \to C$}, it satisfies: for any ''X'', {$\operatorname{Hom}(\varinjlim f(i), X) = \varprojlim \operatorname{Hom}(f(i), X)$}. Explicitly, to give {$\varinjlim f(i) \to X$} is to give a family of morphisms {$f(i) \to X$} such that for any {$i \to j$}, {$f(i) \to X$} is {$f(i) \to f(j) \to X$}. Perhaps the simplest example of a colimit is a '''coequalizer'''. For another example, take ''f'' to be the identity functor on ''C'' and suppose {$L = \varinjlim_{X \in C} f(X)$} exists; then the identity morphism on ''L'' corresponds to a compatible family of morphisms {$\alpha_X: X \to L$} such that {$\alpha_L$} is the identity. If {$f: X \to L$} is any morphism, then {$f = \alpha_L \circ f = \alpha_X$}; i.e., ''L'' is a final object of ''C''.
}}

{{term|1=localization of a category}}
{{defn|1=See '''localization of a category'''.}}
{{glossary end}}

==M==
{{glossary}}
{{term|1=monad}}
{{defn|1=A '''monad (category theory)|
monad''' in a category ''X'' is a '''monoid object''' in the monoidal category of endofunctors of ''X'' with the monoidal structure given by composition. For example, given a group ''G'', define an endofunctor ''T'' on '''Set''' by {$T(X) = G \times X$}. Then define the multiplication ''μ'' on ''T'' as the natural transformation {$\mu: T \circ T \to T$} given by
į:
* The term "'''lax functor'''" is essentially synonymous with "'''pseudo-functor'''".
* An object in an abelian category is said to have <span id="finite length"></span>finite length
if it has a '''composition series'''. The maximum number of proper subobjects in any such composition series is called the '''length''' of ''A''.



'''localization of a category'''

* A '''monad'''
in a category ''X'' is a '''monoid object''' in the monoidal category of endofunctors of ''X'' with the monoidal structure given by composition. For example, given a group ''G'', define an endofunctor ''T'' on '''Set''' by {$T(X) = G \times X$}. Then define the multiplication ''μ'' on ''T'' as the natural transformation {$\mu: T \circ T \to T$} given by
2019 vasario 10 d., 11:29 atliko AndriusKulikauskas -
Pakeista 7 eilutė iš:
*'''Cat''', the '''category of categories|category of (small) categories''', where the objects are categories (which are small with respect to some universe) and the morphisms '''functor'''s.
į:
*'''Cat''', the '''category of (small) categories''', where the objects are categories (which are small with respect to some universe) and the morphisms '''functor'''s.
Pakeistos 21-24 eilutės iš
į:
* An object ''A'' is '''initial''' if there is exactly one morphism from ''A'' to each object; e.g., '''empty set''' in ''''''Category of sets|Set''''''.
* An object ''A'' in an ∞-category ''C'' is initial if {$\operatorname{Map}_C(A, B)$} is '''contractible''' for each object ''B'' in ''C''.
* An object ''A'' in an abelian category is '''injective''' if the functor {$\operatorname{Hom}(-, A)$} is exact. It is the dual of a projective object.
Pakeistos 31-34 eilutės iš
į:
* The '''identity morphism''' ''f'' of an object ''A'' is a morphism from ''A'' to ''A'' such that for any morphisms ''g'' with domain ''A'' and ''h'' with codomain ''A'', {$g\circ f=g$} and {$f\circ h=h$}.
* A morphism ''f'' is an '''inverse''' to a morphism ''g'' if {$g\circ f$} is defined and is equal to the identity morphism on the codomain of ''g'', and {$f\circ g$} is defined and equal to the identity morphism on the domain of ''g''. The inverse of ''g'' is unique and is denoted by ''g''<sup>−1</sup>. ''f'' is a left inverse to ''g'' if {$f\circ g$} is defined and is equal to the identity morphism on the domain of ''g'', and similarly for a right inverse.
* A morphism ''f'' is an '''isomorphism''' if there exists an ''inverse'' of ''f''.
Pakeistos 83-85 eilutės iš
į:
* '''homotopy category'''. It is closely related to a '''localization of a category'''.
* '''1=∞-category''' An '''∞-category''' ''C'' is a '''simplicial set''' satisfying the following condition: for each 0 < ''i'' < ''n'', every map of simplicial sets {$f: \Lambda^n_i \to C$} extends to an ''n''-simplex {$f: \Delta^n \to C$}, where &Delta;<sup>''n''</sup> is the standard ''n''-simplex and {$\Lambda^n_i$} is obtained from &Delta;<sup>''n''</sup> by removing the ''i''-th face and the interior (see '''Kan fibration#Definition'''). For example, the '''nerve of a category''' satisfies the condition and thus can be considered as an ∞-category.
Pakeistos 111-113 eilutės iš
į:
* The '''identity functor''' on a category ''C'' is a functor from ''C'' to ''C'' that sends objects and morphisms to themselves.
* Given a '''monoidal category''' (''C'', ⊗), the '''internal Hom''' is a functor {$[-, -]: C^{\text{op}} \times C \to C$} such that {$[Y, -]$} is the right adjoint to {$- \otimes Y$} for each object ''Y'' in ''C''. For example, the '''category of modules''' over a commutative ring ''R'' has the internal Hom given as {$[M, N] = \operatorname{Hom}_R(M, N)$}, the set of ''R''-linear maps.
Pakeistos 131-133 eilutės iš
į:
* The '''image''' of a morphism ''f'': ''X'' → ''Y'' is the equalizer of {$Y \rightrightarrows Y \sqcup_X Y$}.
* '''ind-limit''' A colimit (or inductive limit) in {$\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}.}}
Pakeistos 165-227 eilutės iš




* Hall algebra of a category. See '''Ringel–Hall algebra'''.



{{term|1=Higher
category theory}}
{{defn|1=
'''Higher category theory''' is a subfield of category theory that concerns the study of '''n-category|''n''-categories''' and '''∞-categories'''.}}

{{term|1=homological
dimension}}
{{defn|1=The
'''homological dimension''' of an abelian category with enough injectives is the least non-negative intege ''n'' such that every object in the category admits an injective resolution of length at most ''n''. The dimension is ∞ if no such integer exists. For example, the homological dimension of '''category of modules|Mod<sub>''R''</sub>''' with a principal ideal domain ''R'' is at most one.}}

{{term|1=homotopy category}}
{{defn|1=See<!-- for now -->
'''homotopy category'''. It is closely related to a '''localization of a category'''.}}

{{term|1=homotopy hypothesis}}
{{defn|1=The
'''homotopy hypothesis''' states an '''∞-groupoid''' is a space (less equivocally, an ''n''-groupoid can be used as a homotopy ''n''-type.)}}
{{glossary end}}

==I==
{{glossary}}
{{term|1=identity}}
{{defn|no=1|1=The
'''identity morphism''' ''f'' of an object ''A'' is a morphism from ''A'' to ''A'' such that for any morphisms ''g'' with domain ''A'' and ''h'' with codomain ''A'', {$g\circ f=g$} and {$f\circ h=h$}.}}
{{defn|no=2|The '''identity functor''' on a category ''C'' is a functor from ''C'' to ''C'' that sends objects and morphisms to themselves.}}
{{defn|no=3|Given a functor ''F'': ''C'' → ''D'', the '''identity natural transformation''' from ''F'' to ''F'' is a natural transformation consisting of the identity morphisms of ''F''(''X'') in ''D'' for the objects ''X'' in ''C''.}}

{{term|1=image}}
{{defn|1=The '''image of a morphism|image''' of a morphism ''f'': ''X'' → ''Y'' is the equalizer of {$Y \rightrightarrows Y \sqcup_X Y$}.}}

{{term|1=ind-limit}}
{{defn|1=A colimit (or inductive limit) in {$\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}.}}

{{term|1=∞-category}}
{{defn|1=An '''∞-category''' ''C'' is a '''simplicial set''' satisfying the following condition: for each 0 < ''i'' < ''n'',
*every map of simplicial sets {$f: \Lambda^n_i \to C$} extends to an ''n''-simplex {$f: \Delta^n \to C$}
where &Delta;<sup>''n''</sup> is the standard ''n''-simplex and {$\Lambda^n_i$} is obtained from &Delta;<sup>''n''</sup> by removing the ''i''-th face and the interior (see '''Kan fibration#Definition'''). For example, the '''nerve of a category''' satisfies the condition and thus can be considered as an ∞-category.}}

{{term|1=initial}}
{{defn|no=1|1=An object ''A'' is '''initial object|initial''' if there is exactly one morphism from ''A'' to each object; e.g., '''empty set''' in ''''''Category of sets|Set''''''.}}
{{defn|no=2|1=An object ''A'' in an ∞-category ''C'' is initial if {$\operatorname{Map}_C(A, B)$} is '''contractible space|contractible''' for each object ''B'' in ''C''.}}

{{term|1=injective}}
{{defn|1=An object ''A'' in an abelian category is '''injective object|injective''' if the functor {$\operatorname{Hom}(-, A)$} is exact. It is the dual of a projective object.}}

{{term|1=internal Hom}}
{{defn|1=Given a '''monoidal category''' (''C'', ⊗), the '''internal Hom''' is a functor {$[-, -]: C^{\text{op}} \times C \to C$} such that {$[Y, -]$} is the right adjoint to {$- \otimes Y$} for each object ''Y'' in ''C''. For example, the '''category of modules''' over a commutative ring ''R'' has the internal Hom given as {$[M, N] = \operatorname{Hom}_R(M, N)$}, the set of ''R''-linear maps.}}

{{term|1=inverse}}
{{defn|1=A morphism ''f'' is an '''inverse function|inverse''' to a morphism ''g'' if {$g\circ f$} is defined and is equal to the identity morphism on the codomain of ''g'', and {$f\circ g$} is defined and equal to the identity morphism on the domain of ''g''. The inverse of ''g'' is unique and is denoted by ''g''<sup>−1</sup>. ''f'' is a left inverse to ''g'' if {$f\circ g$} is defined and is equal to the identity morphism on the domain of ''g'', and similarly for a right inverse.}}

{{term|1=isomorphic}}
{{defn|no=1|1=An object is '''isomorphic''' to another object if there is an isomorphism between them.}}
{{defn|no=2|1=A category is isomorphic to another category if there is an isomorphism between them.}}

{{term|1=isomorphism}}
{{defn|1=A morphism ''f'' is an '''isomorphism''' if there exists an ''inverse'' of ''f''.}}
{{glossary end}}

==K==
{{glossary}}
į:
* '''Hall algebra of a category'''. See '''Ringel–Hall algebra'''.
* '''Higher category theory''' is a subfield of category theory that concerns the study of '''''n''-categories''' and '''∞-categories'''.
* The '''homological dimension''' of an abelian category with enough injectives is the least non-negative integer ''n'' such that every object in the category admits an injective resolution of length at most ''n''. The dimension is ∞ if no such integer exists. For example, the homological dimension of '''Mod<sub>''R''</sub>''' with a principal ideal domain ''R'' is at most one.
* The '''homotopy hypothesis''' states an '''∞-groupoid''' is a space (less equivocally, an ''n''-groupoid can be used as a homotopy ''n''-type.)}}
* Given a functor ''F'': ''C'' → ''D'', the '''identity natural transformation''' from ''F'' to ''F'' is a natural transformation consisting of the identity morphisms of ''F''(''X'') in ''D'' for the objects ''X'' in ''C''.



* An object is
'''isomorphic''' to another object if there is an isomorphism between them.

* A category is
'''isomorphic''' to another category if there is an isomorphism between them.
2019 vasario 10 d., 11:09 atliko AndriusKulikauskas -
Pakeistos 76-77 eilutės iš
į:
* The '''heart''' of a '''t-structure''' ({$D^{\ge 0}$}, {$D^{\le 0}$}) on a triangulated category is the intersection {$D^{\ge 0} \cap D^{\le 0}$}. It is an abelian category.
Pakeistos 158-166 eilutės iš
{{glossary end}}

==H==
{{glossary}}
{{term|1=
Hall algebra of a category}}
{{defn|1=See
'''Ringel–Hall algebra'''.}}

{{term|1=heart}}
{{defn|1=The '''heart (category theory)|heart''' of a '''t-structure''' ({$D^{\ge 0}$}, {$D^{\le 0}$}) on a triangulated category is the intersection {$D^{\ge 0} \cap D^{\le 0}$}. It is an abelian category.}}
į:
* Hall algebra of a category. See '''Ringel–Hall algebra'''.
2019 vasario 10 d., 11:06 atliko AndriusKulikauskas -
Pakeistos 20-21 eilutės iš
į:
* In a category ''C'', a family of objects {$G_i, i \in I$} is a '''system of generators''' of ''C'' if the functor {$X \mapsto \prod_{i \in I} \operatorname{Hom}(G_i, X)$} is conservative. Its dual is called a system of cogenerators.
Pakeistos 73-74 eilutės iš
į:
* A '''Grothendieck category''' is a certain well-behaved kind of an abelian category.
* '''Grothendieck fibration''' A '''fibered category'''.
* A category is called a '''groupoid''' if every morphism in it is an isomorphism. An ∞-category is called an '''∞-groupoid''' if every morphism in it is an equivalence (or equivalently if it is a '''Kan complex'''.
Pakeistos 149-176 eilutės iš


{{glossary end}}

==G==
{{glossary}}
{{term|1=Gabriel–Popescu
theorem}}
{{defn|1=The
'''Gabriel–Popescu theorem''' says an abelian category is a '''Serre quotient category|quotient''' of the category of modules.}}

{{term|1=generator}}
{{defn|1=In a category
''C'', a family of objects {$G_i, i \in I$} is a '''generator (category theory)|system of generators''' of ''C'' if the functor {$X \mapsto \prod_{i \in I} \operatorname{Hom}(G_i, X)$} is conservative. Its dual is called a system of cogenerators.}}

{{term|1=Grothendieck
's Galois theory}}
{{defn|1=A category-theoretic generalization of
'''Galois theory'''; see '''Grothendieck's Galois theory'''.}}

{{term|1=Grothendieck
category}}
{{defn|1=A
'''Grothendieck category''' is a certain well-behaved kind of an abelian category.}}

{{term|1=Grothendieck construction}}
{{defn|1=Given
a functor {$U: C \to \mathbf{Cat}$}, let ''D''<sub>''U''</sub> be the category where the objects are pairs (''x'', ''u'') consisting of an object ''x'' in ''C'' and an object ''u'' in the category ''U''(''x'') and a morphism from (''x'', ''u'') to (''y'', ''v'') is a pair consisting of a morphism ''f'': ''x'' → ''y'' in ''C'' and a morphism ''U''(''f'')(''u'') → ''v'' in ''U''(''y''). The passage from ''U'' to ''D''<sub>''U''</sub> is then called the '''Grothendieck construction'''.}}

{{term|1=Grothendieck fibration}}
{{defn|1=A '''fibered category'''.}}

{{term|1=groupoid}}
{{defn|no=1|1=A category is called a '''groupoid''' if every morphism in it is an isomorphism.}}
{{defn|no=2|1=An ∞-category is called an '''∞-groupoid''' if every morphism in it is an equivalence (or equivalently if it is a '''Kan complex'''.)}}
į:
* The '''Gabriel–Popescu theorem''' says an abelian category is a '''quotient''' of the category of modules.
* '''Grothendieck's Galois theory''' A category-theoretic generalization of '''Galois theory'''; see '''Grothendieck's Galois theory'''.
*
'''Grothendieck construction''' Given a functor {$U: C \to \mathbf{Cat}$}, let ''D''<sub>''U''</sub> be the category where the objects are pairs (''x'', ''u'') consisting of an object ''x'' in ''C'' and an object ''u'' in the category ''U''(''x'') and a morphism from (''x'', ''u'') to (''y'', ''v'') is a pair consisting of a morphism ''f'': ''x'' → ''y'' in ''C'' and a morphism ''U''(''f'')(''u'') → ''v'' in ''U''(''y''). The passage from ''U'' to ''D''<sub>''U''</sub> is then called the '''Grothendieck construction'''.


2019 vasario 10 d., 11:01 atliko AndriusKulikauskas -
Pakeistos 68-73 eilutės iš
į:
* A '''Frobenius category''' is an '''exact category''' that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects.
* '''Fukaya category'''.
* A category ''A'' is a '''full subcategory''' of a category ''B'' if the inclusion functor from ''A'' to ''B'' is full.
* The '''functor category''' '''Fct'''(''C'', ''D'') from a category ''C'' to a category ''D'' is the category where the objects are all the functors from ''C'' to ''D'' and the morphisms are all the natural transformations between the functors.
Ištrintos 147-155 eilutės:

* A '''Frobenius category''' is an '''exact category''' that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects.
* '''Fukaya category'''.
* A category ''A'' is a '''full subcategory''' of a category ''B'' if the inclusion functor from ''A'' to ''B'' is full.



{{term|1=functor category}}
{{defn|1=The '''functor category''' '''Fct'''(''C'', ''D'') from a category ''C'' to a category ''D'' is the category where the objects are all the functors from ''C'' to ''D'' and the morphisms are all the natural transformations between the functors.}}
2019 vasario 10 d., 10:59 atliko AndriusKulikauskas -
Pakeistos 64-68 eilutės iš
į:
* The '''fundamental groupoid''' {$\Pi_1 X$} of a Kan complex ''X'' is the category where an object is a 0-simplex (vertex) {$\Delta^0 \to X$}, a morphism is a homotopy class of a 1-simplex (path) {$\Delta^1 \to X$} and a composition is determined by the Kan property.
* A '''filtered category''' (also called a filtrant category) is a nonempty category with the properties (1) given objects ''i'' and ''j'', there are an object ''k'' and morphisms ''i'' → ''k'' and ''j'' → ''k'' and (2) given morphisms ''u'', ''v'': ''i'' → ''j'', there are an object ''k'' and a morphism ''w'': ''j'' → ''k'' such that ''w'' ∘ ''u'' = ''w'' ∘ ''v''. A category ''I'' is filtered if and only if, for each finite category ''J'' and functor ''f'': ''J'' → ''I'', the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''.
* Given a cardinal number π, a category is said to be π-filtrant if, for each category ''J'' whose set of morphisms has cardinal number strictly less than π, the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''.
* A category is '''finite''' if it has only finitely many morphisms.
Pridėtos 71-74 eilutės:
* Given categories ''C'', ''D'', a '''functor''' ''F'' from ''C'' to ''D'' is a structure-preserving map from ''C'' to ''D''; i.e., it consists of an object ''F''(''x'') in ''D'' for each object ''x'' in ''C'' and a morphism ''F''(''f'') in ''D'' for each morphism ''f'' in ''C'' satisfying the conditions: (1) {$F(f \circ g) = F(f) \circ F(g)$} whenever {$f \circ g$} is defined and (2) {$F(\operatorname{id}_x) = \operatorname{id}_{F(x)}$}. For example,
:{$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}, \, S \mapsto \mathfrak{P}(S)$},
where {$\mathfrak{P}(S)$} is the '''power set''' of ''S'' is a functor if we define: for each function {$f: S \to T$}, {$\mathfrak{P}(f): \mathfrak{P}(S) \to \mathfrak{P}(T)$} by {$\mathfrak{P}(f)(A) = f(A)$}.
Pakeistos 80-81 eilutės iš
* A '''contravariant functor''' ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a '''presheaf (category theory)|presheaf''' especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let {$\mathfrak{P}(S)$} be the power set of ''S'' and for each function {$f: S \to T$}, define:{$\mathfrak{P}(f): \mathfrak{P}(T) \to \mathfrak{P}(S)$}
by sending a subset ''A'' of ''T'' to the pre-image {$f^{-1}(A)$}. With this, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}$} is a contravariant functor.}}
į:
* A '''contravariant functor''' ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a '''presheaf (category theory)|presheaf''' especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let {$\mathfrak{P}(S)$} be the power set of ''S'' and for each function {$f: S \to T$}, define:{$\mathfrak{P}(f): \mathfrak{P}(T) \to \mathfrak{P}(S)$} by sending a subset ''A'' of ''T'' to the pre-image {$f^{-1}(A)$}. With this, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}$} is a contravariant functor.}}
Pakeistos 89-90 eilutės iš
į:
* The '''fundamental category functor''' {$\tau_1: s\mathbf{Set} \to \mathbf{Cat}$} is the left adjoint to the nerve functor ''N''. For every category ''C'', {$\tau_1 NC = C$}.
* A functor π: ''C'' → ''D'' is said to exhibit ''C'' as a '''category fibered over''' ''D'' if, for each morphism ''g'': ''x'' → π(''y'') in ''D'', there exists a π-cartesian morphism ''f'': ''x<nowiki>'</nowiki>'' → ''y'' in ''C'' such that π(''f'') = ''g''. If ''D'' is the category of affine schemes (say of finite type over some field), then π is more commonly called a '''prestack'''. '''Note''': π is often a forgetful functor and in fact the '''Grothendieck construction''' implies that every fibered category can be taken to be that form (up to equivalences in a suitable sense).
* The '''forgetful functor''' is, roughly, a functor that loses some of data of the objects; for example, the functor {$\mathbf{Grp} \to \mathbf{Set}$} that sends a group to its underlying set and a group homomorphism to itself is a forgetful functor.
* A '''free functor''' is a left adjoint to a forgetful functor. For example, for a ring ''R'', the functor that sends a set ''X'' to the '''free module|free ''R''-module''' generated by ''X'' is a free functor (whence the name).
* A functor is '''full''' if it is surjective when restricted to each '''hom-set'''.
Pakeistos 111-112 eilutės iš
į:
* Given a category ''C'' and a set ''I'', the '''fiber product''' over an object ''S'' of a family of objects ''X''<sub>''i''</sub> in ''C'' indexed by ''I'' is the product of the family in the '''slice category''' {$C_{/S}$} of ''C'' over ''S'' (provided there are {$X_i \to S$}). The fiber product of two objects ''X'' and ''Y'' over an object ''S'' is denoted by {$X \times_S Y$} and is also called a '''Cartesian square'''.
Pakeistos 140-184 eilutės iš


{{term|1=fundamental category}}
{{defn|1=The
'''fundamental category functor''' {$\tau_1: s\mathbf{Set} \to \mathbf{Cat}$} is the left adjoint to the nerve functor ''N''. For every category ''C'', {$\tau_1 NC = C$}.}}

{{term|1=fundamental groupoid}}
{{defn|1=The
'''fundamental groupoid''' {$\Pi_1 X$} of a Kan complex ''X'' is the category where an object is a 0-simplex (vertex) {$\Delta^0 \to X$}, a morphism is a homotopy class of a 1-simplex (path) {$\Delta^1 \to X$} and a composition is determined by the Kan property.<!-- check this: Equivalently, it is the groupoid completion of the fundamental category {$\tau_1 X$} of ''X''.-->}}

{{term|1=fibered category}}
{{defn|1=A functor π: ''C'' → ''D'' is said to exhibit ''C'' as a '''fibered category|category fibered over''' ''D'' if, for each morphism ''g'': ''x'' → π(''y'') in ''D'', there exists a π-cartesian morphism ''f'': ''x<nowiki>'</nowiki>'' → ''y'' in ''C'' such that π(''f'') = ''g''. If ''D'' is the category of affine schemes (say of finite type over some field), then π is more commonly called a '''prestack'''. '''Note''': π is often a forgetful functor and in fact the '''Grothendieck construction''' implies that every fibered category can be taken to be that form (up to equivalences in a suitable sense).}}

{{term|1=fiber product}}
{{defn|1=Given a category ''C'' and a set ''I'', the '''fiber product''' over an object ''S'' of a family of objects ''X''<sub>''i''</sub> in ''C'' indexed by ''I'' is the product of the family in the '''slice category''' {$C_{/S}$} of ''C'' over ''S'' (provided there are {$X_i \to S$}). The fiber product of two objects ''X'' and ''Y'' over an object ''S'' is denoted by {$X \times_S Y$} and is also called a '''Cartesian square'''.}}

{{term|1=filtered}}
{{defn|no=1|1=A '''filtered category''' (also called a filtrant category) is a nonempty category with the properties (1) given objects ''i'' and ''j'', there are an object ''k'' and morphisms ''i'' → ''k'' and ''j'' → ''k'' and (2) given morphisms ''u'', ''v'': ''i'' → ''j'', there are an object ''k'' and a morphism ''w'': ''j'' → ''k'' such that ''w'' ∘ ''u'' = ''w'' ∘ ''v''. A category ''I'' is filtered if and only if, for each finite category ''J'' and functor ''f'': ''J'' → ''I'', the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''.}}
{{defn|no=2|1=Given a cardinal number π, a category is said to be π-filtrant if, for each category ''J'' whose set of morphisms has cardinal number strictly less than π, the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''.}}

{{term|1=finitary monad}}
{{defn|1=A '''finitary monad''' or an algebraic monad is a monad on '''Set''' whose underlying endofunctor commutes with filtered colimits.}}

{{term|1=finite}}
{{defn|1=A category is finite if it has only finitely many morphisms.}}

{{term|1=forgetful functor}}
{{defn|The '''forgetful functor''' is, roughly, a functor that loses some of data of the objects; for example, the functor {$\mathbf{Grp} \to \mathbf{Set}$} that sends a group to its underlying set and a group homomorphism to itself is a forgetful functor.}}

{{term|1=free functor}}
{{defn|1=A '''free functor''' is a left adjoint to a forgetful functor. For example, for a ring ''R'', the functor that sends a set ''X'' to the '''free module|free ''R''-module''' generated by ''X'' is a free functor (whence the name).}}

{{term|1=Frobenius category}}
{{defn|1=A '''Frobenius category''' is an '''exact category''' that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects.}}

{{term|1=Fukaya category}}
{{defn|1=See '''Fukaya category'''.}}

{{term|1=full}}
{{defn|no=1|1=A functor is '''full functor|full''' if it is surjective when restricted to each '''hom-set'''.}}
{{defn|no=2|1=A category ''A'' is a '''full subcategory''' of a category ''B'' if the inclusion functor from ''A'' to ''B'' is full.}}

{{term|1=functor}}
{{defn|1=Given categories ''C'', ''D'', a '''functor''' ''F'' from ''C'' to ''D'' is a structure-preserving map from ''C'' to ''D''; i.e., it consists of an object ''F''(''x'') in ''D'' for each object ''x'' in ''C'' and a morphism ''F''(''f'') in ''D'' for each morphism ''f'' in ''C'' satisfying the conditions: (1) {$F(f \circ g) = F(f) \circ F(g)$} whenever {$f \circ g$} is defined and (2) {$F(\operatorname{id}_x) = \operatorname{id}_{F(x)}$}. For example,
:{$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}, \, S \mapsto \mathfrak{P}(S)$},
where {$\mathfrak{P}(S)$} is the '''power set''' of ''S'' is a functor if we define: for each function {$f: S \to T$}, {$\mathfrak{P}(f): \mathfrak{P}(S) \to \mathfrak{P}(T)$} by {$\mathfrak{P}(f)(A) = f(A)$}.}}
į:
* A '''finitary monad''' or an algebraic monad is a monad on '''Set''' whose underlying endofunctor commutes with filtered colimits.



* A
'''Frobenius category''' is an '''exact category''' that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects.
* '''Fukaya category'''.
* A category ''A'' is
a '''full subcategory''' of a category ''B'' if the inclusion functor from ''A'' to ''B'' is full.
2019 vasario 10 d., 10:44 atliko AndriusKulikauskas -
Pakeistos 26-27 eilutės iš
į:
* A morphism ''f'' is an '''epimorphism''' if {$g=h$} whenever {$g\circ f=h\circ f$}. In other words, ''f'' is the dual of a monomorphism.
Pakeistos 63-64 eilutės iš
į:
* The '''empty category''' is a category with no object. It is the same thing as the '''empty set''' when the empty set is viewed as a discrete category.
Pakeistos 77-83 eilutės iš
į:
* A functor is an '''equivalence''' if it is faithful, full and essentially surjective. A morphism in an ∞-category ''C'' is an equivalence if it gives an isomorphism in the homotopy category of ''C''.
* A category is equivalent to another category if there is an '''equivalence''' between them.
* A functor ''F'' is called '''essentially surjective''' (or isomorphism-dense) if for every object ''B'' there exists an object ''A'' such that ''F''(''A'') is isomorphic to ''B''.
* Given categories ''C'', ''D'' and an object ''A'' in ''C'', the '''evaluation''' at ''A'' is the functor:{$\mathbf{Fct}(C, D) \to D, \,\, F \mapsto F(A).$} For example, the '''Eilenberg–Steenrod axioms''' give an instance when the functor is an equivalence.
* A functor is '''faithful''' if it is injective when restricted to each '''hom-set'''.
Pakeistos 99-100 eilutės iš
į:
* The '''equalizer (mathematics)|equalizer''' of a pair of morphisms {$f, g: A \to B$} is the limit of the pair. It is the dual of a coequalizer.
Pakeistos 126-128 eilutės iš
* The '''density theorem (category theory)|density theorem''' states that every presheaf (a set-valued contravariant functor) is a colimit of representable presheaves. Yoneda's lemma embeds a category ''C'' into the category of presheaves on ''C''. The density theorem then says the image is "dense", so to say. The name "density" is because of the analogy with the '''Jacobson density theorem''' (or other variants) in abstract algebra.
į:
* The '''density theorem''' states that every presheaf (a set-valued contravariant functor) is a colimit of representable presheaves. Yoneda's lemma embeds a category ''C'' into the category of presheaves on ''C''. The density theorem then says the image is "dense", so to say. The name "density" is because of the analogy with the '''Jacobson density theorem''' (or other variants) in abstract algebra.
Ištrintos 129-159 eilutės:


{{term|1=empty}}
{{defn|The '''empty category (category theory)|empty category''' is a category with no object. It is the same thing as the '''empty set''' when the empty set is viewed as a discrete category.}}

{{term|1=epimorphism}}
{{defn|1=A morphism ''f'' is an '''epimorphism''' if {$g=h$} whenever {$g\circ f=h\circ f$}. In other words, ''f'' is the dual of a monomorphism.}}

{{term|1=equalizer}}
{{defn|1=The '''equalizer (mathematics)|equalizer''' of a pair of morphisms {$f, g: A \to B$} is the limit of the pair. It is the dual of a coequalizer.}}

{{term|1=equivalence}}
{{defn|no=1|A functor is an '''equivalence of categories|equivalence''' if it is faithful, full and essentially surjective.}}
{{defn|no=2|A morphism in an ∞-category ''C'' is an equivalence if it gives an isomorphism in the homotopy category of ''C''.}}

{{term|1=equivalent}}
{{defn|1=A category is equivalent to another category if there is an '''equivalence of categories|equivalence''' between them.}}

{{term|1=essentially surjective}}
{{defn|1=A functor ''F'' is called '''essentially surjective''' (or isomorphism-dense) if for every object ''B'' there exists an object ''A'' such that ''F''(''A'') is isomorphic to ''B''.}}

{{term|1=evaluation}}
{{defn|1=Given categories ''C'', ''D'' and an object ''A'' in ''C'', the '''evaluation (category theory)|evaluation''' at ''A'' is the functor
:{$\mathbf{Fct}(C, D) \to D, \,\, F \mapsto F(A).$}
For example, the '''Eilenberg–Steenrod axioms''' give an instance when the functor is an equivalence.}}
{{glossary end}}

==F==
{{glossary}}
{{term|1=faithful}}
{{defn|1=A functor is '''faithful functor|faithful''' if it is injective when restricted to each '''hom-set'''.}}
2019 vasario 10 d., 10:30 atliko AndriusKulikauskas -
Pakeistos 53-62 eilutės iš
į:
* Eilenberg–Moore category. Another name for the category of '''algebras for a given monad'''.
* '''enriched category''' Given a monoidal category (''C'', ⊗, 1), a '''category enriched''' over ''C'' is, informally, a category whose Hom sets are in ''C''. More precisely, a category ''D'' enriched over ''C'' is a data consisting of
# A class of objects,
# For each pair of objects ''X'', ''Y'' in ''D'', an object {$\operatorname{Map}_D(X, Y)$} in ''C'', called the '''mapping object''' from ''X'' to ''Y'',
# For each triple of objects ''X'', ''Y'', ''Z'' in ''D'', a morphism in ''C'',
#:{$\circ: \operatorname{Map}_D(Y, Z) \otimes \operatorname{Map}_D(X, Y) \to \operatorname{Map}_D(X, Z)$},
#:called the composition,
#For each object ''X'' in ''D'', a morphism {$1_X: 1 \to \operatorname{Map}_D(X, X)$} in ''C'', called the unit morphism of ''X'' subject to the conditions that (roughly) the compositions are associative and the unit morphisms act as the multiplicative identity.
For example, a category enriched over sets is an ordinary category.
Pakeistos 74-75 eilutės iš
į:
* '''endofunctor.''' A functor between the same category.
Pakeistos 86-91 eilutės iš
į:
* The '''end''' of a functor {$F: C^{\text{op}} \times C \to X$} is the limit
:{$\int_{c \in C} F(c, c) = \varprojlim (F^{\#}: C^{\#} \to X)$}
where {$C^{\#}$} is the category (called the '''subdivision category''' of ''C'') whose objects are symbols {$c^{\#}, u^{\#}$} for all objects ''c'' and all morphisms ''u'' in ''C'' and whose morphisms are {$b^{\#} \to u^{\#}$} and {$u^{\#} \to c^{\#}$} if {$u: b \to c$} and where {$F^{\#}$} is induced by ''F'' so that {$c^{\#}$} would go to {$F(c, c)$} and {$u^{\#}, u: b \to c$} would go to {$F(b, c)$}. For example, for functors {$F, G: C \to X$},
:{$\int_{c \in C} \operatorname{Hom}(F(c), G(c))$}
is the set of natural transformations from ''F'' to ''G''. For more examples, see [http://mathoverflow.net/questions/78471/intuition-for-coends this mathoverflow thread]. The dual of an end is a coend.
Pakeistos 122-144 eilutės iš
* Eilenberg–Moore category. Another name for the category of '''algebras for a given monad'''.

{{term|1=end}}
{{defn|1=The '''end (category theory)|end''' of a functor {$F: C^{\text{op}} \times C \to X$} is the limit
:{$\int_{c \in C} F(c, c) = \varprojlim (F^{\#}: C^{\#} \to X)$}
where {$C^{\#}$} is the category (called the '''subdivision category''' of ''C'') whose objects are symbols {$c^{\#}, u^{\#}$} for all objects ''c'' and all morphisms ''u'' in ''C'' and whose morphisms are {$b^{\#} \to u^{\#}$} and {$u^{\#} \to c^{\#}$} if {$u: b \to c$} and where {$F^{\#}$} is induced by ''F'' so that {$c^{\#}$} would go to {$F(c, c)$} and {$u^{\#}, u: b \to c$} would go to {$F(b, c)$}. For example, for functors {$F, G: C \to X$},
:{$\int_{c \in C} \operatorname{Hom}(F(c), G(c))$}
is the set of natural transformations from ''F'' to ''G''. For more examples, see [http://mathoverflow.net/questions/78471/intuition-for-coends this mathoverflow thread]. The dual of an end is a coend.}}

{{term|1=endofunctor}}
{{defn|1=A functor between the same category.}}

{{term|1=enriched category}}
{{defn|1=Given a monoidal category (''C'', ⊗, 1), a '''category enriched''' over ''C'' is, informally, a category whose Hom sets are in ''C''. More precisely, a category ''D'' enriched over ''C'' is a data consisting of
# A class of objects,
# For each pair of objects ''X'', ''Y'' in ''D'', an object {$\operatorname{Map}_D(X, Y)$} in ''C'', called the '''mapping object''' from ''X'' to ''Y'',
# For each triple of objects ''X'', ''Y'', ''Z'' in ''D'', a morphism in ''C'',
#:{$\circ: \operatorname{Map}_D(Y, Z) \otimes \operatorname{Map}_D(X, Y) \to \operatorname{Map}_D(X, Z)$},
#:called the composition,
#For each object ''X'' in ''D'', a morphism {$1_X: 1 \to \operatorname{Map}_D(X, X)$} in ''C'', called the unit morphism of ''X''
subject to the conditions that (roughly) the compositions are associative and the unit morphisms act as the multiplicative identity.

For example, a category enriched over sets is an ordinary category.}}
į:
2019 vasario 10 d., 10:19 atliko AndriusKulikauskas -
Pakeistos 64-65 eilutės iš
į:
* '''distributor'''. Another term for "profunctor".
Pakeistos 105-119 eilutės iš



{{term|1=distributor}}
{{defn|1=Another term for "profunctor".}}

{{term|1=Dwyer–Kan
equivalence}}
{{defn|1=A
'''Dwyer–Kan equivalence''' is a generalization of an equivalence of categories to the simplicial context.<ref>{{cite arxiv|last=Hinich|first=V.|date=2013-11-17|title=Dwyer-Kan localization revisited|eprint=1311.4128|class=math.QA}}</ref>}}
{{glossary end}}

==E==
{{glossary}}
{{term|1=Eilenberg–Moore
category}}
{{defn|1=Another name
for the category of '''algebra for a monad|algebras for a given monad'''.}}
į:
* A '''Dwyer–Kan equivalence''' is a generalization of an equivalence of categories to the simplicial context.

* Eilenberg–Moore category
. Another name for the category of '''algebras for a given monad'''.
2019 vasario 10 d., 10:14 atliko AndriusKulikauskas -
Pakeistos 51-53 eilutės iš
į:
* A '''differential graded category''' is a category whose Hom sets are equipped with structures of '''differential graded module'''s. In particular, if the category has only one object, it is the same as a differential graded module.
* A category is '''discrete''' if each morphism is an identity morphism (of some object). For example, a set can be viewed as a discrete category.
Pakeistos 74-75 eilutės iš
į:
* Given a category ''C'', a '''diagram''' in ''C'' is a functor {$f: I \to C$} from a small category ''I''.
Pakeistos 105-112 eilutės iš
{{term|1=diagram}}
{{defn|1=Given a category ''C'', a '''diagram (category theory)|diagram''' in ''C'' is a functor {$f: I \to C$} from a small category ''I''.}}

{{term|1=differential graded category}}
{{defn|1=A '''differential graded category''' is a category whose Hom sets are equipped with structures of '''differential graded module'''s. In particular, if the category has only one object, it is the same as a differential graded module.}}

{{term|1=discrete}}
{{defn|1=A category is '''discrete category|discrete''' if each morphism is an identity morphism (of some object). For example, a set can be viewed as a discrete category.}}
į:
2019 vasario 10 d., 10:11 atliko AndriusKulikauskas -
Pakeistos 46-51 eilutės iš
į:
* The '''category of (small) categories''', denoted by '''Cat''', is a category where the objects are all the categories which are small with respect to some fixed universe and the morphisms are all the '''functor'''s.
* Given functors {$f: C \to B, g: D \to B$}, the '''comma category''' {$(f \downarrow g)$} is a category where (1) the objects are morphisms {$f(c) \to g(d)$} and (2) a morphism from {$\alpha: f(c) \to g(d)$} to {$\beta: f(c') \to g(d')$} consists of {$c \to c'$} and {$d \to d'$} such that {$f(c) \to f(c') \overset{\beta}\to g(d')$} is {$f(c) \overset{\alpha}\to g(d) \to g(d').$} For example, if ''f'' is the identity functor and ''g'' is the constant functor with a value ''b'', then it is the slice category of ''B'' over an object ''b''.
* A category is '''complete''' if all small limits exist.
* A '''concrete category''' ''C'' is a category such that there is a faithful functor from ''C'' to ''''''Category of sets|Set''''''; e.g., ''''''category of vector spaces|Vec'''''', ''''''category of groups|Grp'''''' and ''''''category of topological spaces|Top''''''.
* A category is '''connected''' if, for each pair of objects ''x'', ''y'', there exists a finite sequence of objects ''z''<sub>''i''</sub> such that {$z_0 = x, z_n = y$} and either {$\operatorname{Hom}(z_i, z_{i+1})$} or {$\operatorname{Hom}(z_{i+1}, z_i)$} is nonempty for any ''i''.
Pakeistos 57-64 eilutės iš
[+Other+]

* Given a monad ''T'' in a category ''X'', an '''algebra for a algebra for ''T''''' or a ''T''-algebra is an object in ''X'' with a '''monoid action''' of ''T'' ("algebra" is misleading and "''T''-object" is perhaps a better term.) For example, given a group ''G'' that determines a monad ''T'' in '''Set''' in the standard way, a ''T''-algebra is a set with an '''action''' of ''G''.
* '''Beck's theorem''' characterizes the category of
'''algebras for a given monad'''.
* A
'''bicategory''' is a model of a weak '''2-category'''.
* '''Bousfield localization'''
* The '''calculus of functors''' is a technique of studying functors in the manner similar to the way a '''function (mathematics)|function''' is studied via its '''Taylor series''' expansion; whence, the term "calculus".
į:
* A '''conservative functor''' is a functor that reflects isomorphisms. Many forgetful functors are conservative, but the forgetful functor from '''Top''' to '''Set''' is not conservative.
* A functor is
'''constant''' if it maps every object in a category to the same object ''A'' and every morphism to the identity on ''A''. Put in another way, a functor {$f: C \to D$} is constant if it factors as: {$C \to \{ A \} \overset{i}\to D$} for some object ''A'' in ''D'', where ''i'' is the inclusion of the discrete category { ''A'' }.
* A '''contravariant functor''' ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a '''presheaf (category theory)|presheaf''' especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let {$\mathfrak{P}(S)$} be the power set of ''S'' and for each function {$f: S \to T$}, define:{$\mathfrak{P}(f): \mathfrak{P}(T) \to \mathfrak{P}(S)$}
by sending a subset ''A'' of ''T'' to the pre-image {$f^{-1}(A)$}. With this, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}$} is a contravariant functor.}}
* Given categories ''I'', ''C'', the '''diagonal functor''' is the functor:{$\Delta: C \to \mathbf{Fct}(I, C), \, A \mapsto \Delta_A$} that sends each object ''A'' to the constant functor with value ''A'' and each morphism {$f: A \to B$} to the natural transformation {$\Delta_{f, i}: \Delta_A(i) = A \to \Delta_B(i) =B$} that is ''f'' at each ''i''.

[+Types of diagram+]
Pridėtos 66-79 eilutės:
* The '''coequalizer''' of a pair of morphisms {$f, g: A \to B$} is the colimit of the pair. It is the dual of an equalizer.
* The '''coimage''' of a morphism ''f'': ''X'' → ''Y'' is the coequalizer of {$X \times_Y X \rightrightarrows X$}.
* A '''cone (category theory)|cone''' is a way to express the '''universal property''' of a colimit (or dually a limit). One can show<ref>{{harvnb|Kashiwara|Schapira|2006|loc=Ch. 2, Exercise 2.8.}}</ref> that the colimit {$\varinjlim$} is the left adjoint to the diagonal functor {$\Delta: C \to \operatorname{Fct}(I, C)$}, which sends an object ''X'' to the constant functor with value ''X''; that is, for any ''X'' and any functor {$f: I \to C$},
:{$\operatorname{Hom}(\varinjlim f, X) \simeq \operatorname{Hom}(f, \Delta_X),$}
provided the colimit in question exists. The right-hand side is then the set of cones with vertex ''X''.
* The '''coproduct''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the inductive limit {$\varinjlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where ''I'' is viewed as a discrete category. It is the dual of the product of the family. For example, a coproduct in ''''''category of groups|Grp'''''' is a '''free product'''.

[+Other+]

* Given a monad ''T'' in a category ''X'', an '''algebra for a algebra for ''T''''' or a ''T''-algebra is an object in ''X'' with a '''monoid action''' of ''T'' ("algebra" is misleading and "''T''-object" is perhaps a better term.) For example, given a group ''G'' that determines a monad ''T'' in '''Set''' in the standard way, a ''T''-algebra is a set with an '''action''' of ''G''.
* '''Beck's theorem''' characterizes the category of '''algebras for a given monad'''.
* A '''bicategory''' is a model of a weak '''2-category'''.
* '''Bousfield localization'''
* The '''calculus of functors''' is a technique of studying functors in the manner similar to the way a '''function (mathematics)|function''' is studied via its '''Taylor series''' expansion; whence, the term "calculus".
Pakeistos 82-94 eilutės iš

{{term|1=category of categories}}
{{defn|1=The
'''category of categories|category of (small) categories''', denoted by '''Cat''', is a category where the objects are all the categories which are small with respect to some fixed universe and the morphisms are all the '''functor'''s.}}

{{term|1=classifying space}}
{{defn|1=The '''classifying space of
a category''' ''C'' is the geometric realization of the nerve of ''C''.}}

{{term|1=co-}}
{{defn|1=Often used synonymous with op-; for example, a '''colimit''' refers to an op-limit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an op-fibration is not the same thing as a '''cofibration'''.}}

{{term|1=coend}}
{{defn|1=
The coend of a functor {$F: C^{\text{op}} \times C \to X$} is the dual of the '''end (category theory)|end''' of ''F'' and is denoted by
į:
* The '''classifying space of a category''' ''C'' is the geometric realization of the nerve of ''C''.
*
'''co-''' Often used synonymous with op-; for example, a '''colimit''' refers to an op-limit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an op-fibration is not the same thing as a '''cofibration'''.}}
* The coend of a functor {$F: C^{\text{op}} \times C \to X$} is the dual of the '''end (category theory)|end''' of ''F'' and is denoted by
Pakeistos 88-158 eilutės iš
where ''R'' is viewed as a category with one object whose morphisms are the elements of ''R''.}}

{{term|1=coequalizer}}
{{defn|1=The
'''coequalizer''' of a pair of morphisms {$f, g: A \to B$} is the colimit of the pair. It is the dual of an equalizer.}}

{{term|1=coimage}}
{{defn|1=The '''coimage'''
of a morphism ''f'': ''X'' → ''Y'' is the coequalizer of {$X \times_Y X \rightrightarrows X$}.}}

{{term|1=colored operad}}
{{defn|1=Another term for
'''multicategory''', a generalized category where a morphism can have several domains. The notion of "colored operad" is more primitive than that of operad: in fact, an operad can be defined as a colored operad with a single object.}}

{{term|1=comma}}
{{defn|1=Given functors {$f: C
\to B, g: D \to B$}, the '''comma category''' {$(f \downarrow g)$} is a category where (1) the objects are morphisms {$f(c) \to g(d)$} and (2) a morphism from {$\alpha: f(c) \to g(d)$} to {$\beta: f(c') \to g(d')$} consists of {$c \to c'$} and {$d \to d'$} such that {$f(c) \to f(c') \overset{\beta}\to g(d')$} is {$f(c) \overset{\alpha}\to g(d) \to g(d').$} For example, if ''f'' is the identity functor and ''g'' is the constant functor with a value ''b'', then it is the slice category of ''B'' over an object ''b''.
}}

{{term|1=comonad}}
{{defn|1=A
'''comonad''' in a category ''X'' is a '''comonid''' in the monoidal category of endofunctors of ''X''.}}

{{term|1=compact}}
{{defn|1=Probably synonymous with '''#accessible'''
.}}

{{term|1=complete}}
{{defn|1=A category is '''complete category|complete''' if all small limits exist.}}

{{term|1=composition}}
{{defn|no=1|1=A composition of morphisms in a category is part of the datum defining the category
.}}
{{defn|no=2|1=If {$f: C \to D, \, g: D \to E$} are functors, then the composition {$g \circ f$} or {$gf$} is the functor defined by: for an object ''x'' and a morphism ''u'' in ''C'', {$(g \circ f)(x) = g(f(x)), \, (g \circ f)(u) = g(f(u))$}.}}
{{defn|no=3|1=Natural transformations are composed pointwise: if {$\varphi: f \to g, \, \psi: g \to h$} are natural transformations, then {$\psi \circ \varphi$} is the natural transformation given by {$(\psi \circ \varphi)_x = \psi_x \circ \varphi_x$}.}}

{{term|1=concrete}}
{{defn|1=A '''concrete category''' ''C'' is a category such that there is a faithful functor from ''C'' to ''''''Category of sets|Set''''''; e.g., ''''''category of vector spaces|Vec'''''', ''''''category of groups|Grp'''''' and ''''''category of topological spaces|Top''''''.}}

{{term|1=cone}}
{{defn|1=A '''cone (category theory)|cone''' is a way to express the '''universal property''' of a colimit (or dually a limit). One can show<ref>{{harvnb|Kashiwara|Schapira|2006|loc=Ch. 2, Exercise 2.8.}}</ref> that the colimit {$\varinjlim$} is the left adjoint to the diagonal functor {$\Delta: C \to \operatorname{Fct}(I, C)$}, which sends an object ''X'' to the constant functor with value ''X''; that is, for any ''X'' and any functor {$f: I \to C$},
:{$\operatorname{Hom}(\varinjlim f, X) \simeq \operatorname{Hom}(f, \Delta_X),$}
provided the colimit in question exists. The right-hand side is then the set of cones with vertex ''X''.<ref>{{harvnb|Mac Lane|1998|loc=Ch. III, § 3.}}.</ref><!--For example, let {$f: \mathbb{N} \to \mathbf{Set}$} be a functor that maps each {$i \to j$} to an inclusion. Then the cone is a map from the union of {$f(i)$} over all ''i'' to any -->}}

{{term|1=connected}}
{{defn|1=A category is '''connected category|connected''' if, for each pair of objects ''x'', ''y'', there exists a finite sequence of objects ''z''<sub>''i''</sub> such that {$z_0 = x, z_n = y$} and either {$\operatorname{Hom}(z_i, z_{i+1})$} or {$\operatorname{Hom}(z_{i+1}, z_i)$} is nonempty for any ''i''.}}

{{term|1=conservative functor}}
{{defn|1=A '''conservative functor''' is a functor that reflects isomorphisms. Many forgetful functors are conservative, but the forgetful functor from '''Top''' to '''Set''' is not conservative.}}

{{term|1=constant}}
{{defn|1=A functor is '''constant functor|constant''' if it maps every object in a category to the same object ''A'' and every morphism to the identity on ''A''. Put in another way, a functor {$f: C \to D$} is constant if it factors as: {$C \to \{ A \} \overset{i}\to D$} for some object ''A'' in ''D'', where ''i'' is the inclusion of the discrete category { ''A'' }.}}

{{term|1=contravariant functor}}
{{defn|1=A '''contravariant functor''' ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a '''presheaf (category theory)|presheaf''' especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let {$\mathfrak{P}(S)$} be the power set of ''S'' and for each function {$f: S \to T$}, define
:{$\mathfrak{P}(f): \mathfrak{P}(T) \to \mathfrak{P}(S)$}
by sending a subset ''A'' of ''T'' to the pre-image {$f^{-1}(A)$}. With this, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}$} is a contravariant functor.}}

{{term|1=coproduct}}
{{defn|1=The '''coproduct''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the inductive limit {$\varinjlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where ''I'' is viewed as a discrete category. It is the dual of the product of the family. For example, a coproduct in ''''''category of groups|Grp'''''' is a '''free product'''.}}

{{term|1=core}}
{{defn|1=The '''core (category theory)|core''' of a category is the maximal groupoid contained in the category.}}

{{glossary end}}

==D==
{{glossary}}
{{term|1=Day convolution}}
{{defn|Given a group or monoid ''M'', the '''Day convolution''' is the tensor product in {$\mathbf{Fct}(M, \mathbf{Set})$}.<ref>http://ncatlab.org/nlab/show/Day+convolution</ref>}}

{{term|1=density theorem}}
{{defn|1=The '''density theorem (category theory)|density theorem''' states that every presheaf (a set-valued contravariant functor) is a colimit of representable presheaves. Yoneda's lemma embeds a category ''C'' into the category of presheaves on ''C''. The density theorem then says the image is "dense", so to say. The name "density" is because of the analogy with the '''Jacobson density theorem''' (or other variants) in abstract algebra.}}

{{term|1=diagonal functor}}
{{defn|1=Given categories ''I'', ''C'', the '''diagonal functor''' is the functor
:{$\Delta: C \to \mathbf{Fct}(I, C), \, A \mapsto \Delta_A$}
that sends each object ''A'' to the constant functor with value ''A'' and each morphism {$f: A \to B$} to the natural transformation {$\Delta_{f, i}: \Delta_A(i) = A \to \Delta_B(i) =B$} that is ''f'' at each ''i''.}}
į:
where ''R'' is viewed as a category with one object whose morphisms are the elements of ''R''.
* '''colored operad''' Another term for '''multicategory''', a generalized category where a morphism can have several domains. The notion of "colored operad" is more primitive than that of operad: in fact, an operad can be defined as a colored operad with a single object.}}
* A '''comonad''' in a category ''X'' is a '''comonid''' in the monoidal category of endofunctors of ''X''.
* '''compact''' Probably synonymous with '''#accessible'''.

* '''composition''' A composition of morphisms in a category is part of the datum defining the category
.
# If {$f: C \to D, \, g: D \to E$} are functors, then the composition {$g \circ f$} or {$gf$} is the functor defined by: for an object ''x'' and a morphism ''u'' in ''C'', {$(g \circ f)(x) = g(f(x)), \, (g \circ f)(u) = g(f(u))$}.
# Natural transformations are composed pointwise: if
{$\varphi: f \to g, \, \psi: g \to h$} are natural transformations, then {$\psi \circ \varphi$} is the natural transformation given by {$(\psi \circ \varphi)_x = \psi_x \circ \varphi_x$}.
* The
'''core''' of a category is the maximal groupoid contained in the category.
* Given a group or monoid
''M'', the '''Day convolution''' is the tensor product in {$\mathbf{Fct}(M, \mathbf{Set})$}.
* The
'''density theorem (category theory)|density theorem''' states that every presheaf (a set-valued contravariant functor) is a colimit of representable presheaves. Yoneda's lemma embeds a category ''C'' into the category of presheaves on ''C''. The density theorem then says the image is "dense", so to say. The name "density" is because of the analogy with the '''Jacobson density theorem''' (or other variants) in abstract algebra.
2019 vasario 10 d., 09:38 atliko AndriusKulikauskas -
Pakeistos 24-26 eilutės iš
į:
* Given a functor π: ''C'' → ''D'' (e.g., a '''prestack''' over schemes), a morphism ''f'': ''x'' → ''y'' in ''C'' is '''π-cartesian''' if, for each object ''z'' in ''C'', each morphism ''g'': ''z'' → ''y'' in ''C'' and each morphism ''v'': π(''z'') → π(''x'') in ''D'' such that π(''g'') = π(''f'') ∘ ''v'', there exists a unique morphism ''u'': ''z'' → ''x'' such that π(''u'') = ''v'' and ''g'' = ''f'' ∘ ''u''.
* Given a functor π: ''C'' → ''D'' (e.g., a '''prestack''' over rings), a morphism ''f'': ''x'' → ''y'' in ''C'' is '''π-coCartesian''' if, for each object ''z'' in ''C'', each morphism ''g'': ''x'' → ''z'' in ''C'' and each morphism ''v'': π(''y'') → π(''z'') in ''D'' such that π(''g'') = ''v'' ∘ π(''f''), there exists a unique morphism ''u'': ''y'' → ''z'' such that π(''u'') = ''v'' and ''g'' = ''u'' ∘ ''f''. (In short, ''f'' is the dual of a π-cartesian morphism.)
Pridėtos 29-39 eilutės:
* A '''category (mathematics)|category''' consists of the following data
#A class of objects,
#For each pair of objects ''X'', ''Y'', a set {$\operatorname{Hom}(X, Y)$}, whose elements are called morphisms from ''X'' to ''Y'',
#For each triple of objects ''X'', ''Y'', ''Z'', a map (called composition)
#:{$\circ: \operatorname{Hom}(Y, Z) \times \operatorname{Hom}(X, Y) \to \operatorname{Hom}(X, Z), \, (g, f) \mapsto g \circ f$},
#For each object ''X'', an identity morphism {$\operatorname{id}_X \in \operatorname{Hom}(X, X)$}
subject to the conditions: for any morphisms {$f: X \to Y$}, {$g: Y \to Z$} and {$h: Z \to W$},
*{$(h \circ g) \circ f = h \circ (g \circ f)$} and {$\operatorname{id}_Y \circ f = f \circ \operatorname{id}_X = f$}.
For example, a '''partially ordered set''' can be viewed as a category: the objects are the elements of the set and for each pair of objects ''x'', ''y'', there is a unique morphism {$x \to y$} if and only if {$x \le y$}; the associativity of composition means transitivity.}}
Pakeistos 51-52 eilutės iš
į:
* Given relative categories {$p: F \to C, q: G \to C$} over the same base category ''C'', a functor {$f: F \to G$} over ''C'' is cartesian if it sends cartesian morphisms to cartesian morphisms.
Pakeistos 60-87 eilutės iš

{{term|1=cartesian functor}}
{{defn|1=Given relative categories {$p: F \to C, q: G \to C$} over the same base category
''C'', a functor {$f: F \to G$} over ''C'' is cartesian if it sends cartesian morphisms to cartesian morphisms.}}

{{term|1=cartesian morphism}}
{{defn|no=1|1=Given a functor π: ''C'' → ''D'' (e.g., a '''prestack''' over schemes), a morphism ''f'': ''x'' → ''y'' in ''C'' is '''cartesian morphism|π-cartesian''' if, for each object ''z'' in ''C'', each morphism ''g'': ''z'' → ''y'' in ''C'' and each morphism ''v'': π(''z'') → π(''x'') in ''D'' such that π(''g'') = π(''f'') ∘ ''v'', there exists a unique morphism ''u'': ''z'' → ''x'' such that π(''u'') = ''v'' and ''g'' = ''f'' ∘ ''u''
.}}
{{defn|no=2|1=Given a functor π: ''C'' → ''D'' (e.g., a '''prestack''' over rings), a morphism ''f'': ''x'' → ''y'' in ''C'' is '''cartesian morphism|π-coCartesian''' if, for each object ''z'' in ''C'', each morphism ''g'': ''x'' → ''z'' in ''C'' and each morphism ''v'': π(''y'') → π(''z'') in ''D'' such that π(''g'') = ''v'' ∘ π(''f''), there exists a unique morphism ''u'': ''y'' → ''z'' such that π(''u'') = ''v'' and ''g'' = ''u'' ∘ ''f''. (In short, ''f'' is the dual of a π-cartesian morphism.)}}

{{term|1=Cartesian square}}
{{defn|1=A commutative diagram that is isomorphic to the diagram given as a fiber product.<!-- really need a diagram here -->}}

{{term|1=categorical logic}}
{{defn|1='''Categorical logic''' is an approach to '''mathematical logic''' that uses category theory.}}

{{term|1=categorification}}
{{defn|1='''Categorification''' is a process of replacing sets and set-theoretic concepts with categories and category-theoretic concepts in some nontrivial way to capture categoric flavors. Decategorification is the reverse of categorification.}}

{{term|1=category}}
{{defn|1=A '''category (mathematics)|category''' consists of the following data
#A class of objects,
#For each pair of objects ''X'', ''Y'', a set {$\operatorname{Hom}(X, Y)$}, whose elements are called morphisms from ''X'' to ''Y'',
#For each triple of objects ''X'', ''Y'', ''Z'', a map (called composition)
#:{$\circ: \operatorname{Hom}(Y, Z) \times \operatorname{Hom}(X, Y) \to \operatorname{Hom}(X, Z), \, (g, f) \mapsto g \circ f$},
#For each object ''X'', an identity morphism {$\operatorname{id}_X \in \operatorname{Hom}(X, X)$}
subject to the conditions: for any morphisms {$f: X \to Y$}, {$g: Y \to Z$} and {$h: Z \to W$},
*{$(h \circ g) \circ f = h \circ (g \circ f)$} and {$\operatorname{id}_Y \circ f = f \circ \operatorname{id}_X = f$}.
For example, a '''partially ordered set''' can be viewed as a category: the objects are the elements of the set and for each pair of objects ''x'', ''y'', there is a unique morphism {$x \to y$} if and only if {$x \le y$}; the associativity of composition means transitivity.}}
į:
* '''Cartesian square''' A commutative diagram that is isomorphic to the diagram given as a fiber product.
*
'''Categorical logic''' is an approach to '''mathematical logic''' that uses category theory.
* '''Categorification''' is a process of replacing sets and set-theoretic concepts with categories and category-theoretic concepts in some nontrivial way to capture categoric flavors. Decategorification is the reverse of categorification
.
2019 vasario 10 d., 09:33 atliko AndriusKulikauskas -
Pakeistos 17-19 eilutės iš
[+Types of category+]

* A category is '''abelian''' if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.
į:
[+Types of object+]
Pridėtos 20-27 eilutės:

[+Types of morphism+]

* A '''bimorphism''' is a morphism that is both an epimorphism and a monomorphism.

[+Types of category+]

* A category is '''abelian''' if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.
Pakeistos 29-73 eilutės iš
{{term|1=additive}}
{{defn|1=A category is
'''additive category|additive''' if it is preadditive (to be precise, has some pre-additive structure) and admits all finite '''coproduct'''s. Although "preadditive" is an additional structure, one can show "additive" is a ''property'' of a category; i.e., one can ask whether a given category is additive or not.<ref>Remark 2.7. of https://ncatlab.org/nlab/show/additive+category</ref>}}

{{term|1=adjunction}}
{{defn|1=An
'''adjoint functor|adjunction''' (also called an adjoint pair) is a pair of functors ''F'': ''C'' → ''D'', ''G'': ''D'' → ''C'' such that there is a "natural" bijection
:{$\operatorname{Hom}_D (F(X), Y) \simeq \operatorname{Hom}_C (X, G(Y))$};
''F'' is said to be left adjoint to ''G'' and ''G'' to right adjoint to ''F''. Here, "natural" means there is a natural isomorphism {$\operatorname{Hom}_D (F(-), -) \simeq \operatorname{Hom}_C (-, G(-))$} of bifunctors (which are contravariant in the first variable.)}}

{{term|1=algebra for a monad}}
{{defn|1=Given
a monad ''T'' in a category ''X'', an '''algebra for a monad|algebra for ''T''''' or a ''T''-algebra is an object in ''X'' with a '''monoid action''' of ''T'' ("algebra" is misleading and "''T''-object" is perhaps a better term.) For example, given a group ''G'' that determines a monad ''T'' in '''Set''' in the standard way, a ''T''-algebra is a set with an '''group action|action''' of ''G''.}}

{{term|1=amnestic}}
{{defn|1=A functor is amnestic if it has the property: if
''k'' is an isomorphism and ''F''(''k'') is an identity, then ''k'' is an identity.}}
{{glossary end}}

==B==
{{glossary}}
{{term|1=balanced}}
{{defn|1=A category is balanced if every bimorphism is
an isomorphism.}}

{{term|1=Beck
's theorem}}
{{defn|1='''Beck's monadicity theorem|Beck's theorem
''' characterizes the category of '''algebra for a monad|algebras for a given monad'''.}}

{{term|1=bicategory}}
{{defn|1=A
'''bicategory''' is a model of a weak '''2-category'''.}}

{{term|1=bifunctor}}
{{defn|1=A '''bifunctor''' from
a pair of categories ''C'' and ''D'' to a category ''E'' is a functor ''C'' × ''D'' → ''E''. For example, for any category ''C'', {$\operatorname{Hom}(-, -)$} is a bifunctor from ''C''<sup>op</sup> and ''C'' to '''Set'''.}}

{{term|1=bimorphism}}
{{defn|1=A
'''bimorphism''' is a morphism that is both an epimorphism and a monomorphism.}}

{{term|1=Bousfield localization}}
{{defn|1=See '''Bousfield localization
'''.}}

{{glossary end}}

==C==
{{glossary}}
{{term
|1=calculus of functors}}
{{defn|1=The
'''calculus of functors''' is a technique of studying functors in the manner similar to the way a '''function (mathematics)|function''' is studied via its '''Taylor series''' expansion; whence, the term "calculus".}}

{{term|1=cartesian closed}}
{{defn|1=A category is '''Cartesian closed category|cartesian closed''' if it has a terminal object and that any two objects have a product and exponential.}}
į:
* A category is '''additive''' if it is preadditive (to be precise, has some pre-additive structure) and admits all finite '''coproduct'''s. Although "preadditive" is an additional structure, one can show "additive" is a ''property'' of a category; i.e., one can ask whether a given category is additive or not.
* An '''adjunction''' (also called an adjoint pair) is a pair of functors ''F'': ''C'' → ''D'', ''G'': ''D'' → ''C'' such that there is a "natural" bijection:{$\operatorname{Hom}_D (F(X), Y) \simeq \operatorname{Hom}_C (X, G(Y))$};''F'' is said to be left adjoint to ''G'' and ''G'' to right adjoint to ''F''. Here, "natural" means there is a natural isomorphism {$\operatorname{Hom}_D (F(-), -) \simeq \operatorname{Hom}_C (-, G(-))$} of bifunctors (which are contravariant in the first variable.)}}
* A category is '''balanced''' if every bimorphism is an isomorphism.
* A category is '''cartesian closed''' if it has
a terminal object and that any two objects have a product and exponential.

[+Types of functor+]

* A functor
is amnestic if it has the property: if ''k'' is an isomorphism and ''F''(''k'') is an identity, then ''k'' is an identity.
* A
'''bifunctor''' from a pair of categories ''C'' and ''D'' to a category ''E'' is a functor ''C'' × ''D'' → ''E''. For example, for any category ''C'', {$\operatorname{Hom}(-, -)$} is a bifunctor from ''C''<sup>op</sup> and ''C'' to '''Set'''.

[+Other+]

* Given a monad ''T'' in a category ''X'',
an '''algebra for a algebra for ''T''''' or a ''T''-algebra is an object in ''X'' with a '''monoid action''' of ''T'' ("algebra" is misleading and "''T''-object" is perhaps a better term.) For example, given a group ''G'' that determines a monad ''T'' in '''Set''' in the standard way, a ''T''-algebra is a set with an '''action''' of ''G''.
* '''Beck's theorem''' characterizes the category of
'''algebras for a given monad'''.
* A
'''bicategory''' is a model of a weak '''2-category'''.
* '''Bousfield localization'''
* The '''calculus of functors''' is a technique of studying functors in the manner similar to the way a '''function (mathematics)|function''' is studied via its '''Taylor series''' expansion; whence, the term "calculus".
2019 vasario 10 d., 09:18 atliko AndriusKulikauskas -
Ištrintos 4-7 eilutės:
*'''Notes on foundations''': In many expositions (e.g., Vistoli), the set-theoretic issues are ignored; this means, for instance, that one does not distinguish between small and large categories and that one can arbitrarily form a localization of a category.<ref>If one believes in the existence of '''strongly inaccessible cardinal'''s, then there can be a rigorous theory where statements and constructions have references to '''Grothendieck universe'''s.</ref> Like those expositions, this glossary also generally ignores the set-theoretic issues, except when they are relevant (e.g., the discussion on accessibility.)

Especially for higher categories, the concepts from algebraic topology are also used in the category theory. For that see also '''glossary of algebraic topology'''.
Pakeista 6 eilutė iš:
*[''n''] = { 0, 1, 2, …, ''n'' }, which is viewed as a category (by writing <math>i \to j \Leftrightarrow i \le j</math>.)
į:
*[''n''] = { 0, 1, 2, …, ''n'' }, which is viewed as a category (by writing {$i \to j \Leftrightarrow i \le j$}.)
Pakeistos 14-24 eilutės iš
{{Compact ToC|short1|sym=yes|x='''#XYZ|XYZ'''|y=|z=|seealso=yes|refs=yes}}

==A==
{{glossary}}
{{term|1=abelian}}
{{defn|1=A category is '''abelian category|abelian''' if
it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.}}

{{term|1=accessible}}
{{defn|no=1|Given a
'''cardinal number''' κ, an object ''X'' in a category is '''accessible object|κ-accessible''' (or κ-compact or κ-presentable) if <math>\operatorname{Hom}(X, -)</math> commutes with κ-filtered colimits.}}
{{defn|no=2|Given
a '''regular cardinal''' κ, a category is '''accessible category|κ-accessible''' if it has κ-filtered colimits and there exists a small set ''S'' of κ-compact objects that generates the category under colimits, meaning every object can be written as a colimit of diagrams of objects in ''S''.}}
į:
----------------

[+Types of category+]

* A category is
'''abelian''' if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.
* Given a '''cardinal number''' κ, an object
''X'' in a category is '''κ-accessible''' (or κ-compact or κ-presentable) if {$\operatorname{Hom}(X, -)$} commutes with κ-filtered colimits.
* Given a '''regular cardinal'''
κ, a category is '''κ-accessible''' if it has κ-filtered colimits and there exists a small set ''S'' of κ-compact objects that generates the category under colimits, meaning every object can be written as a colimit of diagrams of objects in ''S''.
Pakeistos 28-30 eilutės iš
:<math>\operatorname{Hom}_D (F(X), Y) \simeq \operatorname{Hom}_C (X, G(Y))</math>;
''F'' is said to be left adjoint to ''G'' and ''G'' to right adjoint to ''F''. Here, "natural" means there is a natural isomorphism <math>\operatorname{Hom}_D (F(-), -) \simeq \operatorname{Hom}_C (-, G(-))</math> of bifunctors (which are contravariant in the first variable.)}}
į:
:{$\operatorname{Hom}_D (F(X), Y) \simeq \operatorname{Hom}_C (X, G(Y))$};
''F'' is said to be left adjoint to ''G'' and ''G'' to right adjoint to ''F''. Here, "natural" means there is a natural isomorphism {$\operatorname{Hom}_D (F(-), -) \simeq \operatorname{Hom}_C (-, G(-))$} of bifunctors (which are contravariant in the first variable.)}}
Pakeistos 50-51 eilutės iš
{{defn|1=A '''bifunctor''' from a pair of categories ''C'' and ''D'' to a category ''E'' is a functor ''C'' × ''D'' → ''E''. For example, for any category ''C'', <math>\operatorname{Hom}(-, -)</math> is a bifunctor from ''C''<sup>op</sup> and ''C'' to '''Set'''.}}
į:
{{defn|1=A '''bifunctor''' from a pair of categories ''C'' and ''D'' to a category ''E'' is a functor ''C'' × ''D'' → ''E''. For example, for any category ''C'', {$\operatorname{Hom}(-, -)$} is a bifunctor from ''C''<sup>op</sup> and ''C'' to '''Set'''.}}
Pakeistos 69-70 eilutės iš
{{defn|1=Given relative categories <math>p: F \to C, q: G \to C</math> over the same base category ''C'', a functor <math>f: F \to G</math> over ''C'' is cartesian if it sends cartesian morphisms to cartesian morphisms.}}
į:
{{defn|1=Given relative categories {$p: F \to C, q: G \to C$} over the same base category ''C'', a functor {$f: F \to G$} over ''C'' is cartesian if it sends cartesian morphisms to cartesian morphisms.}}
Pakeista 87 eilutė iš:
#For each pair of objects ''X'', ''Y'', a set <math>\operatorname{Hom}(X, Y)</math>, whose elements are called morphisms from ''X'' to ''Y'',
į:
#For each pair of objects ''X'', ''Y'', a set {$\operatorname{Hom}(X, Y)$}, whose elements are called morphisms from ''X'' to ''Y'',
Pakeistos 89-94 eilutės iš
#:<math>\circ: \operatorname{Hom}(Y, Z) \times \operatorname{Hom}(X, Y) \to \operatorname{Hom}(X, Z), \, (g, f) \mapsto g \circ f</math>,
#For each object ''X'', an identity morphism <math>\operatorname{id}_X \in \operatorname{Hom}(X, X)</math>
subject to the conditions: for any morphisms <math>f: X \to Y</math>, <math>g: Y \to Z</math> and <math>h: Z \to W</math>,
*<math>(h \circ g) \circ f = h \circ (g \circ f)</math> and <math>\operatorname{id}_Y \circ f = f \circ \operatorname{id}_X = f</math>.
For example, a '''partially ordered set''' can be viewed as a category: the objects are the elements of the set and for each pair of objects ''x'', ''y'', there is a unique morphism <math>x \to y</math> if and only if <math>x \le y</math>; the associativity of composition means transitivity.}}
į:
#:{$\circ: \operatorname{Hom}(Y, Z) \times \operatorname{Hom}(X, Y) \to \operatorname{Hom}(X, Z), \, (g, f) \mapsto g \circ f$},
#For each object ''X'', an identity morphism {$\operatorname{id}_X \in \operatorname{Hom}(X, X)$}
subject to the conditions: for any morphisms {$f: X \to Y$}, {$g: Y \to Z$} and {$h: Z \to W$},
*{$(h \circ g) \circ f = h \circ (g \circ f)$} and {$\operatorname{id}_Y \circ f = f \circ \operatorname{id}_X = f$}.
For example, a '''partially ordered set''' can be viewed as a category: the objects are the elements of the set and for each pair of objects ''x'', ''y'', there is a unique morphism {$x \to y$} if and only if {$x \le y$}; the associativity of composition means transitivity.}}
Pakeistos 105-106 eilutės iš
{{defn|1=The coend of a functor <math>F: C^{\text{op}} \times C \to X</math> is the dual of the '''end (category theory)|end''' of ''F'' and is denoted by
:<math>\int^{c \in C} F(c, c)</math>.
į:
{{defn|1=The coend of a functor {$F: C^{\text{op}} \times C \to X$} is the dual of the '''end (category theory)|end''' of ''F'' and is denoted by
:{$\int^{c \in C} F(c, c)$}.
Pakeista 108 eilutė iš:
:<math>M \otimes_R N = \int^{R} M \otimes_{\mathbb{Z}} N</math>
į:
:{$M \otimes_R N = \int^{R} M \otimes_{\mathbb{Z}} N$}
Pakeistos 112-113 eilutės iš
{{defn|1=The '''coequalizer''' of a pair of morphisms <math>f, g: A \to B</math> is the colimit of the pair. It is the dual of an equalizer.}}
į:
{{defn|1=The '''coequalizer''' of a pair of morphisms {$f, g: A \to B$} is the colimit of the pair. It is the dual of an equalizer.}}
Pakeistos 115-116 eilutės iš
{{defn|1=The '''coimage''' of a morphism ''f'': ''X'' → ''Y'' is the coequalizer of <math>X \times_Y X \rightrightarrows X</math>.}}
į:
{{defn|1=The '''coimage''' of a morphism ''f'': ''X'' → ''Y'' is the coequalizer of {$X \times_Y X \rightrightarrows X$}.}}
Pakeista 121 eilutė iš:
{{defn|1=Given functors <math>f: C \to B, g: D \to B</math>, the '''comma category''' <math>(f \downarrow g)</math> is a category where (1) the objects are morphisms <math>f(c) \to g(d)</math> and (2) a morphism from <math>\alpha: f(c) \to g(d)</math> to <math>\beta: f(c') \to g(d')</math> consists of <math>c \to c'</math> and <math>d \to d'</math> such that <math>f(c) \to f(c') \overset{\beta}\to g(d')</math> is <math>f(c) \overset{\alpha}\to g(d) \to g(d').</math> For example, if ''f'' is the identity functor and ''g'' is the constant functor with a value ''b'', then it is the slice category of ''B'' over an object ''b''.
į:
{{defn|1=Given functors {$f: C \to B, g: D \to B$}, the '''comma category''' {$(f \downarrow g)$} is a category where (1) the objects are morphisms {$f(c) \to g(d)$} and (2) a morphism from {$\alpha: f(c) \to g(d)$} to {$\beta: f(c') \to g(d')$} consists of {$c \to c'$} and {$d \to d'$} such that {$f(c) \to f(c') \overset{\beta}\to g(d')$} is {$f(c) \overset{\alpha}\to g(d) \to g(d').$} For example, if ''f'' is the identity functor and ''g'' is the constant functor with a value ''b'', then it is the slice category of ''B'' over an object ''b''.
Pakeistos 135-137 eilutės iš
{{defn|no=2|1=If <math>f: C \to D, \, g: D \to E</math> are functors, then the composition <math>g \circ f</math> or <math>gf</math> is the functor defined by: for an object ''x'' and a morphism ''u'' in ''C'', <math>(g \circ f)(x) = g(f(x)), \, (g \circ f)(u) = g(f(u))</math>.}}
{{defn|no=3|1=Natural transformations are composed pointwise: if <math>\varphi: f \to g, \, \psi: g \to h</math> are natural transformations, then <math>\psi \circ \varphi</math> is the natural transformation given by <math>(\psi \circ \varphi)_x = \psi_x \circ \varphi_x</math>.}}
į:
{{defn|no=2|1=If {$f: C \to D, \, g: D \to E$} are functors, then the composition {$g \circ f$} or {$gf$} is the functor defined by: for an object ''x'' and a morphism ''u'' in ''C'', {$(g \circ f)(x) = g(f(x)), \, (g \circ f)(u) = g(f(u))$}.}}
{{defn|no=3|1=Natural transformations are composed pointwise: if {$\varphi: f \to g, \, \psi: g \to h$} are natural transformations, then {$\psi \circ \varphi$} is the natural transformation given by {$(\psi \circ \varphi)_x = \psi_x \circ \varphi_x$}.}}
Pakeistos 142-145 eilutės iš
{{defn|1=A '''cone (category theory)|cone''' is a way to express the '''universal property''' of a colimit (or dually a limit). One can show<ref>{{harvnb|Kashiwara|Schapira|2006|loc=Ch. 2, Exercise 2.8.}}</ref> that the colimit <math>\varinjlim</math> is the left adjoint to the diagonal functor <math>\Delta: C \to \operatorname{Fct}(I, C)</math>, which sends an object ''X'' to the constant functor with value ''X''; that is, for any ''X'' and any functor <math>f: I \to C</math>,
:<math>\operatorname{Hom}(\varinjlim f, X) \simeq \operatorname{Hom}(f, \Delta_X),</math>
provided the colimit in question exists. The right-hand side is then the set of cones with vertex ''X''.<ref>{{harvnb|Mac Lane|1998|loc=Ch. III, § 3.}}.</ref><!--For example, let <math>f: \mathbb{N} \to \mathbf{Set}</math> be a functor that maps each <math>i \to j</math> to an inclusion. Then the cone is a map from the union of <math>f(i)</math> over all ''i'' to any -->}}
į:
{{defn|1=A '''cone (category theory)|cone''' is a way to express the '''universal property''' of a colimit (or dually a limit). One can show<ref>{{harvnb|Kashiwara|Schapira|2006|loc=Ch. 2, Exercise 2.8.}}</ref> that the colimit {$\varinjlim$} is the left adjoint to the diagonal functor {$\Delta: C \to \operatorname{Fct}(I, C)$}, which sends an object ''X'' to the constant functor with value ''X''; that is, for any ''X'' and any functor {$f: I \to C$},
:{$\operatorname{Hom}(\varinjlim f, X) \simeq \operatorname{Hom}(f, \Delta_X),$}
provided the colimit in question exists. The right-hand side is then the set of cones with vertex ''X''.<ref>{{harvnb|Mac Lane|1998|loc=Ch. III, § 3.}}.</ref><!--For example, let {$f: \mathbb{N} \to \mathbf{Set}$} be a functor that maps each {$i \to j$} to an inclusion. Then the cone is a map from the union of {$f(i)$} over all ''i'' to any -->}}
Pakeistos 147-148 eilutės iš
{{defn|1=A category is '''connected category|connected''' if, for each pair of objects ''x'', ''y'', there exists a finite sequence of objects ''z''<sub>''i''</sub> such that <math>z_0 = x, z_n = y</math> and either <math>\operatorname{Hom}(z_i, z_{i+1})</math> or <math>\operatorname{Hom}(z_{i+1}, z_i)</math> is nonempty for any ''i''.}}
į:
{{defn|1=A category is '''connected category|connected''' if, for each pair of objects ''x'', ''y'', there exists a finite sequence of objects ''z''<sub>''i''</sub> such that {$z_0 = x, z_n = y$} and either {$\operatorname{Hom}(z_i, z_{i+1})$} or {$\operatorname{Hom}(z_{i+1}, z_i)$} is nonempty for any ''i''.}}
Pakeistos 153-154 eilutės iš
{{defn|1=A functor is '''constant functor|constant''' if it maps every object in a category to the same object ''A'' and every morphism to the identity on ''A''. Put in another way, a functor <math>f: C \to D</math> is constant if it factors as: <math>C \to \{ A \} \overset{i}\to D</math> for some object ''A'' in ''D'', where ''i'' is the inclusion of the discrete category { ''A'' }.}}
į:
{{defn|1=A functor is '''constant functor|constant''' if it maps every object in a category to the same object ''A'' and every morphism to the identity on ''A''. Put in another way, a functor {$f: C \to D$} is constant if it factors as: {$C \to \{ A \} \overset{i}\to D$} for some object ''A'' in ''D'', where ''i'' is the inclusion of the discrete category { ''A'' }.}}
Pakeistos 156-159 eilutės iš
{{defn|1=A '''contravariant functor''' ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a '''presheaf (category theory)|presheaf''' especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let <math>\mathfrak{P}(S)</math> be the power set of ''S'' and for each function <math>f: S \to T</math>, define
:<math>\mathfrak{P}(f): \mathfrak{P}(T) \to \mathfrak{P}(S)</math>
by sending a subset ''A'' of ''T'' to the pre-image <math>f^{-1}(A)</math>. With this, <math>\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}</math> is a contravariant functor.}}
į:
{{defn|1=A '''contravariant functor''' ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a '''presheaf (category theory)|presheaf''' especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let {$\mathfrak{P}(S)$} be the power set of ''S'' and for each function {$f: S \to T$}, define
:{$\mathfrak{P}(f): \mathfrak{P}(T) \to \mathfrak{P}(S)$}
by sending a subset ''A'' of ''T'' to the pre-image {$f^{-1}(A)$}. With this, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}$} is a contravariant functor.}}
Pakeistos 161-162 eilutės iš
{{defn|1=The '''coproduct''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the inductive limit <math>\varinjlim</math> of the functor <math>I \to C, \, i \mapsto X_i</math>, where ''I'' is viewed as a discrete category. It is the dual of the product of the family. For example, a coproduct in ''''''category of groups|Grp'''''' is a '''free product'''.}}
į:
{{defn|1=The '''coproduct''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the inductive limit {$\varinjlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where ''I'' is viewed as a discrete category. It is the dual of the product of the family. For example, a coproduct in ''''''category of groups|Grp'''''' is a '''free product'''.}}
Pakeistos 171-172 eilutės iš
{{defn|Given a group or monoid ''M'', the '''Day convolution''' is the tensor product in <math>\mathbf{Fct}(M, \mathbf{Set})</math>.<ref>http://ncatlab.org/nlab/show/Day+convolution</ref>}}
į:
{{defn|Given a group or monoid ''M'', the '''Day convolution''' is the tensor product in {$\mathbf{Fct}(M, \mathbf{Set})$}.<ref>http://ncatlab.org/nlab/show/Day+convolution</ref>}}
Pakeistos 178-180 eilutės iš
:<math>\Delta: C \to \mathbf{Fct}(I, C), \, A \mapsto \Delta_A</math>
that sends each object ''A'' to the constant functor with value ''A'' and each morphism <math>f: A \to B</math> to the natural transformation <math>\Delta_{f, i}: \Delta_A(i) = A \to \Delta_B(i) =B</math> that is ''f'' at each ''i''.}}
į:
:{$\Delta: C \to \mathbf{Fct}(I, C), \, A \mapsto \Delta_A$}
that sends each object ''A'' to the constant functor with value ''A'' and each morphism {$f: A \to B$} to the natural transformation {$\Delta_{f, i}: \Delta_A(i) = A \to \Delta_B(i) =B$} that is ''f'' at each ''i''.}}
Pakeistos 182-183 eilutės iš
{{defn|1=Given a category ''C'', a '''diagram (category theory)|diagram''' in ''C'' is a functor <math>f: I \to C</math> from a small category ''I''.}}
į:
{{defn|1=Given a category ''C'', a '''diagram (category theory)|diagram''' in ''C'' is a functor {$f: I \to C$} from a small category ''I''.}}
Pakeistos 203-206 eilutės iš
{{defn|1=The '''end (category theory)|end''' of a functor <math>F: C^{\text{op}} \times C \to X</math> is the limit
:<math>\int_{c \in C} F(c, c) = \varprojlim (F^{\#}: C^{\#} \to X)</math>
where <math>C^{\#}</math> is the category (called the '''subdivision category''' of ''C'') whose objects are symbols <math>c^{\#}, u^{\#}</math> for all objects ''c'' and all morphisms ''u'' in ''C'' and whose morphisms are <math>b^{\#} \to u^{\#}</math> and <math>u^{\#} \to c^{\#}</math> if <math>u: b \to c</math> and where <math>F^{\#}</math> is induced by ''F'' so that <math>c^{\#}</math> would go to <math>F(c, c)</math> and <math>u^{\#}, u: b \to c</math> would go to <math>F(b, c)</math>. For example, for functors <math>F, G: C \to X</math>,
:<math>\int_{c \in C} \operatorname{Hom}(F(c), G(c))</math>
į:
{{defn|1=The '''end (category theory)|end''' of a functor {$F: C^{\text{op}} \times C \to X$} is the limit
:{$\int_{c \in C} F(c, c) = \varprojlim (F^{\#}: C^{\#} \to X)$}
where {$C^{\#}$} is the category (called the '''subdivision category''' of ''C'') whose objects are symbols {$c^{\#}, u^{\#}$} for all objects ''c'' and all morphisms ''u'' in ''C'' and whose morphisms are {$b^{\#} \to u^{\#}$} and {$u^{\#} \to c^{\#}$} if {$u: b \to c$} and where {$F^{\#}$} is induced by ''F'' so that {$c^{\#}$} would go to {$F(c, c)$} and {$u^{\#}, u: b \to c$} would go to {$F(b, c)$}. For example, for functors {$F, G: C \to X$},
:{$\int_{c \in C} \operatorname{Hom}(F(c), G(c))$}
Pakeista 215 eilutė iš:
# For each pair of objects ''X'', ''Y'' in ''D'', an object <math>\operatorname{Map}_D(X, Y)</math> in ''C'', called the '''mapping object''' from ''X'' to ''Y'',
į:
# For each pair of objects ''X'', ''Y'' in ''D'', an object {$\operatorname{Map}_D(X, Y)$} in ''C'', called the '''mapping object''' from ''X'' to ''Y'',
Pakeista 217 eilutė iš:
#:<math>\circ: \operatorname{Map}_D(Y, Z) \otimes \operatorname{Map}_D(X, Y) \to \operatorname{Map}_D(X, Z)</math>,
į:
#:{$\circ: \operatorname{Map}_D(Y, Z) \otimes \operatorname{Map}_D(X, Y) \to \operatorname{Map}_D(X, Z)$},
Pakeista 219 eilutė iš:
#For each object ''X'' in ''D'', a morphism <math>1_X: 1 \to \operatorname{Map}_D(X, X)</math> in ''C'', called the unit morphism of ''X''
į:
#For each object ''X'' in ''D'', a morphism {$1_X: 1 \to \operatorname{Map}_D(X, X)$} in ''C'', called the unit morphism of ''X''
Pakeistos 228-229 eilutės iš
{{defn|1=A morphism ''f'' is an '''epimorphism''' if <math>g=h</math> whenever <math>g\circ f=h\circ f</math>. In other words, ''f'' is the dual of a monomorphism.}}
į:
{{defn|1=A morphism ''f'' is an '''epimorphism''' if {$g=h$} whenever {$g\circ f=h\circ f$}. In other words, ''f'' is the dual of a monomorphism.}}
Pakeistos 231-232 eilutės iš
{{defn|1=The '''equalizer (mathematics)|equalizer''' of a pair of morphisms <math>f, g: A \to B</math> is the limit of the pair. It is the dual of a coequalizer.}}
į:
{{defn|1=The '''equalizer (mathematics)|equalizer''' of a pair of morphisms {$f, g: A \to B$} is the limit of the pair. It is the dual of a coequalizer.}}
Pakeista 245 eilutė iš:
:<math>\mathbf{Fct}(C, D) \to D, \,\, F \mapsto F(A).</math>
į:
:{$\mathbf{Fct}(C, D) \to D, \,\, F \mapsto F(A).$}
Pakeistos 255-256 eilutės iš
{{defn|1=The '''fundamental category functor''' <math>\tau_1: s\mathbf{Set} \to \mathbf{Cat}</math> is the left adjoint to the nerve functor ''N''. For every category ''C'', <math>\tau_1 NC = C</math>.}}
į:
{{defn|1=The '''fundamental category functor''' {$\tau_1: s\mathbf{Set} \to \mathbf{Cat}$} is the left adjoint to the nerve functor ''N''. For every category ''C'', {$\tau_1 NC = C$}.}}
Pakeistos 258-259 eilutės iš
{{defn|1=The '''fundamental groupoid''' <math>\Pi_1 X</math> of a Kan complex ''X'' is the category where an object is a 0-simplex (vertex) <math>\Delta^0 \to X</math>, a morphism is a homotopy class of a 1-simplex (path) <math>\Delta^1 \to X</math> and a composition is determined by the Kan property.<!-- check this: Equivalently, it is the groupoid completion of the fundamental category <math>\tau_1 X</math> of ''X''.-->}}
į:
{{defn|1=The '''fundamental groupoid''' {$\Pi_1 X$} of a Kan complex ''X'' is the category where an object is a 0-simplex (vertex) {$\Delta^0 \to X$}, a morphism is a homotopy class of a 1-simplex (path) {$\Delta^1 \to X$} and a composition is determined by the Kan property.<!-- check this: Equivalently, it is the groupoid completion of the fundamental category {$\tau_1 X$} of ''X''.-->}}
Pakeistos 264-265 eilutės iš
{{defn|1=Given a category ''C'' and a set ''I'', the '''fiber product''' over an object ''S'' of a family of objects ''X''<sub>''i''</sub> in ''C'' indexed by ''I'' is the product of the family in the '''slice category''' <math>C_{/S}</math> of ''C'' over ''S'' (provided there are <math>X_i \to S</math>). The fiber product of two objects ''X'' and ''Y'' over an object ''S'' is denoted by <math>X \times_S Y</math> and is also called a '''Cartesian square'''.}}
į:
{{defn|1=Given a category ''C'' and a set ''I'', the '''fiber product''' over an object ''S'' of a family of objects ''X''<sub>''i''</sub> in ''C'' indexed by ''I'' is the product of the family in the '''slice category''' {$C_{/S}$} of ''C'' over ''S'' (provided there are {$X_i \to S$}). The fiber product of two objects ''X'' and ''Y'' over an object ''S'' is denoted by {$X \times_S Y$} and is also called a '''Cartesian square'''.}}
Pakeistos 267-269 eilutės iš
{{defn|no=1|1=A '''filtered category''' (also called a filtrant category) is a nonempty category with the properties (1) given objects ''i'' and ''j'', there are an object ''k'' and morphisms ''i'' → ''k'' and ''j'' → ''k'' and (2) given morphisms ''u'', ''v'': ''i'' → ''j'', there are an object ''k'' and a morphism ''w'': ''j'' → ''k'' such that ''w'' ∘ ''u'' = ''w'' ∘ ''v''. A category ''I'' is filtered if and only if, for each finite category ''J'' and functor ''f'': ''J'' → ''I'', the set <math>\varprojlim \operatorname{Hom}(f(j), i)</math> is nonempty for some object ''i'' in ''I''.}}
{{defn|no=2|1=Given a cardinal number π, a category is said to be π-filtrant if, for each category ''J'' whose set of morphisms has cardinal number strictly less than π, the set <math>\varprojlim \operatorname{Hom}(f(j), i)</math> is nonempty for some object ''i'' in ''I''.}}
į:
{{defn|no=1|1=A '''filtered category''' (also called a filtrant category) is a nonempty category with the properties (1) given objects ''i'' and ''j'', there are an object ''k'' and morphisms ''i'' → ''k'' and ''j'' → ''k'' and (2) given morphisms ''u'', ''v'': ''i'' → ''j'', there are an object ''k'' and a morphism ''w'': ''j'' → ''k'' such that ''w'' ∘ ''u'' = ''w'' ∘ ''v''. A category ''I'' is filtered if and only if, for each finite category ''J'' and functor ''f'': ''J'' → ''I'', the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''.}}
{{defn|no=2|1=Given a cardinal number π, a category is said to be π-filtrant if, for each category ''J'' whose set of morphisms has cardinal number strictly less than π, the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''.}}
Pakeistos 277-278 eilutės iš
{{defn|The '''forgetful functor''' is, roughly, a functor that loses some of data of the objects; for example, the functor <math>\mathbf{Grp} \to \mathbf{Set}</math> that sends a group to its underlying set and a group homomorphism to itself is a forgetful functor.}}
į:
{{defn|The '''forgetful functor''' is, roughly, a functor that loses some of data of the objects; for example, the functor {$\mathbf{Grp} \to \mathbf{Set}$} that sends a group to its underlying set and a group homomorphism to itself is a forgetful functor.}}
Pakeistos 293-296 eilutės iš
{{defn|1=Given categories ''C'', ''D'', a '''functor''' ''F'' from ''C'' to ''D'' is a structure-preserving map from ''C'' to ''D''; i.e., it consists of an object ''F''(''x'') in ''D'' for each object ''x'' in ''C'' and a morphism ''F''(''f'') in ''D'' for each morphism ''f'' in ''C'' satisfying the conditions: (1) <math>F(f \circ g) = F(f) \circ F(g)</math> whenever <math>f \circ g</math> is defined and (2) <math>F(\operatorname{id}_x) = \operatorname{id}_{F(x)}</math>. For example,
:<math>\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}, \, S \mapsto \mathfrak{P}(S)</math>,
where <math>\mathfrak{P}(S)</math> is the '''power set''' of ''S'' is a functor if we define: for each function <math>f: S \to T</math>, <math>\mathfrak{P}(f): \mathfrak{P}(S) \to \mathfrak{P}(T)</math> by <math>\mathfrak{P}(f)(A) = f(A)</math>.}}
į:
{{defn|1=Given categories ''C'', ''D'', a '''functor''' ''F'' from ''C'' to ''D'' is a structure-preserving map from ''C'' to ''D''; i.e., it consists of an object ''F''(''x'') in ''D'' for each object ''x'' in ''C'' and a morphism ''F''(''f'') in ''D'' for each morphism ''f'' in ''C'' satisfying the conditions: (1) {$F(f \circ g) = F(f) \circ F(g)$} whenever {$f \circ g$} is defined and (2) {$F(\operatorname{id}_x) = \operatorname{id}_{F(x)}$}. For example,
:{$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}, \, S \mapsto \mathfrak{P}(S)$},
where {$\mathfrak{P}(S)$} is the '''power set''' of ''S'' is a functor if we define: for each function {$f: S \to T$}, {$\mathfrak{P}(f): \mathfrak{P}(S) \to \mathfrak{P}(T)$} by {$\mathfrak{P}(f)(A) = f(A)$}.}}
Pakeistos 308-309 eilutės iš
{{defn|1=In a category ''C'', a family of objects <math>G_i, i \in I</math> is a '''generator (category theory)|system of generators''' of ''C'' if the functor <math>X \mapsto \prod_{i \in I} \operatorname{Hom}(G_i, X)</math> is conservative. Its dual is called a system of cogenerators.}}
į:
{{defn|1=In a category ''C'', a family of objects {$G_i, i \in I$} is a '''generator (category theory)|system of generators''' of ''C'' if the functor {$X \mapsto \prod_{i \in I} \operatorname{Hom}(G_i, X)$} is conservative. Its dual is called a system of cogenerators.}}
Pakeistos 317-318 eilutės iš
{{defn|1=Given a functor <math>U: C \to \mathbf{Cat}</math>, let ''D''<sub>''U''</sub> be the category where the objects are pairs (''x'', ''u'') consisting of an object ''x'' in ''C'' and an object ''u'' in the category ''U''(''x'') and a morphism from (''x'', ''u'') to (''y'', ''v'') is a pair consisting of a morphism ''f'': ''x'' → ''y'' in ''C'' and a morphism ''U''(''f'')(''u'') → ''v'' in ''U''(''y''). The passage from ''U'' to ''D''<sub>''U''</sub> is then called the '''Grothendieck construction'''.}}
į:
{{defn|1=Given a functor {$U: C \to \mathbf{Cat}$}, let ''D''<sub>''U''</sub> be the category where the objects are pairs (''x'', ''u'') consisting of an object ''x'' in ''C'' and an object ''u'' in the category ''U''(''x'') and a morphism from (''x'', ''u'') to (''y'', ''v'') is a pair consisting of a morphism ''f'': ''x'' → ''y'' in ''C'' and a morphism ''U''(''f'')(''u'') → ''v'' in ''U''(''y''). The passage from ''U'' to ''D''<sub>''U''</sub> is then called the '''Grothendieck construction'''.}}
Pakeistos 334-335 eilutės iš
{{defn|1=The '''heart (category theory)|heart''' of a '''t-structure''' (<math>D^{\ge 0}</math>, <math>D^{\le 0}</math>) on a triangulated category is the intersection <math>D^{\ge 0} \cap D^{\le 0}</math>. It is an abelian category.}}
į:
{{defn|1=The '''heart (category theory)|heart''' of a '''t-structure''' ({$D^{\ge 0}$}, {$D^{\le 0}$}) on a triangulated category is the intersection {$D^{\ge 0} \cap D^{\le 0}$}. It is an abelian category.}}
Pakeista 352 eilutė iš:
{{defn|no=1|1=The '''identity morphism''' ''f'' of an object ''A'' is a morphism from ''A'' to ''A'' such that for any morphisms ''g'' with domain ''A'' and ''h'' with codomain ''A'', <math>g\circ f=g</math> and <math>f\circ h=h</math>.}}
į:
{{defn|no=1|1=The '''identity morphism''' ''f'' of an object ''A'' is a morphism from ''A'' to ''A'' such that for any morphisms ''g'' with domain ''A'' and ''h'' with codomain ''A'', {$g\circ f=g$} and {$f\circ h=h$}.}}
Pakeistos 357-358 eilutės iš
{{defn|1=The '''image of a morphism|image''' of a morphism ''f'': ''X'' → ''Y'' is the equalizer of <math>Y \rightrightarrows Y \sqcup_X Y</math>.}}
į:
{{defn|1=The '''image of a morphism|image''' of a morphism ''f'': ''X'' → ''Y'' is the equalizer of {$Y \rightrightarrows Y \sqcup_X Y$}.}}
Pakeistos 360-361 eilutės iš
{{defn|1=A colimit (or inductive limit) in <math>\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})</math>.}}
į:
{{defn|1=A colimit (or inductive limit) in {$\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}.}}
Pakeistos 364-366 eilutės iš
*every map of simplicial sets <math>f: \Lambda^n_i \to C</math> extends to an ''n''-simplex <math>f: \Delta^n \to C</math>
where &Delta;<sup>''n''</sup> is the standard ''n''-simplex and <math>\Lambda^n_i</math> is obtained from &Delta;<sup>''n''</sup> by removing the ''i''-th face and the interior (see '''Kan fibration#Definition'''). For example, the '''nerve of a category''' satisfies the condition and thus can be considered as an ∞-category.}}
į:
*every map of simplicial sets {$f: \Lambda^n_i \to C$} extends to an ''n''-simplex {$f: \Delta^n \to C$}
where &Delta;<sup>''n''</sup> is the standard ''n''-simplex and {$\Lambda^n_i$} is obtained from &Delta;<sup>''n''</sup> by removing the ''i''-th face and the interior (see '''Kan fibration#Definition'''). For example, the '''nerve of a category''' satisfies the condition and thus can be considered as an ∞-category.}}
Pakeistos 369-370 eilutės iš
{{defn|no=2|1=An object ''A'' in an ∞-category ''C'' is initial if <math>\operatorname{Map}_C(A, B)</math> is '''contractible space|contractible''' for each object ''B'' in ''C''.}}
į:
{{defn|no=2|1=An object ''A'' in an ∞-category ''C'' is initial if {$\operatorname{Map}_C(A, B)$} is '''contractible space|contractible''' for each object ''B'' in ''C''.}}
Pakeistos 372-373 eilutės iš
{{defn|1=An object ''A'' in an abelian category is '''injective object|injective''' if the functor <math>\operatorname{Hom}(-, A)</math> is exact. It is the dual of a projective object.}}
į:
{{defn|1=An object ''A'' in an abelian category is '''injective object|injective''' if the functor {$\operatorname{Hom}(-, A)$} is exact. It is the dual of a projective object.}}
Pakeistos 375-376 eilutės iš
{{defn|1=Given a '''monoidal category''' (''C'', ⊗), the '''internal Hom''' is a functor <math>[-, -]: C^{\text{op}} \times C \to C</math> such that <math>[Y, -]</math> is the right adjoint to <math>- \otimes Y</math> for each object ''Y'' in ''C''. For example, the '''category of modules''' over a commutative ring ''R'' has the internal Hom given as <math>[M, N] = \operatorname{Hom}_R(M, N)</math>, the set of ''R''-linear maps.}}
į:
{{defn|1=Given a '''monoidal category''' (''C'', ⊗), the '''internal Hom''' is a functor {$[-, -]: C^{\text{op}} \times C \to C$} such that {$[Y, -]$} is the right adjoint to {$- \otimes Y$} for each object ''Y'' in ''C''. For example, the '''category of modules''' over a commutative ring ''R'' has the internal Hom given as {$[M, N] = \operatorname{Hom}_R(M, N)$}, the set of ''R''-linear maps.}}
Pakeistos 378-379 eilutės iš
{{defn|1=A morphism ''f'' is an '''inverse function|inverse''' to a morphism ''g'' if <math>g\circ f</math> is defined and is equal to the identity morphism on the codomain of ''g'', and <math>f\circ g</math> is defined and equal to the identity morphism on the domain of ''g''. The inverse of ''g'' is unique and is denoted by ''g''<sup>−1</sup>. ''f'' is a left inverse to ''g'' if <math>f\circ g</math> is defined and is equal to the identity morphism on the domain of ''g'', and similarly for a right inverse.}}
į:
{{defn|1=A morphism ''f'' is an '''inverse function|inverse''' to a morphism ''g'' if {$g\circ f$} is defined and is equal to the identity morphism on the codomain of ''g'', and {$f\circ g$} is defined and equal to the identity morphism on the domain of ''g''. The inverse of ''g'' is unique and is denoted by ''g''<sup>−1</sup>. ''f'' is a left inverse to ''g'' if {$f\circ g$} is defined and is equal to the identity morphism on the domain of ''g'', and similarly for a right inverse.}}
Pakeistos 394-398 eilutės iš
{{defn|no=1|Given a category ''C'', the left '''Kan extension''' functor along a functor <math>f: I \to J</math> is the left adjoint (if it exists) to <math>f^* = - \circ f: \operatorname{Fct}(J, C) \to \operatorname{Fct}(I, C)</math> and is denoted by <math>f_!</math>. For any <math>\alpha: I \to C</math>, the functor <math>f_! \alpha: J \to C</math> is called the left Kan extension of α along ''f''.<ref>http://www.math.harvard.edu/~lurie/282ynotes/LectureXI-Homological.pdf</ref> One can show:
:<math>(f_! \alpha)(j) = \varinjlim_{f(i) \to j} \alpha(i)</math>
where the colimit runs over all objects <math>f(i) \to j</math> in the comma category.}}
{{defn|no=2|The right Kan extension functor is the right adjoint (if it exists) to <math>f^*</math>.}}
į:
{{defn|no=1|Given a category ''C'', the left '''Kan extension''' functor along a functor {$f: I \to J$} is the left adjoint (if it exists) to {$f^* = - \circ f: \operatorname{Fct}(J, C) \to \operatorname{Fct}(I, C)$} and is denoted by {$f_!$}. For any {$\alpha: I \to C$}, the functor {$f_! \alpha: J \to C$} is called the left Kan extension of α along ''f''.<ref>http://www.math.harvard.edu/~lurie/282ynotes/LectureXI-Homological.pdf</ref> One can show:
:{$(f_! \alpha)(j) = \varinjlim_{f(i) \to j} \alpha(i)$}
where the colimit runs over all objects {$f(i) \to j$} in the comma category.}}
{{defn|no=2|The right Kan extension functor is the right adjoint (if it exists) to {$f^*$}.}}
Pakeistos 412-415 eilutės iš
{{defn|no=1|The '''limit (category theory)|limit''' (or '''projective limit''') of a functor <math>f: I^{\text{op}} \to \mathbf{Set}</math> is
::<math>\varprojlim_{i \in I} f(i) = \{ (x_i|i) \in \prod_{i} f(i) | f(s)(x_j) = x_i \text{ for any } s: i \to j \}.</math>}}
{{defn|no=2|The limit <math>\varprojlim_{i \in I} f(i)</math> of a functor <math>f: I^{\text{op}} \to C</math> is an object, if any, in ''C'' that satisfies: for any object ''X'' in ''C'', <math>\operatorname{Hom}(X, \varprojlim_{i \in I} f(i)) = \varprojlim_{i \in I} \operatorname{Hom}(X, f(i))</math>; i.e., it is an object representing the functor <math>X \mapsto \varprojlim_i \operatorname{Hom}(X, f(i)).</math>}}
{{defn|no=3|The '''colimit''' (or '''inductive limit''') <math>\varinjlim_{i \in I} f(i)</math> is the dual of a limit; i.e., given a functor <math>f: I \to C</math>, it satisfies: for any ''X'', <math>\operatorname{Hom}(\varinjlim f(i), X) = \varprojlim \operatorname{Hom}(f(i), X)</math>. Explicitly, to give <math>\varinjlim f(i) \to X</math> is to give a family of morphisms <math>f(i) \to X</math> such that for any <math>i \to j</math>, <math>f(i) \to X</math> is <math>f(i) \to f(j) \to X</math>. Perhaps the simplest example of a colimit is a '''coequalizer'''. For another example, take ''f'' to be the identity functor on ''C'' and suppose <math>L = \varinjlim_{X \in C} f(X)</math> exists; then the identity morphism on ''L'' corresponds to a compatible family of morphisms <math>\alpha_X: X \to L</math> such that <math>\alpha_L</math> is the identity. If <math>f: X \to L</math> is any morphism, then <math>f = \alpha_L \circ f = \alpha_X</math>; i.e., ''L'' is a final object of ''C''.
į:
{{defn|no=1|The '''limit (category theory)|limit''' (or '''projective limit''') of a functor {$f: I^{\text{op}} \to \mathbf{Set}$} is
::{$\varprojlim_{i \in I} f(i) = \{ (x_i|i) \in \prod_{i} f(i) | f(s)(x_j) = x_i \text{ for any } s: i \to j \}.$}}}
{{defn|no=2|The limit {$\varprojlim_{i \in I} f(i)$} of a functor {$f: I^{\text{op}} \to C$} is an object, if any, in ''C'' that satisfies: for any object ''X'' in ''C'', {$\operatorname{Hom}(X, \varprojlim_{i \in I} f(i)) = \varprojlim_{i \in I} \operatorname{Hom}(X, f(i))$}; i.e., it is an object representing the functor {$X \mapsto \varprojlim_i \operatorname{Hom}(X, f(i)).$}}}
{{defn|no=3|The '''colimit''' (or '''inductive limit''') {$\varinjlim_{i \in I} f(i)$} is the dual of a limit; i.e., given a functor {$f: I \to C$}, it satisfies: for any ''X'', {$\operatorname{Hom}(\varinjlim f(i), X) = \varprojlim \operatorname{Hom}(f(i), X)$}. Explicitly, to give {$\varinjlim f(i) \to X$} is to give a family of morphisms {$f(i) \to X$} such that for any {$i \to j$}, {$f(i) \to X$} is {$f(i) \to f(j) \to X$}. Perhaps the simplest example of a colimit is a '''coequalizer'''. For another example, take ''f'' to be the identity functor on ''C'' and suppose {$L = \varinjlim_{X \in C} f(X)$} exists; then the identity morphism on ''L'' corresponds to a compatible family of morphisms {$\alpha_X: X \to L$} such that {$\alpha_L$} is the identity. If {$f: X \to L$} is any morphism, then {$f = \alpha_L \circ f = \alpha_X$}; i.e., ''L'' is a final object of ''C''.
Pakeistos 425-428 eilutės iš
{{defn|1=A '''monad (category theory)|monad''' in a category ''X'' is a '''monoid object''' in the monoidal category of endofunctors of ''X'' with the monoidal structure given by composition. For example, given a group ''G'', define an endofunctor ''T'' on '''Set''' by <math>T(X) = G \times X</math>. Then define the multiplication ''μ'' on ''T'' as the natural transformation <math>\mu: T \circ T \to T</math> given by
:<math>\mu_X: G \times (G \times X) \to G \times X, \,\, (g, (h, x)) \mapsto (gh, x)</math>
and also define the identity map ''η'' in the analogous fashion. Then (''T'', ''μ'', ''η'') constitutes a monad in '''Set'''. More substantially, an adjunction between functors <math>F: X \rightleftarrows A : G</math> determines a monad in ''X''; namely, one takes <math>T = G \circ F</math>, the identity map ''η'' on ''T'' to be a unit of the adjunction and also defines ''μ'' using the adjunction.}}
į:
{{defn|1=A '''monad (category theory)|monad''' in a category ''X'' is a '''monoid object''' in the monoidal category of endofunctors of ''X'' with the monoidal structure given by composition. For example, given a group ''G'', define an endofunctor ''T'' on '''Set''' by {$T(X) = G \times X$}. Then define the multiplication ''μ'' on ''T'' as the natural transformation {$\mu: T \circ T \to T$} given by
:{$\mu_X: G \times (G \times X) \to G \times X, \,\, (g, (h, x)) \mapsto (gh, x)$}
and also define the identity map ''η'' in the analogous fashion. Then (''T'', ''μ'', ''η'') constitutes a monad in '''Set'''. More substantially, an adjunction between functors {$F: X \rightleftarrows A : G$} determines a monad in ''X''; namely, one takes {$T = G \circ F$}, the identity map ''η'' on ''T'' to be a unit of the adjunction and also defines ''μ'' using the adjunction.}}
Pakeistos 434-435 eilutės iš
{{defn|1=A '''monoidal category''', also called a tensor category, is a category ''C'' equipped with (1) a '''bifunctor''' <math>\otimes: C \times C \to C</math>, (2) an identity object and (3) natural isomorphisms that make ⊗ associative and the identity object an identity for ⊗, subject to certain coherence conditions.}}
į:
{{defn|1=A '''monoidal category''', also called a tensor category, is a category ''C'' equipped with (1) a '''bifunctor''' {$\otimes: C \times C \to C$}, (2) an identity object and (3) natural isomorphisms that make ⊗ associative and the identity object an identity for ⊗, subject to certain coherence conditions.}}
Pakeistos 440-441 eilutės iš
{{defn|1=A morphism ''f'' is a '''monomorphism''' (also called monic) if <math>g=h</math> whenever <math>f\circ g=f\circ h</math>; e.g., an '''Injective function|injection''' in ''''''Category of sets|Set''''''. In other words, ''f'' is the dual of an epimorphism.}}
į:
{{defn|1=A morphism ''f'' is a '''monomorphism''' (also called monic) if {$g=h$} whenever {$f\circ g=f\circ h$}; e.g., an '''Injective function|injection''' in ''''''Category of sets|Set''''''. In other words, ''f'' is the dual of an epimorphism.}}
Pakeistos 462-463 eilutės iš
:<math>\{ \phi_x: F(x) \to G(x) \mid x \in \operatorname{Ob}(C) \}</math>
satisfying the condition: for each morphism ''f'': ''x'' → ''y'' in ''C'', <math>\phi_y \circ F(f) = G(f) \circ \phi_x</math>. For example, writing <math>GL_n(R)</math> for the group of invertible ''n''-by-''n'' matrices with coefficients in a commutative ring ''R'', we can view <math>GL_n</math> as a functor from the category '''CRing''' of commutative rings to the category '''Grp''' of groups. Similarly, <math>R \mapsto R^*</math> is a functor from '''CRing''' to '''Grp'''. Then the '''determinant''' det is a natural transformation from <math>GL_n</math> to -<sup>*</sup>.}}
į:
:{$\{ \phi_x: F(x) \to G(x) \mid x \in \operatorname{Ob}(C) \}$}
satisfying the condition: for each morphism ''f'': ''x'' → ''y'' in ''C'', {$\phi_y \circ F(f) = G(f) \circ \phi_x$}. For example, writing {$GL_n(R)$} for the group of invertible ''n''-by-''n'' matrices with coefficients in a commutative ring ''R'', we can view {$GL_n$} as a functor from the category '''CRing''' of commutative rings to the category '''Grp''' of groups. Similarly, {$R \mapsto R^*$} is a functor from '''CRing''' to '''Grp'''. Then the '''determinant''' det is a natural transformation from {$GL_n$} to -<sup>*</sup>.}}
Pakeistos 468-469 eilutės iš
{{defn|1=The '''nerve functor''' ''N'' is the functor from '''Cat''' to ''s'''''Set''' given by <math>N(C)_n = \operatorname{Hom}_{\mathbf{Cat}}([n], C)</math>. For example, if <math>\varphi</math> is a functor in <math>N(C)_2</math> (called a 2-simplex), let <math>x_i = \varphi(i), \, 0 \le i \le 2</math>. Then <math>\varphi(0 \to 1)</math> is a morphism <math>f: x_0 \to x_1</math> in ''C'' and also <math>\varphi(1 \to 2) = g: x_1 \to x_2</math> for some ''g'' in ''C''. Since <math>0 \to 2</math> is <math>0 \to 1</math> followed by <math>1 \to 2</math> and since <math>\varphi</math> is a functor, <math>\varphi(0 \to 2) = g \circ f</math>. In other words, <math>\varphi</math> encodes ''f'', ''g'' and their compositions.}}
į:
{{defn|1=The '''nerve functor''' ''N'' is the functor from '''Cat''' to ''s'''''Set''' given by {$N(C)_n = \operatorname{Hom}_{\mathbf{Cat}}([n], C)$}. For example, if {$\varphi$} is a functor in {$N(C)_2$} (called a 2-simplex), let {$x_i = \varphi(i), \, 0 \le i \le 2$}. Then {$\varphi(0 \to 1)$} is a morphism {$f: x_0 \to x_1$} in ''C'' and also {$\varphi(1 \to 2) = g: x_1 \to x_2$} for some ''g'' in ''C''. Since {$0 \to 2$} is {$0 \to 1$} followed by {$1 \to 2$} and since {$\varphi$} is a functor, {$\varphi(0 \to 2) = g \circ f$}. In other words, {$\varphi$} encodes ''f'', ''g'' and their compositions.}}
Pakeistos 508-510 eilutės iš
{{defn|no=1|The '''product (category theory)|product''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the projective limit <math>\varprojlim</math> of the functor <math>I \to C, \, i \mapsto X_i</math>, where ''I'' is viewed as a discrete category. It is denoted by <math>\prod_i X_i</math> and is the dual of the coproduct of the family.}}
{{defn|no=2|The '''product of categories|product of a family of categories''' ''C''<sub>''i''</sub>'s indexed by a set ''I'' is the category denoted by <math>\prod_i C_i</math> whose class of objects is the product of the classes of objects of ''C''<sub>''i''</sub>'s and whose hom-sets are <math>\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)</math>; the morphisms are composed component-wise. It is the dual of the disjoint union.}}
į:
{{defn|no=1|The '''product (category theory)|product''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the projective limit {$\varprojlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where ''I'' is viewed as a discrete category. It is denoted by {$\prod_i X_i$} and is the dual of the coproduct of the family.}}
{{defn|no=2|The '''product of categories|product of a family of categories''' ''C''<sub>''i''</sub>'s indexed by a set ''I'' is the category denoted by {$\prod_i C_i$} whose class of objects is the product of the classes of objects of ''C''<sub>''i''</sub>'s and whose hom-sets are {$\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)$}; the morphisms are composed component-wise. It is the dual of the disjoint union.}}
Pakeistos 512-513 eilutės iš
{{defn|1=Given categories ''C'' and ''D'', a '''profunctor''' (or a distributor) from ''C'' to ''D'' is a functor of the form <math>D^{\text{op}} \times C \to \mathbf{Set}</math>.}}
į:
{{defn|1=Given categories ''C'' and ''D'', a '''profunctor''' (or a distributor) from ''C'' to ''D'' is a functor of the form {$D^{\text{op}} \times C \to \mathbf{Set}$}.}}
Pakeistos 515-516 eilutės iš
{{defn|1=An object ''A'' in an abelian category is '''projective object|projective''' if the functor <math>\operatorname{Hom}(A, -)</math> is exact. It is the dual of an injective object.}}
į:
{{defn|1=An object ''A'' in an abelian category is '''projective object|projective''' if the functor {$\operatorname{Hom}(A, -)$} is exact. It is the dual of an injective object.}}
Pakeistos 534-535 eilutės iš
{{defn|1=A set-valued contravariant functor ''F'' on a category ''C'' is said to be '''representable functor|representable''' if it belongs to the essential image of the '''Yoneda embedding''' <math>C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})</math>; i.e., <math>F \simeq \operatorname{Hom}_C(-, Z)</math> for some object ''Z''. The object ''Z'' is said to be the representing object of ''F''.}}
į:
{{defn|1=A set-valued contravariant functor ''F'' on a category ''C'' is said to be '''representable functor|representable''' if it belongs to the essential image of the '''Yoneda embedding''' {$C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}; i.e., {$F \simeq \operatorname{Hom}_C(-, Z)$} for some object ''Z''. The object ''Z'' is said to be the representing object of ''F''.}}
Pakeista 552 eilutė iš:
{{defn|1=Given a ''k''-linear category ''C'' over a field ''k'', a '''Serre functor''' <math>f: C \to C</math> is an auto-equivalence such that <math>\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*</math> for any objects ''A'', ''B''.}}
į:
{{defn|1=Given a ''k''-linear category ''C'' over a field ''k'', a '''Serre functor''' {$f: C \to C$} is an auto-equivalence such that {$\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*$} for any objects ''A'', ''B''.}}
Pakeistos 568-569 eilutės iš
{{defn|1=A '''simplicial object''' in a category ''C'' is roughly a sequence of objects <math>X_0, X_1, X_2, \dots</math> in ''C'' that forms a simplicial set. In other words, it is a covariant or contravariant functor Δ → ''C''. For example, a '''simplicial presheaf''' is a simplicial object in the category of presheaves.}}
į:
{{defn|1=A '''simplicial object''' in a category ''C'' is roughly a sequence of objects {$X_0, X_1, X_2, \dots$} in ''C'' that forms a simplicial set. In other words, it is a covariant or contravariant functor Δ → ''C''. For example, a '''simplicial presheaf''' is a simplicial object in the category of presheaves.}}
Pakeistos 571-572 eilutės iš
{{defn|1=A '''simplicial set''' is a contravariant functor from &Delta; to '''Set''', where &Delta; is the '''simplex category''', a category whose objects are the sets [''n''] = { 0, 1, …, ''n'' } and whose morphisms are order-preserving functions. One writes <math>X_n = X([n])</math> and an element of the set <math>X_n</math> is called an ''n''-simplex. For example, <math>\Delta^n = \operatorname{Hom}_{\Delta}(-, [n])</math> is a simplicial set called the standard ''n''-simplex. By Yoneda's lemma, <math>X_n \simeq \operatorname{Nat}(\Delta^n, X)</math>.}}
į:
{{defn|1=A '''simplicial set''' is a contravariant functor from &Delta; to '''Set''', where &Delta; is the '''simplex category''', a category whose objects are the sets [''n''] = { 0, 1, …, ''n'' } and whose morphisms are order-preserving functions. One writes {$X_n = X([n])$} and an element of the set {$X_n$} is called an ''n''-simplex. For example, {$\Delta^n = \operatorname{Hom}_{\Delta}(-, [n])$} is a simplicial set called the standard ''n''-simplex. By Yoneda's lemma, {$X_n \simeq \operatorname{Nat}(\Delta^n, X)$}.}}
Pakeistos 580-581 eilutės iš
{{defn|1=Given a category ''C'' and an object ''A'' in it, the '''slice category''' ''C''<sub>/''A''</sub> of ''C'' over ''A'' is the category whose objects are all the morphisms in ''C'' with codomain ''A'', whose morphisms are morphisms in ''C'' such that if ''f'' is a morphism from <math>p_X: X \to A</math> to <math>p_Y: Y \to A</math>, then <math>p_Y \circ f = p_X</math> in ''C'' and whose composition is that of ''C''.}}
į:
{{defn|1=Given a category ''C'' and an object ''A'' in it, the '''slice category''' ''C''<sub>/''A''</sub> of ''C'' over ''A'' is the category whose objects are all the morphisms in ''C'' with codomain ''A'', whose morphisms are morphisms in ''C'' such that if ''f'' is a morphism from {$p_X: X \to A$} to {$p_Y: Y \to A$}, then {$p_Y \circ f = p_X$} in ''C'' and whose composition is that of ''C''.}}
Pakeistos 593-594 eilutės iš
{{defn|A morphism ''f'' in a category admitting finite limits and finite colimits is '''strict morphism|strict''' if the natural morphism <math>\operatorname{Coim}(f) \to \operatorname{Im}(f)</math> is an isomorphism.}}
į:
{{defn|A morphism ''f'' in a category admitting finite limits and finite colimits is '''strict morphism|strict''' if the natural morphism {$\operatorname{Coim}(f) \to \operatorname{Im}(f)$} is an isomorphism.}}
Pakeistos 627-628 eilutės iš
{{defn|The '''Tannakian duality''' states that, in an appropriate setup, to give a morphism <math>f: X \to Y</math> is to give a pullback functor <math>f^*</math> along it. In other words, the Hom set <math>\operatorname{Hom}(X, Y)</math> can be identified with the functor category <math>\operatorname{Fct}(D(Y), D(X))</math>, perhaps in the '''derived algebraic geometry|derived sense''', where <math>D(X)</math> is the category associated to ''X'' (e.g., the derived category).<ref>Jacob Lurie. Tannaka duality for geometric stacks. http://math.harvard.edu/~lurie/, 2004.</ref><ref>{{cite arxiv|last=Bhatt|first=Bhargav|date=2014-04-29|title=Algebraization and Tannaka duality|eprint=1404.7483|class=math.AG}}</ref>}}
į:
{{defn|The '''Tannakian duality''' states that, in an appropriate setup, to give a morphism {$f: X \to Y$} is to give a pullback functor {$f^*$} along it. In other words, the Hom set {$\operatorname{Hom}(X, Y)$} can be identified with the functor category {$\operatorname{Fct}(D(Y), D(X))$}, perhaps in the '''derived algebraic geometry|derived sense''', where {$D(X)$} is the category associated to ''X'' (e.g., the derived category).<ref>Jacob Lurie. Tannaka duality for geometric stacks. http://math.harvard.edu/~lurie/, 2004.</ref><ref>{{cite arxiv|last=Bhatt|first=Bhargav|date=2014-04-29|title=Algebraization and Tannaka duality|eprint=1404.7483|class=math.AG}}</ref>}}
Pakeistos 636-638 eilutės iš
{{defn|Given a monoidal category ''B'', the '''tensor product of functors''' <math>F: C^{\text{op}} \to B</math> and <math>G: C \to B</math> is the coend:
:<math>F \otimes_C G = \int^{c \in C} F(c) \otimes G(c).</math>}}
į:
{{defn|Given a monoidal category ''B'', the '''tensor product of functors''' {$F: C^{\text{op}} \to B$} and {$G: C \to B$} is the coend:
:{$F \otimes_C G = \int^{c \in C} F(c) \otimes G(c).$}}}
Pakeistos 641-642 eilutės iš
{{defn|no=2|An object ''A'' in an ∞-category ''C'' is terminal if <math>\operatorname{Map}_C(B, A)</math> is '''contractible space|contractible''' for every object ''B'' in ''C''.}}
į:
{{defn|no=2|An object ''A'' in an ∞-category ''C'' is terminal if {$\operatorname{Map}_C(B, A)$} is '''contractible space|contractible''' for every object ''B'' in ''C''.}}
Pakeistos 657-660 eilutės iš
{{defn|no=1|Given a functor <math>f: C \to D</math> and an object ''X'' in ''D'', a '''universal morphism''' from ''X'' to ''f'' is an initial object in the '''comma category''' <math>(X \downarrow f)</math>. (Its dual is also called a universal morphism.) For example, take ''f'' to be the forgetful functor <math>\mathbf{Vec}_k \to \mathbf{Set}</math> and ''X'' a set. An initial object of <math>(X \downarrow f)</math> is a function <math>j: X \to f(V_X)</math>. That it is initial means that if <math>k: X \to f(W)</math> is another morphism, then there is a unique morphism from ''j'' to ''k'', which consists of a linear map <math>V_X \to W</math> that extends ''k'' via ''j''; that is to say, <math>V_X</math> is the '''free vector space''' generated by ''X''.}}
{{defn|no=2|Stated more explicitly, given ''f'' as above, a morphism <math>X \to f(u_X)</math> in ''D'' is universal if and only if the natural map
:<math>\operatorname{Hom}_C(u_X, c) \to \operatorname{Hom}_D(X, f(c)), \, \alpha \mapsto (X \to f(u_x) \overset{f(\alpha)}\to f(c))</math>
is bijective. In particular, if <math>\operatorname{Hom}_C(u_X, -) \simeq \operatorname{Hom}_D(X, f(-))</math>, then taking ''c'' to be ''u''<sub>''X''</sub> one gets a universal morphism by sending the identity morphism. In other words, having a universal morphism is equivalent to the representability of the functor <math>\operatorname{Hom}_D(X, f(-))</math>.}}
į:
{{defn|no=1|Given a functor {$f: C \to D$} and an object ''X'' in ''D'', a '''universal morphism''' from ''X'' to ''f'' is an initial object in the '''comma category''' {$(X \downarrow f)$}. (Its dual is also called a universal morphism.) For example, take ''f'' to be the forgetful functor {$\mathbf{Vec}_k \to \mathbf{Set}$} and ''X'' a set. An initial object of {$(X \downarrow f)$} is a function {$j: X \to f(V_X)$}. That it is initial means that if {$k: X \to f(W)$} is another morphism, then there is a unique morphism from ''j'' to ''k'', which consists of a linear map {$V_X \to W$} that extends ''k'' via ''j''; that is to say, {$V_X$} is the '''free vector space''' generated by ''X''.}}
{{defn|no=2|Stated more explicitly, given ''f'' as above, a morphism {$X \to f(u_X)$} in ''D'' is universal if and only if the natural map
:{$\operatorname{Hom}_C(u_X, c) \to \operatorname{Hom}_D(X, f(c)), \, \alpha \mapsto (X \to f(u_x) \overset{f(\alpha)}\to f(c))$}
is bijective. In particular, if {$\operatorname{Hom}_C(u_X, -) \simeq \operatorname{Hom}_D(X, f(-))$}, then taking ''c'' to be ''u''<sub>''X''</sub> one gets a universal morphism by sending the identity morphism. In other words, having a universal morphism is equivalent to the representability of the functor {$\operatorname{Hom}_D(X, f(-))$}.}}
Pakeista 680 eilutė iš:
:<math>F(X) \simeq \operatorname{Nat}(\operatorname{Hom}_C(-, X), F)</math>
į:
:{$F(X) \simeq \operatorname{Nat}(\operatorname{Hom}_C(-, X), F)$}
Pakeista 682 eilutė iš:
:<math>y: C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(-, X)</math>
į:
:{$y: C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(-, X)$}
Pakeista 684 eilutė iš:
{{defn|no=2|1=If <math>F: C \to D</math> is a functor and ''y'' is the Yoneda embedding of ''C'', then the '''Yoneda extension''' of ''F'' is the left Kan extension of ''F'' along ''y''.}}
į:
{{defn|no=2|1=If {$F: C \to D$} is a functor and ''y'' is the Yoneda embedding of ''C'', then the '''Yoneda extension''' of ''F'' is the left Kan extension of ''F'' along ''y''.}}
2019 vasario 10 d., 09:14 atliko AndriusKulikauskas -
Pakeistos 708-717 eilutės iš
| title = Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1
|series=Lecture Notes in Mathematics |volume=269
| year = 1972
| publisher = '''Springer Science+Business Media|Springer-Verlag'''
| location = Berlin; New York
| language = French
| pages = xix+525
| nopp = true
|doi= 10.1007/BFb0081551
|isbn= 978-3-540-05896-0
į:
title = Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1
2019 vasario 10 d., 09:12 atliko AndriusKulikauskas -
Pakeistos 3-8 eilutės iš
This is a glossary of properties and concepts in [[category theory]] in [[mathematics]].

*
'''Notes on foundations''': In many expositions (e.g., Vistoli), the set-theoretic issues are ignored; this means, for instance, that one does not distinguish between small and large categories and that one can arbitrarily form a localization of a category.<ref>If one believes in the existence of [[strongly inaccessible cardinal]]s, then there can be a rigorous theory where statements and constructions have references to [[Grothendieck universe]]s.</ref> Like those expositions, this glossary also generally ignores the set-theoretic issues, except when they are relevant (e.g., the discussion on accessibility.)

Especially for higher categories, the concepts from algebraic topology are also used in the category theory. For that see also [[glossary of algebraic topology]].
į:
This is a glossary of properties and concepts in '''category theory''' in '''mathematics'''.

*'''
Notes on foundations''': In many expositions (e.g., Vistoli), the set-theoretic issues are ignored; this means, for instance, that one does not distinguish between small and large categories and that one can arbitrarily form a localization of a category.<ref>If one believes in the existence of '''strongly inaccessible cardinal'''s, then there can be a rigorous theory where statements and constructions have references to '''Grothendieck universe'''s.</ref> Like those expositions, this glossary also generally ignores the set-theoretic issues, except when they are relevant (e.g., the discussion on accessibility.)

Especially for higher categories, the concepts from algebraic topology are also used in the category theory. For that see also '''glossary of algebraic topology'''.
Pakeistos 11-12 eilutės iš
*'''Cat''', the [[category of categories|category of (small) categories]], where the objects are categories (which are small with respect to some universe) and the morphisms [[functor]]s.
*'''Fct'''(''C'', ''D''), the [[functor category]]: the category of [[functor]]s from a category ''C'' to a category ''D''.
į:
*'''Cat''', the '''category of categories|category of (small) categories''', where the objects are categories (which are small with respect to some universe) and the morphisms '''functor'''s.
*'''Fct'''(''C'', ''D''), the '''functor category''': the category of '''functor'''s from a category ''C'' to a category ''D''.
Pakeista 14 eilutė iš:
*''s'''''Set''', the category of [[simplicial set]]s.
į:
*''s'''''Set''', the category of '''simplicial set'''s.
Pakeistos 16-19 eilutės iš
*By an [[∞-category]], we mean a [[quasi-category]], the most popular model, unless other models are being discussed.
*The number [[zero]] 0 is a natural number.
{{Compact ToC|short1|sym=yes|x=[[#XYZ|XYZ]]|y=|z=|seealso=yes|refs=yes}}
į:
*By an '''∞-category''', we mean a '''quasi-category''', the most popular model, unless other models are being discussed.
*The number '''zero''' 0 is a natural number.
{{Compact ToC|short1|sym=yes|x='''#XYZ|XYZ'''|y=|z=|seealso=yes|refs=yes}}
Pakeistos 23-24 eilutės iš
{{defn|1=A category is [[abelian category|abelian]] if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.}}
į:
{{defn|1=A category is '''abelian category|abelian''' if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.}}
Pakeistos 26-28 eilutės iš
{{defn|no=1|Given a [[cardinal number]] κ, an object ''X'' in a category is [[accessible object|κ-accessible]] (or κ-compact or κ-presentable) if <math>\operatorname{Hom}(X, -)</math> commutes with κ-filtered colimits.}}
{{defn|no=2|Given a [[regular cardinal]] κ, a category is [[accessible category|κ-accessible]] if it has κ-filtered colimits and there exists a small set ''S'' of κ-compact objects that generates the category under colimits, meaning every object can be written as a colimit of diagrams of objects in ''S''.}}
į:
{{defn|no=1|Given a '''cardinal number''' κ, an object ''X'' in a category is '''accessible object|κ-accessible''' (or κ-compact or κ-presentable) if <math>\operatorname{Hom}(X, -)</math> commutes with κ-filtered colimits.}}
{{defn|no=2|Given a '''regular cardinal''' κ, a category is '''accessible category|κ-accessible''' if it has κ-filtered colimits and there exists a small set ''S'' of κ-compact objects that generates the category under colimits, meaning every object can be written as a colimit of diagrams of objects in ''S''.}}
Pakeistos 30-31 eilutės iš
{{defn|1=A category is [[additive category|additive]] if it is preadditive (to be precise, has some pre-additive structure) and admits all finite [[coproduct]]s. Although "preadditive" is an additional structure, one can show "additive" is a ''property'' of a category; i.e., one can ask whether a given category is additive or not.<ref>Remark 2.7. of https://ncatlab.org/nlab/show/additive+category</ref>}}
į:
{{defn|1=A category is '''additive category|additive''' if it is preadditive (to be precise, has some pre-additive structure) and admits all finite '''coproduct'''s. Although "preadditive" is an additional structure, one can show "additive" is a ''property'' of a category; i.e., one can ask whether a given category is additive or not.<ref>Remark 2.7. of https://ncatlab.org/nlab/show/additive+category</ref>}}
Pakeista 33 eilutė iš:
{{defn|1=An [[adjoint functor|adjunction]] (also called an adjoint pair) is a pair of functors ''F'': ''C'' → ''D'', ''G'': ''D'' → ''C'' such that there is a "natural" bijection
į:
{{defn|1=An '''adjoint functor|adjunction''' (also called an adjoint pair) is a pair of functors ''F'': ''C'' → ''D'', ''G'': ''D'' → ''C'' such that there is a "natural" bijection
Pakeistos 38-39 eilutės iš
{{defn|1=Given a monad ''T'' in a category ''X'', an [[algebra for a monad|algebra for ''T'']] or a ''T''-algebra is an object in ''X'' with a [[monoid action]] of ''T'' ("algebra" is misleading and "''T''-object" is perhaps a better term.) For example, given a group ''G'' that determines a monad ''T'' in '''Set''' in the standard way, a ''T''-algebra is a set with an [[group action|action]] of ''G''.}}
į:
{{defn|1=Given a monad ''T'' in a category ''X'', an '''algebra for a monad|algebra for ''T''''' or a ''T''-algebra is an object in ''X'' with a '''monoid action''' of ''T'' ("algebra" is misleading and "''T''-object" is perhaps a better term.) For example, given a group ''G'' that determines a monad ''T'' in '''Set''' in the standard way, a ''T''-algebra is a set with an '''group action|action''' of ''G''.}}
Pakeistos 50-51 eilutės iš
{{defn|1=[[Beck's monadicity theorem|Beck's theorem]] characterizes the category of [[algebra for a monad|algebras for a given monad]].}}
į:
{{defn|1='''Beck's monadicity theorem|Beck's theorem''' characterizes the category of '''algebra for a monad|algebras for a given monad'''.}}
Pakeistos 53-54 eilutės iš
{{defn|1=A [[bicategory]] is a model of a weak [[2-category]].}}
į:
{{defn|1=A '''bicategory''' is a model of a weak '''2-category'''.}}
Pakeistos 56-57 eilutės iš
{{defn|1=A [[bifunctor]] from a pair of categories ''C'' and ''D'' to a category ''E'' is a functor ''C'' × ''D'' → ''E''. For example, for any category ''C'', <math>\operatorname{Hom}(-, -)</math> is a bifunctor from ''C''<sup>op</sup> and ''C'' to '''Set'''.}}
į:
{{defn|1=A '''bifunctor''' from a pair of categories ''C'' and ''D'' to a category ''E'' is a functor ''C'' × ''D'' → ''E''. For example, for any category ''C'', <math>\operatorname{Hom}(-, -)</math> is a bifunctor from ''C''<sup>op</sup> and ''C'' to '''Set'''.}}
Pakeistos 59-60 eilutės iš
{{defn|1=A [[bimorphism]] is a morphism that is both an epimorphism and a monomorphism.}}
į:
{{defn|1=A '''bimorphism''' is a morphism that is both an epimorphism and a monomorphism.}}
Pakeistos 62-63 eilutės iš
{{defn|1=See [[Bousfield localization]].}}
į:
{{defn|1=See '''Bousfield localization'''.}}
Pakeistos 69-70 eilutės iš
{{defn|1=The [[calculus of functors]] is a technique of studying functors in the manner similar to the way a [[function (mathematics)|function]] is studied via its [[Taylor series]] expansion; whence, the term "calculus".}}
į:
{{defn|1=The '''calculus of functors''' is a technique of studying functors in the manner similar to the way a '''function (mathematics)|function''' is studied via its '''Taylor series''' expansion; whence, the term "calculus".}}
Pakeistos 72-73 eilutės iš
{{defn|1=A category is [[Cartesian closed category|cartesian closed]] if it has a terminal object and that any two objects have a product and exponential.}}
į:
{{defn|1=A category is '''Cartesian closed category|cartesian closed''' if it has a terminal object and that any two objects have a product and exponential.}}
Pakeistos 78-80 eilutės iš
{{defn|no=1|1=Given a functor π: ''C'' → ''D'' (e.g., a [[prestack]] over schemes), a morphism ''f'': ''x'' → ''y'' in ''C'' is [[cartesian morphism|π-cartesian]] if, for each object ''z'' in ''C'', each morphism ''g'': ''z'' → ''y'' in ''C'' and each morphism ''v'': π(''z'') → π(''x'') in ''D'' such that π(''g'') = π(''f'') ∘ ''v'', there exists a unique morphism ''u'': ''z'' → ''x'' such that π(''u'') = ''v'' and ''g'' = ''f'' ∘ ''u''.}}
{{defn|no=2|1=Given a functor π: ''C'' → ''D'' (e.g., a [[prestack]] over rings), a morphism ''f'': ''x'' → ''y'' in ''C'' is [[cartesian morphism|π-coCartesian]] if, for each object ''z'' in ''C'', each morphism ''g'': ''x'' → ''z'' in ''C'' and each morphism ''v'': π(''y'') → π(''z'') in ''D'' such that π(''g'') = ''v'' ∘ π(''f''), there exists a unique morphism ''u'': ''y'' → ''z'' such that π(''u'') = ''v'' and ''g'' = ''u'' ∘ ''f''. (In short, ''f'' is the dual of a π-cartesian morphism.)}}
į:
{{defn|no=1|1=Given a functor π: ''C'' → ''D'' (e.g., a '''prestack''' over schemes), a morphism ''f'': ''x'' → ''y'' in ''C'' is '''cartesian morphism|π-cartesian''' if, for each object ''z'' in ''C'', each morphism ''g'': ''z'' → ''y'' in ''C'' and each morphism ''v'': π(''z'') → π(''x'') in ''D'' such that π(''g'') = π(''f'') ∘ ''v'', there exists a unique morphism ''u'': ''z'' → ''x'' such that π(''u'') = ''v'' and ''g'' = ''f'' ∘ ''u''.}}
{{defn|no=2|1=Given a functor π: ''C'' → ''D'' (e.g., a '''prestack''' over rings), a morphism ''f'': ''x'' → ''y'' in ''C'' is '''cartesian morphism|π-coCartesian''' if, for each object ''z'' in ''C'', each morphism ''g'': ''x'' → ''z'' in ''C'' and each morphism ''v'': π(''y'') → π(''z'') in ''D'' such that π(''g'') = ''v'' ∘ π(''f''), there exists a unique morphism ''u'': ''y'' → ''z'' such that π(''u'') = ''v'' and ''g'' = ''u'' ∘ ''f''. (In short, ''f'' is the dual of a π-cartesian morphism.)}}
Pakeistos 85-86 eilutės iš
{{defn|1=[[Categorical logic]] is an approach to [[mathematical logic]] that uses category theory.}}
į:
{{defn|1='''Categorical logic''' is an approach to '''mathematical logic''' that uses category theory.}}
Pakeistos 88-89 eilutės iš
{{defn|1=[[Categorification]] is a process of replacing sets and set-theoretic concepts with categories and category-theoretic concepts in some nontrivial way to capture categoric flavors. Decategorification is the reverse of categorification.}}
į:
{{defn|1='''Categorification''' is a process of replacing sets and set-theoretic concepts with categories and category-theoretic concepts in some nontrivial way to capture categoric flavors. Decategorification is the reverse of categorification.}}
Pakeista 91 eilutė iš:
{{defn|1=A [[category (mathematics)|category]] consists of the following data
į:
{{defn|1=A '''category (mathematics)|category''' consists of the following data
Pakeistos 99-100 eilutės iš
For example, a [[partially ordered set]] can be viewed as a category: the objects are the elements of the set and for each pair of objects ''x'', ''y'', there is a unique morphism <math>x \to y</math> if and only if <math>x \le y</math>; the associativity of composition means transitivity.}}
į:
For example, a '''partially ordered set''' can be viewed as a category: the objects are the elements of the set and for each pair of objects ''x'', ''y'', there is a unique morphism <math>x \to y</math> if and only if <math>x \le y</math>; the associativity of composition means transitivity.}}
Pakeistos 102-103 eilutės iš
{{defn|1=The [[category of categories|category of (small) categories]], denoted by '''Cat''', is a category where the objects are all the categories which are small with respect to some fixed universe and the morphisms are all the [[functor]]s.}}
į:
{{defn|1=The '''category of categories|category of (small) categories''', denoted by '''Cat''', is a category where the objects are all the categories which are small with respect to some fixed universe and the morphisms are all the '''functor'''s.}}
Pakeistos 105-106 eilutės iš
{{defn|1=The [[classifying space of a category]] ''C'' is the geometric realization of the nerve of ''C''.}}
į:
{{defn|1=The '''classifying space of a category''' ''C'' is the geometric realization of the nerve of ''C''.}}
Pakeistos 108-109 eilutės iš
{{defn|1=Often used synonymous with op-; for example, a [[colimit]] refers to an op-limit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an op-fibration is not the same thing as a [[cofibration]].}}
į:
{{defn|1=Often used synonymous with op-; for example, a '''colimit''' refers to an op-limit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an op-fibration is not the same thing as a '''cofibration'''.}}
Pakeista 111 eilutė iš:
{{defn|1=The coend of a functor <math>F: C^{\text{op}} \times C \to X</math> is the dual of the [[end (category theory)|end]] of ''F'' and is denoted by
į:
{{defn|1=The coend of a functor <math>F: C^{\text{op}} \times C \to X</math> is the dual of the '''end (category theory)|end''' of ''F'' and is denoted by
Pakeista 113 eilutė iš:
For example, if ''R'' is a ring, ''M'' a right ''R''-module and ''N'' a left ''R''-module, then the [[tensor product of modules|tensor product]] of ''M'' and ''N'' is
į:
For example, if ''R'' is a ring, ''M'' a right ''R''-module and ''N'' a left ''R''-module, then the '''tensor product of modules|tensor product''' of ''M'' and ''N'' is
Pakeistos 118-119 eilutės iš
{{defn|1=The [[coequalizer]] of a pair of morphisms <math>f, g: A \to B</math> is the colimit of the pair. It is the dual of an equalizer.}}
į:
{{defn|1=The '''coequalizer''' of a pair of morphisms <math>f, g: A \to B</math> is the colimit of the pair. It is the dual of an equalizer.}}
Pakeistos 121-122 eilutės iš
{{defn|1=The [[coimage]] of a morphism ''f'': ''X'' → ''Y'' is the coequalizer of <math>X \times_Y X \rightrightarrows X</math>.}}
į:
{{defn|1=The '''coimage''' of a morphism ''f'': ''X'' → ''Y'' is the coequalizer of <math>X \times_Y X \rightrightarrows X</math>.}}
Pakeistos 124-125 eilutės iš
{{defn|1=Another term for [[multicategory]], a generalized category where a morphism can have several domains. The notion of "colored operad" is more primitive than that of operad: in fact, an operad can be defined as a colored operad with a single object.}}
į:
{{defn|1=Another term for '''multicategory''', a generalized category where a morphism can have several domains. The notion of "colored operad" is more primitive than that of operad: in fact, an operad can be defined as a colored operad with a single object.}}
Pakeista 127 eilutė iš:
{{defn|1=Given functors <math>f: C \to B, g: D \to B</math>, the [[comma category]] <math>(f \downarrow g)</math> is a category where (1) the objects are morphisms <math>f(c) \to g(d)</math> and (2) a morphism from <math>\alpha: f(c) \to g(d)</math> to <math>\beta: f(c') \to g(d')</math> consists of <math>c \to c'</math> and <math>d \to d'</math> such that <math>f(c) \to f(c') \overset{\beta}\to g(d')</math> is <math>f(c) \overset{\alpha}\to g(d) \to g(d').</math> For example, if ''f'' is the identity functor and ''g'' is the constant functor with a value ''b'', then it is the slice category of ''B'' over an object ''b''.
į:
{{defn|1=Given functors <math>f: C \to B, g: D \to B</math>, the '''comma category''' <math>(f \downarrow g)</math> is a category where (1) the objects are morphisms <math>f(c) \to g(d)</math> and (2) a morphism from <math>\alpha: f(c) \to g(d)</math> to <math>\beta: f(c') \to g(d')</math> consists of <math>c \to c'</math> and <math>d \to d'</math> such that <math>f(c) \to f(c') \overset{\beta}\to g(d')</math> is <math>f(c) \overset{\alpha}\to g(d) \to g(d').</math> For example, if ''f'' is the identity functor and ''g'' is the constant functor with a value ''b'', then it is the slice category of ''B'' over an object ''b''.
Pakeistos 131-132 eilutės iš
{{defn|1=A [[comonad]] in a category ''X'' is a [[comonid]] in the monoidal category of endofunctors of ''X''.}}
į:
{{defn|1=A '''comonad''' in a category ''X'' is a '''comonid''' in the monoidal category of endofunctors of ''X''.}}
Pakeistos 134-135 eilutės iš
{{defn|1=Probably synonymous with [[#accessible]].}}
į:
{{defn|1=Probably synonymous with '''#accessible'''.}}
Pakeistos 137-138 eilutės iš
{{defn|1=A category is [[complete category|complete]] if all small limits exist.}}
į:
{{defn|1=A category is '''complete category|complete''' if all small limits exist.}}
Pakeistos 145-146 eilutės iš
{{defn|1=A [[concrete category]] ''C'' is a category such that there is a faithful functor from ''C'' to '''[[Category of sets|Set]]'''; e.g., '''[[category of vector spaces|Vec]]''', '''[[category of groups|Grp]]''' and '''[[category of topological spaces|Top]]'''.}}
į:
{{defn|1=A '''concrete category''' ''C'' is a category such that there is a faithful functor from ''C'' to ''''''Category of sets|Set''''''; e.g., ''''''category of vector spaces|Vec'''''', ''''''category of groups|Grp'''''' and ''''''category of topological spaces|Top''''''.}}
Pakeista 148 eilutė iš:
{{defn|1=A [[cone (category theory)|cone]] is a way to express the [[universal property]] of a colimit (or dually a limit). One can show<ref>{{harvnb|Kashiwara|Schapira|2006|loc=Ch. 2, Exercise 2.8.}}</ref> that the colimit <math>\varinjlim</math> is the left adjoint to the diagonal functor <math>\Delta: C \to \operatorname{Fct}(I, C)</math>, which sends an object ''X'' to the constant functor with value ''X''; that is, for any ''X'' and any functor <math>f: I \to C</math>,
į:
{{defn|1=A '''cone (category theory)|cone''' is a way to express the '''universal property''' of a colimit (or dually a limit). One can show<ref>{{harvnb|Kashiwara|Schapira|2006|loc=Ch. 2, Exercise 2.8.}}</ref> that the colimit <math>\varinjlim</math> is the left adjoint to the diagonal functor <math>\Delta: C \to \operatorname{Fct}(I, C)</math>, which sends an object ''X'' to the constant functor with value ''X''; that is, for any ''X'' and any functor <math>f: I \to C</math>,
Pakeistos 153-154 eilutės iš
{{defn|1=A category is [[connected category|connected]] if, for each pair of objects ''x'', ''y'', there exists a finite sequence of objects ''z''<sub>''i''</sub> such that <math>z_0 = x, z_n = y</math> and either <math>\operatorname{Hom}(z_i, z_{i+1})</math> or <math>\operatorname{Hom}(z_{i+1}, z_i)</math> is nonempty for any ''i''.}}
į:
{{defn|1=A category is '''connected category|connected''' if, for each pair of objects ''x'', ''y'', there exists a finite sequence of objects ''z''<sub>''i''</sub> such that <math>z_0 = x, z_n = y</math> and either <math>\operatorname{Hom}(z_i, z_{i+1})</math> or <math>\operatorname{Hom}(z_{i+1}, z_i)</math> is nonempty for any ''i''.}}
Pakeistos 156-157 eilutės iš
{{defn|1=A [[conservative functor]] is a functor that reflects isomorphisms. Many forgetful functors are conservative, but the forgetful functor from '''Top''' to '''Set''' is not conservative.}}
į:
{{defn|1=A '''conservative functor''' is a functor that reflects isomorphisms. Many forgetful functors are conservative, but the forgetful functor from '''Top''' to '''Set''' is not conservative.}}
Pakeistos 159-160 eilutės iš
{{defn|1=A functor is [[constant functor|constant]] if it maps every object in a category to the same object ''A'' and every morphism to the identity on ''A''. Put in another way, a functor <math>f: C \to D</math> is constant if it factors as: <math>C \to \{ A \} \overset{i}\to D</math> for some object ''A'' in ''D'', where ''i'' is the inclusion of the discrete category { ''A'' }.}}
į:
{{defn|1=A functor is '''constant functor|constant''' if it maps every object in a category to the same object ''A'' and every morphism to the identity on ''A''. Put in another way, a functor <math>f: C \to D</math> is constant if it factors as: <math>C \to \{ A \} \overset{i}\to D</math> for some object ''A'' in ''D'', where ''i'' is the inclusion of the discrete category { ''A'' }.}}
Pakeista 162 eilutė iš:
{{defn|1=A [[contravariant functor]] ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a [[presheaf (category theory)|presheaf]] especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let <math>\mathfrak{P}(S)</math> be the power set of ''S'' and for each function <math>f: S \to T</math>, define
į:
{{defn|1=A '''contravariant functor''' ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a '''presheaf (category theory)|presheaf''' especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let <math>\mathfrak{P}(S)</math> be the power set of ''S'' and for each function <math>f: S \to T</math>, define
Pakeistos 167-168 eilutės iš
{{defn|1=The [[coproduct]] of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the inductive limit <math>\varinjlim</math> of the functor <math>I \to C, \, i \mapsto X_i</math>, where ''I'' is viewed as a discrete category. It is the dual of the product of the family. For example, a coproduct in '''[[category of groups|Grp]]''' is a [[free product]].}}
į:
{{defn|1=The '''coproduct''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the inductive limit <math>\varinjlim</math> of the functor <math>I \to C, \, i \mapsto X_i</math>, where ''I'' is viewed as a discrete category. It is the dual of the product of the family. For example, a coproduct in ''''''category of groups|Grp'''''' is a '''free product'''.}}
Pakeistos 170-171 eilutės iš
{{defn|1=The [[core (category theory)|core]] of a category is the maximal groupoid contained in the category.}}
į:
{{defn|1=The '''core (category theory)|core''' of a category is the maximal groupoid contained in the category.}}
Pakeistos 177-178 eilutės iš
{{defn|Given a group or monoid ''M'', the [[Day convolution]] is the tensor product in <math>\mathbf{Fct}(M, \mathbf{Set})</math>.<ref>http://ncatlab.org/nlab/show/Day+convolution</ref>}}
į:
{{defn|Given a group or monoid ''M'', the '''Day convolution''' is the tensor product in <math>\mathbf{Fct}(M, \mathbf{Set})</math>.<ref>http://ncatlab.org/nlab/show/Day+convolution</ref>}}
Pakeistos 180-181 eilutės iš
{{defn|1=The [[density theorem (category theory)|density theorem]] states that every presheaf (a set-valued contravariant functor) is a colimit of representable presheaves. Yoneda's lemma embeds a category ''C'' into the category of presheaves on ''C''. The density theorem then says the image is "dense", so to say. The name "density" is because of the analogy with the [[Jacobson density theorem]] (or other variants) in abstract algebra.}}
į:
{{defn|1=The '''density theorem (category theory)|density theorem''' states that every presheaf (a set-valued contravariant functor) is a colimit of representable presheaves. Yoneda's lemma embeds a category ''C'' into the category of presheaves on ''C''. The density theorem then says the image is "dense", so to say. The name "density" is because of the analogy with the '''Jacobson density theorem''' (or other variants) in abstract algebra.}}
Pakeista 183 eilutė iš:
{{defn|1=Given categories ''I'', ''C'', the [[diagonal functor]] is the functor
į:
{{defn|1=Given categories ''I'', ''C'', the '''diagonal functor''' is the functor
Pakeistos 188-189 eilutės iš
{{defn|1=Given a category ''C'', a [[diagram (category theory)|diagram]] in ''C'' is a functor <math>f: I \to C</math> from a small category ''I''.}}
į:
{{defn|1=Given a category ''C'', a '''diagram (category theory)|diagram''' in ''C'' is a functor <math>f: I \to C</math> from a small category ''I''.}}
Pakeistos 191-192 eilutės iš
{{defn|1=A [[differential graded category]] is a category whose Hom sets are equipped with structures of [[differential graded module]]s. In particular, if the category has only one object, it is the same as a differential graded module.}}
į:
{{defn|1=A '''differential graded category''' is a category whose Hom sets are equipped with structures of '''differential graded module'''s. In particular, if the category has only one object, it is the same as a differential graded module.}}
Pakeistos 194-195 eilutės iš
{{defn|1=A category is [[discrete category|discrete]] if each morphism is an identity morphism (of some object). For example, a set can be viewed as a discrete category.}}
į:
{{defn|1=A category is '''discrete category|discrete''' if each morphism is an identity morphism (of some object). For example, a set can be viewed as a discrete category.}}
Pakeista 200 eilutė iš:
{{defn|1=A [[Dwyer–Kan equivalence]] is a generalization of an equivalence of categories to the simplicial context.<ref>{{cite arxiv|last=Hinich|first=V.|date=2013-11-17|title=Dwyer-Kan localization revisited|eprint=1311.4128|class=math.QA}}</ref>}}
į:
{{defn|1=A '''Dwyer–Kan equivalence''' is a generalization of an equivalence of categories to the simplicial context.<ref>{{cite arxiv|last=Hinich|first=V.|date=2013-11-17|title=Dwyer-Kan localization revisited|eprint=1311.4128|class=math.QA}}</ref>}}
Pakeistos 206-207 eilutės iš
{{defn|1=Another name for the category of [[algebra for a monad|algebras for a given monad]].}}
į:
{{defn|1=Another name for the category of '''algebra for a monad|algebras for a given monad'''.}}
Pakeista 209 eilutė iš:
{{defn|1=The [[end (category theory)|end]] of a functor <math>F: C^{\text{op}} \times C \to X</math> is the limit
į:
{{defn|1=The '''end (category theory)|end''' of a functor <math>F: C^{\text{op}} \times C \to X</math> is the limit
Pakeista 211 eilutė iš:
where <math>C^{\#}</math> is the category (called the [[subdivision category]] of ''C'') whose objects are symbols <math>c^{\#}, u^{\#}</math> for all objects ''c'' and all morphisms ''u'' in ''C'' and whose morphisms are <math>b^{\#} \to u^{\#}</math> and <math>u^{\#} \to c^{\#}</math> if <math>u: b \to c</math> and where <math>F^{\#}</math> is induced by ''F'' so that <math>c^{\#}</math> would go to <math>F(c, c)</math> and <math>u^{\#}, u: b \to c</math> would go to <math>F(b, c)</math>. For example, for functors <math>F, G: C \to X</math>,
į:
where <math>C^{\#}</math> is the category (called the '''subdivision category''' of ''C'') whose objects are symbols <math>c^{\#}, u^{\#}</math> for all objects ''c'' and all morphisms ''u'' in ''C'' and whose morphisms are <math>b^{\#} \to u^{\#}</math> and <math>u^{\#} \to c^{\#}</math> if <math>u: b \to c</math> and where <math>F^{\#}</math> is induced by ''F'' so that <math>c^{\#}</math> would go to <math>F(c, c)</math> and <math>u^{\#}, u: b \to c</math> would go to <math>F(b, c)</math>. For example, for functors <math>F, G: C \to X</math>,
Pakeista 219 eilutė iš:
{{defn|1=Given a monoidal category (''C'', ⊗, 1), a [[category enriched]] over ''C'' is, informally, a category whose Hom sets are in ''C''. More precisely, a category ''D'' enriched over ''C'' is a data consisting of
į:
{{defn|1=Given a monoidal category (''C'', ⊗, 1), a '''category enriched''' over ''C'' is, informally, a category whose Hom sets are in ''C''. More precisely, a category ''D'' enriched over ''C'' is a data consisting of
Pakeista 221 eilutė iš:
# For each pair of objects ''X'', ''Y'' in ''D'', an object <math>\operatorname{Map}_D(X, Y)</math> in ''C'', called the [[mapping object]] from ''X'' to ''Y'',
į:
# For each pair of objects ''X'', ''Y'' in ''D'', an object <math>\operatorname{Map}_D(X, Y)</math> in ''C'', called the '''mapping object''' from ''X'' to ''Y'',
Pakeistos 231-232 eilutės iš
{{defn|The [[empty category (category theory)|empty category]] is a category with no object. It is the same thing as the [[empty set]] when the empty set is viewed as a discrete category.}}
į:
{{defn|The '''empty category (category theory)|empty category''' is a category with no object. It is the same thing as the '''empty set''' when the empty set is viewed as a discrete category.}}
Pakeistos 234-235 eilutės iš
{{defn|1=A morphism ''f'' is an [[epimorphism]] if <math>g=h</math> whenever <math>g\circ f=h\circ f</math>. In other words, ''f'' is the dual of a monomorphism.}}
į:
{{defn|1=A morphism ''f'' is an '''epimorphism''' if <math>g=h</math> whenever <math>g\circ f=h\circ f</math>. In other words, ''f'' is the dual of a monomorphism.}}
Pakeistos 237-238 eilutės iš
{{defn|1=The [[equalizer (mathematics)|equalizer]] of a pair of morphisms <math>f, g: A \to B</math> is the limit of the pair. It is the dual of a coequalizer.}}
į:
{{defn|1=The '''equalizer (mathematics)|equalizer''' of a pair of morphisms <math>f, g: A \to B</math> is the limit of the pair. It is the dual of a coequalizer.}}
Pakeista 240 eilutė iš:
{{defn|no=1|A functor is an [[equivalence of categories|equivalence]] if it is faithful, full and essentially surjective.}}
į:
{{defn|no=1|A functor is an '''equivalence of categories|equivalence''' if it is faithful, full and essentially surjective.}}
Pakeistos 244-245 eilutės iš
{{defn|1=A category is equivalent to another category if there is an [[equivalence of categories|equivalence]] between them.}}
į:
{{defn|1=A category is equivalent to another category if there is an '''equivalence of categories|equivalence''' between them.}}
Pakeistos 247-248 eilutės iš
{{defn|1=A functor ''F'' is called [[essentially surjective]] (or isomorphism-dense) if for every object ''B'' there exists an object ''A'' such that ''F''(''A'') is isomorphic to ''B''.}}
į:
{{defn|1=A functor ''F'' is called '''essentially surjective''' (or isomorphism-dense) if for every object ''B'' there exists an object ''A'' such that ''F''(''A'') is isomorphic to ''B''.}}
Pakeista 250 eilutė iš:
{{defn|1=Given categories ''C'', ''D'' and an object ''A'' in ''C'', the [[evaluation (category theory)|evaluation]] at ''A'' is the functor
į:
{{defn|1=Given categories ''C'', ''D'' and an object ''A'' in ''C'', the '''evaluation (category theory)|evaluation''' at ''A'' is the functor
Pakeista 252 eilutė iš:
For example, the [[Eilenberg–Steenrod axioms]] give an instance when the functor is an equivalence.}}
į:
For example, the '''Eilenberg–Steenrod axioms''' give an instance when the functor is an equivalence.}}
Pakeistos 258-259 eilutės iš
{{defn|1=A functor is [[faithful functor|faithful]] if it is injective when restricted to each [[hom-set]].}}
į:
{{defn|1=A functor is '''faithful functor|faithful''' if it is injective when restricted to each '''hom-set'''.}}
Pakeistos 261-262 eilutės iš
{{defn|1=The [[fundamental category functor]] <math>\tau_1: s\mathbf{Set} \to \mathbf{Cat}</math> is the left adjoint to the nerve functor ''N''. For every category ''C'', <math>\tau_1 NC = C</math>.}}
į:
{{defn|1=The '''fundamental category functor''' <math>\tau_1: s\mathbf{Set} \to \mathbf{Cat}</math> is the left adjoint to the nerve functor ''N''. For every category ''C'', <math>\tau_1 NC = C</math>.}}
Pakeistos 264-265 eilutės iš
{{defn|1=The [[fundamental groupoid]] <math>\Pi_1 X</math> of a Kan complex ''X'' is the category where an object is a 0-simplex (vertex) <math>\Delta^0 \to X</math>, a morphism is a homotopy class of a 1-simplex (path) <math>\Delta^1 \to X</math> and a composition is determined by the Kan property.<!-- check this: Equivalently, it is the groupoid completion of the fundamental category <math>\tau_1 X</math> of ''X''.-->}}
į:
{{defn|1=The '''fundamental groupoid''' <math>\Pi_1 X</math> of a Kan complex ''X'' is the category where an object is a 0-simplex (vertex) <math>\Delta^0 \to X</math>, a morphism is a homotopy class of a 1-simplex (path) <math>\Delta^1 \to X</math> and a composition is determined by the Kan property.<!-- check this: Equivalently, it is the groupoid completion of the fundamental category <math>\tau_1 X</math> of ''X''.-->}}
Pakeistos 267-268 eilutės iš
{{defn|1=A functor π: ''C'' → ''D'' is said to exhibit ''C'' as a [[fibered category|category fibered over]] ''D'' if, for each morphism ''g'': ''x'' → π(''y'') in ''D'', there exists a π-cartesian morphism ''f'': ''x<nowiki>'</nowiki>'' → ''y'' in ''C'' such that π(''f'') = ''g''. If ''D'' is the category of affine schemes (say of finite type over some field), then π is more commonly called a [[prestack]]. '''Note''': π is often a forgetful functor and in fact the [[Grothendieck construction]] implies that every fibered category can be taken to be that form (up to equivalences in a suitable sense).}}
į:
{{defn|1=A functor π: ''C'' → ''D'' is said to exhibit ''C'' as a '''fibered category|category fibered over''' ''D'' if, for each morphism ''g'': ''x'' → π(''y'') in ''D'', there exists a π-cartesian morphism ''f'': ''x<nowiki>'</nowiki>'' → ''y'' in ''C'' such that π(''f'') = ''g''. If ''D'' is the category of affine schemes (say of finite type over some field), then π is more commonly called a '''prestack'''. '''Note''': π is often a forgetful functor and in fact the '''Grothendieck construction''' implies that every fibered category can be taken to be that form (up to equivalences in a suitable sense).}}
Pakeistos 270-271 eilutės iš
{{defn|1=Given a category ''C'' and a set ''I'', the [[fiber product]] over an object ''S'' of a family of objects ''X''<sub>''i''</sub> in ''C'' indexed by ''I'' is the product of the family in the [[slice category]] <math>C_{/S}</math> of ''C'' over ''S'' (provided there are <math>X_i \to S</math>). The fiber product of two objects ''X'' and ''Y'' over an object ''S'' is denoted by <math>X \times_S Y</math> and is also called a [[Cartesian square]].}}
į:
{{defn|1=Given a category ''C'' and a set ''I'', the '''fiber product''' over an object ''S'' of a family of objects ''X''<sub>''i''</sub> in ''C'' indexed by ''I'' is the product of the family in the '''slice category''' <math>C_{/S}</math> of ''C'' over ''S'' (provided there are <math>X_i \to S</math>). The fiber product of two objects ''X'' and ''Y'' over an object ''S'' is denoted by <math>X \times_S Y</math> and is also called a '''Cartesian square'''.}}
Pakeista 273 eilutė iš:
{{defn|no=1|1=A [[filtered category]] (also called a filtrant category) is a nonempty category with the properties (1) given objects ''i'' and ''j'', there are an object ''k'' and morphisms ''i'' → ''k'' and ''j'' → ''k'' and (2) given morphisms ''u'', ''v'': ''i'' → ''j'', there are an object ''k'' and a morphism ''w'': ''j'' → ''k'' such that ''w'' ∘ ''u'' = ''w'' ∘ ''v''. A category ''I'' is filtered if and only if, for each finite category ''J'' and functor ''f'': ''J'' → ''I'', the set <math>\varprojlim \operatorname{Hom}(f(j), i)</math> is nonempty for some object ''i'' in ''I''.}}
į:
{{defn|no=1|1=A '''filtered category''' (also called a filtrant category) is a nonempty category with the properties (1) given objects ''i'' and ''j'', there are an object ''k'' and morphisms ''i'' → ''k'' and ''j'' → ''k'' and (2) given morphisms ''u'', ''v'': ''i'' → ''j'', there are an object ''k'' and a morphism ''w'': ''j'' → ''k'' such that ''w'' ∘ ''u'' = ''w'' ∘ ''v''. A category ''I'' is filtered if and only if, for each finite category ''J'' and functor ''f'': ''J'' → ''I'', the set <math>\varprojlim \operatorname{Hom}(f(j), i)</math> is nonempty for some object ''i'' in ''I''.}}
Pakeistos 277-278 eilutės iš
{{defn|1=A [[finitary monad]] or an algebraic monad is a monad on '''Set''' whose underlying endofunctor commutes with filtered colimits.}}
į:
{{defn|1=A '''finitary monad''' or an algebraic monad is a monad on '''Set''' whose underlying endofunctor commutes with filtered colimits.}}
Pakeistos 283-284 eilutės iš
{{defn|The [[forgetful functor]] is, roughly, a functor that loses some of data of the objects; for example, the functor <math>\mathbf{Grp} \to \mathbf{Set}</math> that sends a group to its underlying set and a group homomorphism to itself is a forgetful functor.}}
į:
{{defn|The '''forgetful functor''' is, roughly, a functor that loses some of data of the objects; for example, the functor <math>\mathbf{Grp} \to \mathbf{Set}</math> that sends a group to its underlying set and a group homomorphism to itself is a forgetful functor.}}
Pakeistos 286-287 eilutės iš
{{defn|1=A [[free functor]] is a left adjoint to a forgetful functor. For example, for a ring ''R'', the functor that sends a set ''X'' to the [[free module|free ''R''-module]] generated by ''X'' is a free functor (whence the name).}}
į:
{{defn|1=A '''free functor''' is a left adjoint to a forgetful functor. For example, for a ring ''R'', the functor that sends a set ''X'' to the '''free module|free ''R''-module''' generated by ''X'' is a free functor (whence the name).}}
Pakeistos 289-290 eilutės iš
{{defn|1=A [[Frobenius category]] is an [[exact category]] that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects.}}
į:
{{defn|1=A '''Frobenius category''' is an '''exact category''' that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects.}}
Pakeistos 292-293 eilutės iš
{{defn|1=See [[Fukaya category]].}}
į:
{{defn|1=See '''Fukaya category'''.}}
Pakeistos 295-297 eilutės iš
{{defn|no=1|1=A functor is [[full functor|full]] if it is surjective when restricted to each [[hom-set]].}}
{{defn|no=2|1=A category ''A'' is a [[full subcategory]] of a category ''B'' if the inclusion functor from ''A'' to ''B'' is full.}}
į:
{{defn|no=1|1=A functor is '''full functor|full''' if it is surjective when restricted to each '''hom-set'''.}}
{{defn|no=2|1=A category ''A'' is a '''full subcategory''' of a category ''B'' if the inclusion functor from ''A'' to ''B'' is full.}}
Pakeista 299 eilutė iš:
{{defn|1=Given categories ''C'', ''D'', a [[functor]] ''F'' from ''C'' to ''D'' is a structure-preserving map from ''C'' to ''D''; i.e., it consists of an object ''F''(''x'') in ''D'' for each object ''x'' in ''C'' and a morphism ''F''(''f'') in ''D'' for each morphism ''f'' in ''C'' satisfying the conditions: (1) <math>F(f \circ g) = F(f) \circ F(g)</math> whenever <math>f \circ g</math> is defined and (2) <math>F(\operatorname{id}_x) = \operatorname{id}_{F(x)}</math>. For example,
į:
{{defn|1=Given categories ''C'', ''D'', a '''functor''' ''F'' from ''C'' to ''D'' is a structure-preserving map from ''C'' to ''D''; i.e., it consists of an object ''F''(''x'') in ''D'' for each object ''x'' in ''C'' and a morphism ''F''(''f'') in ''D'' for each morphism ''f'' in ''C'' satisfying the conditions: (1) <math>F(f \circ g) = F(f) \circ F(g)</math> whenever <math>f \circ g</math> is defined and (2) <math>F(\operatorname{id}_x) = \operatorname{id}_{F(x)}</math>. For example,
Pakeistos 301-302 eilutės iš
where <math>\mathfrak{P}(S)</math> is the [[power set]] of ''S'' is a functor if we define: for each function <math>f: S \to T</math>, <math>\mathfrak{P}(f): \mathfrak{P}(S) \to \mathfrak{P}(T)</math> by <math>\mathfrak{P}(f)(A) = f(A)</math>.}}
į:
where <math>\mathfrak{P}(S)</math> is the '''power set''' of ''S'' is a functor if we define: for each function <math>f: S \to T</math>, <math>\mathfrak{P}(f): \mathfrak{P}(S) \to \mathfrak{P}(T)</math> by <math>\mathfrak{P}(f)(A) = f(A)</math>.}}
Pakeistos 304-305 eilutės iš
{{defn|1=The [[functor category]] '''Fct'''(''C'', ''D'') from a category ''C'' to a category ''D'' is the category where the objects are all the functors from ''C'' to ''D'' and the morphisms are all the natural transformations between the functors.}}
į:
{{defn|1=The '''functor category''' '''Fct'''(''C'', ''D'') from a category ''C'' to a category ''D'' is the category where the objects are all the functors from ''C'' to ''D'' and the morphisms are all the natural transformations between the functors.}}
Pakeistos 311-312 eilutės iš
{{defn|1=The [[Gabriel–Popescu theorem]] says an abelian category is a [[Serre quotient category|quotient]] of the category of modules.}}
į:
{{defn|1=The '''Gabriel–Popescu theorem''' says an abelian category is a '''Serre quotient category|quotient''' of the category of modules.}}
Pakeistos 314-315 eilutės iš
{{defn|1=In a category ''C'', a family of objects <math>G_i, i \in I</math> is a [[generator (category theory)|system of generators]] of ''C'' if the functor <math>X \mapsto \prod_{i \in I} \operatorname{Hom}(G_i, X)</math> is conservative. Its dual is called a system of cogenerators.}}
į:
{{defn|1=In a category ''C'', a family of objects <math>G_i, i \in I</math> is a '''generator (category theory)|system of generators''' of ''C'' if the functor <math>X \mapsto \prod_{i \in I} \operatorname{Hom}(G_i, X)</math> is conservative. Its dual is called a system of cogenerators.}}
Pakeistos 317-318 eilutės iš
{{defn|1=A category-theoretic generalization of [[Galois theory]]; see [[Grothendieck's Galois theory]].}}
į:
{{defn|1=A category-theoretic generalization of '''Galois theory'''; see '''Grothendieck's Galois theory'''.}}
Pakeistos 320-321 eilutės iš
{{defn|1=A [[Grothendieck category]] is a certain well-behaved kind of an abelian category.}}
į:
{{defn|1=A '''Grothendieck category''' is a certain well-behaved kind of an abelian category.}}
Pakeistos 323-324 eilutės iš
{{defn|1=Given a functor <math>U: C \to \mathbf{Cat}</math>, let ''D''<sub>''U''</sub> be the category where the objects are pairs (''x'', ''u'') consisting of an object ''x'' in ''C'' and an object ''u'' in the category ''U''(''x'') and a morphism from (''x'', ''u'') to (''y'', ''v'') is a pair consisting of a morphism ''f'': ''x'' → ''y'' in ''C'' and a morphism ''U''(''f'')(''u'') → ''v'' in ''U''(''y''). The passage from ''U'' to ''D''<sub>''U''</sub> is then called the [[Grothendieck construction]].}}
į:
{{defn|1=Given a functor <math>U: C \to \mathbf{Cat}</math>, let ''D''<sub>''U''</sub> be the category where the objects are pairs (''x'', ''u'') consisting of an object ''x'' in ''C'' and an object ''u'' in the category ''U''(''x'') and a morphism from (''x'', ''u'') to (''y'', ''v'') is a pair consisting of a morphism ''f'': ''x'' → ''y'' in ''C'' and a morphism ''U''(''f'')(''u'') → ''v'' in ''U''(''y''). The passage from ''U'' to ''D''<sub>''U''</sub> is then called the '''Grothendieck construction'''.}}
Pakeistos 326-327 eilutės iš
{{defn|1=A [[fibered category]].}}
į:
{{defn|1=A '''fibered category'''.}}
Pakeistos 329-331 eilutės iš
{{defn|no=1|1=A category is called a [[groupoid]] if every morphism in it is an isomorphism.}}
{{defn|no=2|1=An ∞-category is called an [[∞-groupoid]] if every morphism in it is an equivalence (or equivalently if it is a [[Kan complex]].)}}
į:
{{defn|no=1|1=A category is called a '''groupoid''' if every morphism in it is an isomorphism.}}
{{defn|no=2|1=An ∞-category is called an '''∞-groupoid''' if every morphism in it is an equivalence (or equivalently if it is a '''Kan complex'''.)}}
Pakeistos 337-338 eilutės iš
{{defn|1=See [[Ringel–Hall algebra]].}}
į:
{{defn|1=See '''Ringel–Hall algebra'''.}}
Pakeistos 340-341 eilutės iš
{{defn|1=The [[heart (category theory)|heart]] of a [[t-structure]] (<math>D^{\ge 0}</math>, <math>D^{\le 0}</math>) on a triangulated category is the intersection <math>D^{\ge 0} \cap D^{\le 0}</math>. It is an abelian category.}}
į:
{{defn|1=The '''heart (category theory)|heart''' of a '''t-structure''' (<math>D^{\ge 0}</math>, <math>D^{\le 0}</math>) on a triangulated category is the intersection <math>D^{\ge 0} \cap D^{\le 0}</math>. It is an abelian category.}}
Pakeistos 343-344 eilutės iš
{{defn|1=[[Higher category theory]] is a subfield of category theory that concerns the study of [[n-category|''n''-categories]] and [[∞-categories]].}}
į:
{{defn|1='''Higher category theory''' is a subfield of category theory that concerns the study of '''n-category|''n''-categories''' and '''∞-categories'''.}}
Pakeistos 346-347 eilutės iš
{{defn|1=The [[homological dimension]] of an abelian category with enough injectives is the least non-negative intege ''n'' such that every object in the category admits an injective resolution of length at most ''n''. The dimension is ∞ if no such integer exists. For example, the homological dimension of [[category of modules|Mod<sub>''R''</sub>]] with a principal ideal domain ''R'' is at most one.}}
į:
{{defn|1=The '''homological dimension''' of an abelian category with enough injectives is the least non-negative intege ''n'' such that every object in the category admits an injective resolution of length at most ''n''. The dimension is ∞ if no such integer exists. For example, the homological dimension of '''category of modules|Mod<sub>''R''</sub>''' with a principal ideal domain ''R'' is at most one.}}
Pakeistos 349-350 eilutės iš
{{defn|1=See<!-- for now --> [[homotopy category]]. It is closely related to a [[localization of a category]].}}
į:
{{defn|1=See<!-- for now --> '''homotopy category'''. It is closely related to a '''localization of a category'''.}}
Pakeista 352 eilutė iš:
{{defn|1=The [[homotopy hypothesis]] states an [[∞-groupoid]] is a space (less equivocally, an ''n''-groupoid can be used as a homotopy ''n''-type.)}}
į:
{{defn|1=The '''homotopy hypothesis''' states an '''∞-groupoid''' is a space (less equivocally, an ''n''-groupoid can be used as a homotopy ''n''-type.)}}
Pakeistos 358-361 eilutės iš
{{defn|no=1|1=The [[identity morphism]] ''f'' of an object ''A'' is a morphism from ''A'' to ''A'' such that for any morphisms ''g'' with domain ''A'' and ''h'' with codomain ''A'', <math>g\circ f=g</math> and <math>f\circ h=h</math>.}}
{{defn|no=2|The [[identity functor]] on a category
''C'' is a functor from ''C'' to ''C'' that sends objects and morphisms to themselves.}}
{{defn|no=3|Given a functor ''F'': ''C'' → ''D'', the [[identity natural transformation]] from ''F'' to ''F'' is a natural transformation consisting of the identity morphisms of ''F''(''X'') in ''D'' for the objects ''X'' in ''C''.}}
į:
{{defn|no=1|1=The '''identity morphism''' ''f'' of an object ''A'' is a morphism from ''A'' to ''A'' such that for any morphisms ''g'' with domain ''A'' and ''h'' with codomain ''A'', <math>g\circ f=g</math> and <math>f\circ h=h</math>.}}
{{defn|no=2|The
'''identity functor''' on a category ''C'' is a functor from ''C'' to ''C'' that sends objects and morphisms to themselves.}}
{{defn|no=3|Given a functor ''F'': ''C'' → ''D'', the '''identity natural transformation''' from ''F'' to ''F'' is a natural transformation consisting of the identity morphisms of ''F''(''X'') in ''D'' for the objects ''X'' in ''C''.}}
Pakeistos 363-364 eilutės iš
{{defn|1=The [[image of a morphism|image]] of a morphism ''f'': ''X'' → ''Y'' is the equalizer of <math>Y \rightrightarrows Y \sqcup_X Y</math>.}}
į:
{{defn|1=The '''image of a morphism|image''' of a morphism ''f'': ''X'' → ''Y'' is the equalizer of <math>Y \rightrightarrows Y \sqcup_X Y</math>.}}
Pakeista 369 eilutė iš:
{{defn|1=An [[∞-category]] ''C'' is a [[simplicial set]] satisfying the following condition: for each 0 < ''i'' < ''n'',
į:
{{defn|1=An '''∞-category''' ''C'' is a '''simplicial set''' satisfying the following condition: for each 0 < ''i'' < ''n'',
Pakeistos 371-372 eilutės iš
where &Delta;<sup>''n''</sup> is the standard ''n''-simplex and <math>\Lambda^n_i</math> is obtained from &Delta;<sup>''n''</sup> by removing the ''i''-th face and the interior (see [[Kan fibration#Definition]]). For example, the [[nerve of a category]] satisfies the condition and thus can be considered as an ∞-category.}}
į:
where &Delta;<sup>''n''</sup> is the standard ''n''-simplex and <math>\Lambda^n_i</math> is obtained from &Delta;<sup>''n''</sup> by removing the ''i''-th face and the interior (see '''Kan fibration#Definition'''). For example, the '''nerve of a category''' satisfies the condition and thus can be considered as an ∞-category.}}
Pakeistos 374-376 eilutės iš
{{defn|no=1|1=An object ''A'' is [[initial object|initial]] if there is exactly one morphism from ''A'' to each object; e.g., [[empty set]] in '''[[Category of sets|Set]]'''.}}
{{defn
|no=2|1=An object ''A'' in an ∞-category ''C'' is initial if <math>\operatorname{Map}_C(A, B)</math> is [[contractible space|contractible]] for each object ''B'' in ''C''.}}
į:
{{defn|no=1|1=An object ''A'' is '''initial object|initial''' if there is exactly one morphism from ''A'' to each object; e.g., '''empty set''' in ''''''Category of sets|Set''''''.}}
{{defn|no=2|1=An object ''A'' in an ∞-category ''C'' is initial if <math>\operatorname{Map}_C(A, B)</math> is '''contractible space|contractible'''
for each object ''B'' in ''C''.}}
Pakeistos 378-379 eilutės iš
{{defn|1=An object ''A'' in an abelian category is [[injective object|injective]] if the functor <math>\operatorname{Hom}(-, A)</math> is exact. It is the dual of a projective object.}}
į:
{{defn|1=An object ''A'' in an abelian category is '''injective object|injective''' if the functor <math>\operatorname{Hom}(-, A)</math> is exact. It is the dual of a projective object.}}
Pakeistos 381-382 eilutės iš
{{defn|1=Given a [[monoidal category]] (''C'', ⊗), the [[internal Hom]] is a functor <math>[-, -]: C^{\text{op}} \times C \to C</math> such that <math>[Y, -]</math> is the right adjoint to <math>- \otimes Y</math> for each object ''Y'' in ''C''. For example, the [[category of modules]] over a commutative ring ''R'' has the internal Hom given as <math>[M, N] = \operatorname{Hom}_R(M, N)</math>, the set of ''R''-linear maps.}}
į:
{{defn|1=Given a '''monoidal category''' (''C'', ⊗), the '''internal Hom''' is a functor <math>[-, -]: C^{\text{op}} \times C \to C</math> such that <math>[Y, -]</math> is the right adjoint to <math>- \otimes Y</math> for each object ''Y'' in ''C''. For example, the '''category of modules''' over a commutative ring ''R'' has the internal Hom given as <math>[M, N] = \operatorname{Hom}_R(M, N)</math>, the set of ''R''-linear maps.}}
Pakeistos 384-385 eilutės iš
{{defn|1=A morphism ''f'' is an [[inverse function|inverse]] to a morphism ''g'' if <math>g\circ f</math> is defined and is equal to the identity morphism on the codomain of ''g'', and <math>f\circ g</math> is defined and equal to the identity morphism on the domain of ''g''. The inverse of ''g'' is unique and is denoted by ''g''<sup>−1</sup>. ''f'' is a left inverse to ''g'' if <math>f\circ g</math> is defined and is equal to the identity morphism on the domain of ''g'', and similarly for a right inverse.}}
į:
{{defn|1=A morphism ''f'' is an '''inverse function|inverse''' to a morphism ''g'' if <math>g\circ f</math> is defined and is equal to the identity morphism on the codomain of ''g'', and <math>f\circ g</math> is defined and equal to the identity morphism on the domain of ''g''. The inverse of ''g'' is unique and is denoted by ''g''<sup>−1</sup>. ''f'' is a left inverse to ''g'' if <math>f\circ g</math> is defined and is equal to the identity morphism on the domain of ''g'', and similarly for a right inverse.}}
Pakeista 387 eilutė iš:
{{defn|no=1|1=An object is [[isomorphic]] to another object if there is an isomorphism between them.}}
į:
{{defn|no=1|1=An object is '''isomorphic''' to another object if there is an isomorphism between them.}}
Pakeista 391 eilutė iš:
{{defn|1=A morphism ''f'' is an [[isomorphism]] if there exists an ''inverse'' of ''f''.}}
į:
{{defn|1=A morphism ''f'' is an '''isomorphism''' if there exists an ''inverse'' of ''f''.}}
Pakeistos 397-398 eilutės iš
{{defn|1=A [[Kan complex]] is a [[fibrant object]] in the category of simplicial sets.}}
į:
{{defn|1=A '''Kan complex''' is a '''fibrant object''' in the category of simplicial sets.}}
Pakeista 400 eilutė iš:
{{defn|no=1|Given a category ''C'', the left [[Kan extension]] functor along a functor <math>f: I \to J</math> is the left adjoint (if it exists) to <math>f^* = - \circ f: \operatorname{Fct}(J, C) \to \operatorname{Fct}(I, C)</math> and is denoted by <math>f_!</math>. For any <math>\alpha: I \to C</math>, the functor <math>f_! \alpha: J \to C</math> is called the left Kan extension of α along ''f''.<ref>http://www.math.harvard.edu/~lurie/282ynotes/LectureXI-Homological.pdf</ref> One can show:
į:
{{defn|no=1|Given a category ''C'', the left '''Kan extension''' functor along a functor <math>f: I \to J</math> is the left adjoint (if it exists) to <math>f^* = - \circ f: \operatorname{Fct}(J, C) \to \operatorname{Fct}(I, C)</math> and is denoted by <math>f_!</math>. For any <math>\alpha: I \to C</math>, the functor <math>f_! \alpha: J \to C</math> is called the left Kan extension of α along ''f''.<ref>http://www.math.harvard.edu/~lurie/282ynotes/LectureXI-Homological.pdf</ref> One can show:
Pakeista 406 eilutė iš:
{{defn|1=Given a monad ''T'', the [[Kleisli category]] of ''T'' is the full subcategory of the category of ''T''-algebras (called Eilenberg–Moore category) that consists of free ''T''-algebras.}}
į:
{{defn|1=Given a monad ''T'', the '''Kleisli category''' of ''T'' is the full subcategory of the category of ''T''-algebras (called Eilenberg–Moore category) that consists of free ''T''-algebras.}}
Pakeistos 412-413 eilutės iš
{{defn|1=The term "[[lax functor]]" is essentially synonymous with "[[pseudo-functor]]".}}
į:
{{defn|1=The term "'''lax functor'''" is essentially synonymous with "'''pseudo-functor'''".}}
Pakeistos 415-416 eilutės iš
{{defn|1=An object in an abelian category is said to have <span id="finite length"></span>finite length if it has a [[composition series]]. The maximum number of proper subobjects in any such composition series is called the '''length''' of ''A''.<ref>{{harvnb|Kashiwara|Schapira|2006|loc=exercise 8.20}}</ref>}}
į:
{{defn|1=An object in an abelian category is said to have <span id="finite length"></span>finite length if it has a '''composition series'''. The maximum number of proper subobjects in any such composition series is called the '''length''' of ''A''.<ref>{{harvnb|Kashiwara|Schapira|2006|loc=exercise 8.20}}</ref>}}
Pakeista 418 eilutė iš:
{{defn|no=1|The [[limit (category theory)|limit]] (or [[projective limit]]) of a functor <math>f: I^{\text{op}} \to \mathbf{Set}</math> is
į:
{{defn|no=1|The '''limit (category theory)|limit''' (or '''projective limit''') of a functor <math>f: I^{\text{op}} \to \mathbf{Set}</math> is
Pakeista 421 eilutė iš:
{{defn|no=3|The [[colimit]] (or [[inductive limit]]) <math>\varinjlim_{i \in I} f(i)</math> is the dual of a limit; i.e., given a functor <math>f: I \to C</math>, it satisfies: for any ''X'', <math>\operatorname{Hom}(\varinjlim f(i), X) = \varprojlim \operatorname{Hom}(f(i), X)</math>. Explicitly, to give <math>\varinjlim f(i) \to X</math> is to give a family of morphisms <math>f(i) \to X</math> such that for any <math>i \to j</math>, <math>f(i) \to X</math> is <math>f(i) \to f(j) \to X</math>. Perhaps the simplest example of a colimit is a [[coequalizer]]. For another example, take ''f'' to be the identity functor on ''C'' and suppose <math>L = \varinjlim_{X \in C} f(X)</math> exists; then the identity morphism on ''L'' corresponds to a compatible family of morphisms <math>\alpha_X: X \to L</math> such that <math>\alpha_L</math> is the identity. If <math>f: X \to L</math> is any morphism, then <math>f = \alpha_L \circ f = \alpha_X</math>; i.e., ''L'' is a final object of ''C''.
į:
{{defn|no=3|The '''colimit''' (or '''inductive limit''') <math>\varinjlim_{i \in I} f(i)</math> is the dual of a limit; i.e., given a functor <math>f: I \to C</math>, it satisfies: for any ''X'', <math>\operatorname{Hom}(\varinjlim f(i), X) = \varprojlim \operatorname{Hom}(f(i), X)</math>. Explicitly, to give <math>\varinjlim f(i) \to X</math> is to give a family of morphisms <math>f(i) \to X</math> such that for any <math>i \to j</math>, <math>f(i) \to X</math> is <math>f(i) \to f(j) \to X</math>. Perhaps the simplest example of a colimit is a '''coequalizer'''. For another example, take ''f'' to be the identity functor on ''C'' and suppose <math>L = \varinjlim_{X \in C} f(X)</math> exists; then the identity morphism on ''L'' corresponds to a compatible family of morphisms <math>\alpha_X: X \to L</math> such that <math>\alpha_L</math> is the identity. If <math>f: X \to L</math> is any morphism, then <math>f = \alpha_L \circ f = \alpha_X</math>; i.e., ''L'' is a final object of ''C''.
Pakeista 425 eilutė iš:
{{defn|1=See [[localization of a category]].}}
į:
{{defn|1=See '''localization of a category'''.}}
Pakeista 431 eilutė iš:
{{defn|1=A [[monad (category theory)|monad]] in a category ''X'' is a [[monoid object]] in the monoidal category of endofunctors of ''X'' with the monoidal structure given by composition. For example, given a group ''G'', define an endofunctor ''T'' on '''Set''' by <math>T(X) = G \times X</math>. Then define the multiplication ''μ'' on ''T'' as the natural transformation <math>\mu: T \circ T \to T</math> given by
į:
{{defn|1=A '''monad (category theory)|monad''' in a category ''X'' is a '''monoid object''' in the monoidal category of endofunctors of ''X'' with the monoidal structure given by composition. For example, given a group ''G'', define an endofunctor ''T'' on '''Set''' by <math>T(X) = G \times X</math>. Then define the multiplication ''μ'' on ''T'' as the natural transformation <math>\mu: T \circ T \to T</math> given by
Pakeistos 436-438 eilutės iš
{{defn|no=1|1=An adjunction is said to be [[monadic adjunction|monadic]] if it comes from the monad that it determines by means of the [[Eilenberg–Moore category]] (the category of algebras for the monad).}}
{{defn|no=2|1=A functor is said to be [[monadic functor|monadic]] if it is a constituent of a monadic adjunction.}}
į:
{{defn|no=1|1=An adjunction is said to be '''monadic adjunction|monadic''' if it comes from the monad that it determines by means of the '''Eilenberg–Moore category''' (the category of algebras for the monad).}}
{{defn|no=2|1=A functor is said to be '''monadic functor|monadic''' if it is a constituent of a monadic adjunction.}}
Pakeistos 440-441 eilutės iš
{{defn|1=A [[monoidal category]], also called a tensor category, is a category ''C'' equipped with (1) a [[bifunctor]] <math>\otimes: C \times C \to C</math>, (2) an identity object and (3) natural isomorphisms that make ⊗ associative and the identity object an identity for ⊗, subject to certain coherence conditions.}}
į:
{{defn|1=A '''monoidal category''', also called a tensor category, is a category ''C'' equipped with (1) a '''bifunctor''' <math>\otimes: C \times C \to C</math>, (2) an identity object and (3) natural isomorphisms that make ⊗ associative and the identity object an identity for ⊗, subject to certain coherence conditions.}}
Pakeistos 443-444 eilutės iš
{{defn|1=A [[monoid object]] in a monoidal category is an object together with the multiplication map and the identity map that satisfy the expected conditions like associativity. For example, a monoid object in '''Set''' is a usual monoid (unital semigroup) and a monoid object in '''[[category of modules|''R''-mod]]''' is an [[associative algebra]] over a commutative ring ''R''.}}
į:
{{defn|1=A '''monoid object''' in a monoidal category is an object together with the multiplication map and the identity map that satisfy the expected conditions like associativity. For example, a monoid object in '''Set''' is a usual monoid (unital semigroup) and a monoid object in ''''''category of modules|''R''-mod'''''' is an '''associative algebra''' over a commutative ring ''R''.}}
Pakeistos 446-447 eilutės iš
{{defn|1=A morphism ''f'' is a [[monomorphism]] (also called monic) if <math>g=h</math> whenever <math>f\circ g=f\circ h</math>; e.g., an [[Injective function|injection]] in '''[[Category of sets|Set]]'''. In other words, ''f'' is the dual of an epimorphism.}}
į:
{{defn|1=A morphism ''f'' is a '''monomorphism''' (also called monic) if <math>g=h</math> whenever <math>f\circ g=f\circ h</math>; e.g., an '''Injective function|injection''' in ''''''Category of sets|Set''''''. In other words, ''f'' is the dual of an epimorphism.}}
Pakeista 449 eilutė iš:
{{defn|1=A [[multicategory]] is a generalization of a category in which a morphism is allowed to have more than one domain. It is the same thing as a [[colored operad]].<ref>https://ncatlab.org/nlab/show/multicategory</ref>}}
į:
{{defn|1=A '''multicategory''' is a generalization of a category in which a morphism is allowed to have more than one domain. It is the same thing as a '''colored operad'''.<ref>https://ncatlab.org/nlab/show/multicategory</ref>}}
Pakeistos 462-465 eilutės iš
}}{{defn|no=1|1=A [[strict n-category|strict ''n''-category]] is defined inductively: a strict 0-category is a set and a strict ''n''-category is a category whose Hom sets are strict (''n''-1)-categories. Precisely, a strict ''n''-category is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category.}}
{{defn|no=2|1=The notion of a [[weak n-category|weak ''n''-category]] is obtained from the strict one by weakening the conditions like associativity of composition to hold only up to [[coherent isomorphism]]s in the weak sense.}}
{{defn|no=3|1=One can define an ∞-category as a kind of a colim of ''n''-categories. Conversely, if one has the notion of a (weak) ∞-category (say a [[quasi-category]]) in the beginning, then a weak ''n''-category can be defined as a type of a truncated ∞-category.}}
į:
}}{{defn|no=1|1=A '''strict n-category|strict ''n''-category''' is defined inductively: a strict 0-category is a set and a strict ''n''-category is a category whose Hom sets are strict (''n''-1)-categories. Precisely, a strict ''n''-category is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category.}}
{{defn|no=2|1=The notion of a '''weak n-category|weak ''n''-category''' is obtained from the strict one by weakening the conditions like associativity of composition to hold only up to '''coherent isomorphism'''s in the weak sense.}}
{{defn|no=3|1=One can define an ∞-category as a kind of a colim of ''n''-categories. Conversely, if one has the notion of a (weak) ∞-category (say a '''quasi-category''') in the beginning, then a weak ''n''-category can be defined as a type of a truncated ∞-category.}}
Pakeista 467 eilutė iš:
{{defn|no=1|A natural transformation is, roughly, a map between functors. Precisely, given a pair of functors ''F'', ''G'' from a category ''C'' to category ''D'', a [[natural transformation]] φ from ''F'' to ''G'' is a set of morphisms in ''D''
į:
{{defn|no=1|A natural transformation is, roughly, a map between functors. Precisely, given a pair of functors ''F'', ''G'' from a category ''C'' to category ''D'', a '''natural transformation''' φ from ''F'' to ''G'' is a set of morphisms in ''D''
Pakeistos 469-472 eilutės iš
satisfying the condition: for each morphism ''f'': ''x'' → ''y'' in ''C'', <math>\phi_y \circ F(f) = G(f) \circ \phi_x</math>. For example, writing <math>GL_n(R)</math> for the group of invertible ''n''-by-''n'' matrices with coefficients in a commutative ring ''R'', we can view <math>GL_n</math> as a functor from the category '''CRing''' of commutative rings to the category '''Grp''' of groups. Similarly, <math>R \mapsto R^*</math> is a functor from '''CRing''' to '''Grp'''. Then the [[determinant]] det is a natural transformation from <math>GL_n</math> to -<sup>*</sup>.}}
{{defn|no=2|A [[natural isomorphism]] is a natural transformation that is an isomorphism (i.e., admits the inverse).}}

[[Image:Nerve-2-simplex.png|thumb|right|The composition is encoded as a 2-simplex.]]
į:
satisfying the condition: for each morphism ''f'': ''x'' → ''y'' in ''C'', <math>\phi_y \circ F(f) = G(f) \circ \phi_x</math>. For example, writing <math>GL_n(R)</math> for the group of invertible ''n''-by-''n'' matrices with coefficients in a commutative ring ''R'', we can view <math>GL_n</math> as a functor from the category '''CRing''' of commutative rings to the category '''Grp''' of groups. Similarly, <math>R \mapsto R^*</math> is a functor from '''CRing''' to '''Grp'''. Then the '''determinant''' det is a natural transformation from <math>GL_n</math> to -<sup>*</sup>.}}
{{defn|no=2|A '''natural isomorphism''' is a natural transformation that is an isomorphism (i.e., admits the inverse).}}

'''Image:Nerve-2-simplex.png|thumb|right|The composition is encoded as a 2-simplex.'''
Pakeistos 474-475 eilutės iš
{{defn|1=The [[nerve functor]] ''N'' is the functor from '''Cat''' to ''s'''''Set''' given by <math>N(C)_n = \operatorname{Hom}_{\mathbf{Cat}}([n], C)</math>. For example, if <math>\varphi</math> is a functor in <math>N(C)_2</math> (called a 2-simplex), let <math>x_i = \varphi(i), \, 0 \le i \le 2</math>. Then <math>\varphi(0 \to 1)</math> is a morphism <math>f: x_0 \to x_1</math> in ''C'' and also <math>\varphi(1 \to 2) = g: x_1 \to x_2</math> for some ''g'' in ''C''. Since <math>0 \to 2</math> is <math>0 \to 1</math> followed by <math>1 \to 2</math> and since <math>\varphi</math> is a functor, <math>\varphi(0 \to 2) = g \circ f</math>. In other words, <math>\varphi</math> encodes ''f'', ''g'' and their compositions.}}
į:
{{defn|1=The '''nerve functor''' ''N'' is the functor from '''Cat''' to ''s'''''Set''' given by <math>N(C)_n = \operatorname{Hom}_{\mathbf{Cat}}([n], C)</math>. For example, if <math>\varphi</math> is a functor in <math>N(C)_2</math> (called a 2-simplex), let <math>x_i = \varphi(i), \, 0 \le i \le 2</math>. Then <math>\varphi(0 \to 1)</math> is a morphism <math>f: x_0 \to x_1</math> in ''C'' and also <math>\varphi(1 \to 2) = g: x_1 \to x_2</math> for some ''g'' in ''C''. Since <math>0 \to 2</math> is <math>0 \to 1</math> followed by <math>1 \to 2</math> and since <math>\varphi</math> is a functor, <math>\varphi(0 \to 2) = g \circ f</math>. In other words, <math>\varphi</math> encodes ''f'', ''g'' and their compositions.}}
Pakeista 477 eilutė iš:
{{defn|1=A category is [[normal category|normal]] if every monic is normal.{{citation needed|date=October 2015}}}}
į:
{{defn|1=A category is '''normal category|normal''' if every monic is normal.{{citation needed|date=October 2015}}}}
Pakeistos 484-485 eilutės iš
{{defn|no=2|1=An [adjective] object in a category ''C'' is a contravariant functor (or presheaf) from some fixed category corresponding to the "adjective" to ''C''. For example, a [[simplicial object]] in ''C'' is a contravariant functor from the simplicial category to ''C'' and a [[Γ-object]] is a pointed contravariant functor from [[Γ (category theory)|Γ]] (roughly the pointed category of pointed finite sets) to ''C'' provided ''C'' is pointed.}}
į:
{{defn|no=2|1=An [adjective] object in a category ''C'' is a contravariant functor (or presheaf) from some fixed category corresponding to the "adjective" to ''C''. For example, a '''simplicial object''' in ''C'' is a contravariant functor from the simplicial category to ''C'' and a '''Γ-object''' is a pointed contravariant functor from '''Γ (category theory)|Γ''' (roughly the pointed category of pointed finite sets) to ''C'' provided ''C'' is pointed.}}
Pakeistos 487-488 eilutės iš
{{defn|1=A functor π:''C'' → ''D'' is an [[op-fibration]] if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one π-coCartesian morphism ''f'': ''x'' → ''y<nowiki>'</nowiki>'' in ''C'' such that π(''f'') = ''g''. In other words, π is the dual of a [[Grothendieck fibration]].}}
į:
{{defn|1=A functor π:''C'' → ''D'' is an '''op-fibration''' if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one π-coCartesian morphism ''f'': ''x'' → ''y<nowiki>'</nowiki>'' in ''C'' such that π(''f'') = ''g''. In other words, π is the dual of a '''Grothendieck fibration'''.}}
Pakeista 490 eilutė iš:
{{defn|1=The [[opposite category]] of a category is obtained by reversing the arrows. For example, if a partially ordered set is viewed as a category, taking its opposite amounts to reversing the ordering.}}
į:
{{defn|1=The '''opposite category''' of a category is obtained by reversing the arrows. For example, if a partially ordered set is viewed as a category, taking its opposite amounts to reversing the ordering.}}
Pakeistos 496-497 eilutės iš
{{defn|Sometimes synonymous with "compact". See [[perfect complex]].}}
į:
{{defn|Sometimes synonymous with "compact". See '''perfect complex'''.}}
Pakeistos 502-503 eilutės iš
{{defn|1=A functor from the category of finite-dimensional vector spaces to itself is called a [[polynomial functor]] if, for each pair of vector spaces ''V'', ''W'', {{nowrap|''F'': Hom(''V'', ''W'') → Hom(''F''(''V''), ''F''(''W''))}} is a polynomial map between the vector spaces. A [[Schur functor]] is a basic example.}}
į:
{{defn|1=A functor from the category of finite-dimensional vector spaces to itself is called a '''polynomial functor''' if, for each pair of vector spaces ''V'', ''W'', {{nowrap|''F'': Hom(''V'', ''W'') → Hom(''F''(''V''), ''F''(''W''))}} is a polynomial map between the vector spaces. A '''Schur functor''' is a basic example.}}
Pakeistos 505-506 eilutės iš
{{defn|1=A category is [[preadditive category|preadditive]] if it is [[enriched category|enriched]] over the [[monoidal category]] of [[abelian group]]s. More generally, it is [[preadditive category#R-linear categories|''R''-linear]] if it is enriched over the monoidal category of [[module (mathematics)|''R''-modules]], for ''R'' a [[commutative ring]].}}
į:
{{defn|1=A category is '''preadditive category|preadditive''' if it is '''enriched category|enriched''' over the '''monoidal category''' of '''abelian group'''s. More generally, it is '''preadditive category#R-linear categories|''R''-linear''' if it is enriched over the monoidal category of '''module (mathematics)|''R''-modules''', for ''R'' a '''commutative ring'''.}}
Pakeistos 508-509 eilutės iš
{{defn|Given a [[regular cardinal]] κ, a category is [[presentabl category|κ-presentable]] if it admits all small colimits and is [[#accessible|κ-accessible]]. A category is presentable if it is κ-presentable for some regular cardinal κ (hence presentable for any larger cardinal). '''Note''': Some authors call a presentable category a [[locally presentable category]].}}
į:
{{defn|Given a '''regular cardinal''' κ, a category is '''presentabl category|κ-presentable''' if it admits all small colimits and is '''#accessible|κ-accessible'''. A category is presentable if it is κ-presentable for some regular cardinal κ (hence presentable for any larger cardinal). '''Note''': Some authors call a presentable category a '''locally presentable category'''.}}
Pakeistos 511-512 eilutės iš
{{defn|1=Another term for a contravariant functor: a functor from a category ''C''<sup>op</sup> to '''Set''' is a presheaf of sets on ''C'' and a functor from ''C''<sup>op</sup> to ''s'''''Set''' is a presheaf of simplicial sets or [[simplicial presheaf]], etc. A [[Grothendieck topology|topology]] on ''C'', if any, tells which presheaf is a sheaf (with respect to that topology).}}
į:
{{defn|1=Another term for a contravariant functor: a functor from a category ''C''<sup>op</sup> to '''Set''' is a presheaf of sets on ''C'' and a functor from ''C''<sup>op</sup> to ''s'''''Set''' is a presheaf of simplicial sets or '''simplicial presheaf''', etc. A '''Grothendieck topology|topology''' on ''C'', if any, tells which presheaf is a sheaf (with respect to that topology).}}
Pakeistos 514-516 eilutės iš
{{defn|no=1|The [[product (category theory)|product]] of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the projective limit <math>\varprojlim</math> of the functor <math>I \to C, \, i \mapsto X_i</math>, where ''I'' is viewed as a discrete category. It is denoted by <math>\prod_i X_i</math> and is the dual of the coproduct of the family.}}
{{defn|no=2|The [[product of categories|product of a family of categories]] ''C''<sub>''i''</sub>'s indexed by a set ''I'' is the category denoted by <math>\prod_i C_i</math> whose class of objects is the product of the classes of objects of ''C''<sub>''i''</sub>'s and whose hom-sets are <math>\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)</math>; the morphisms are composed component-wise. It is the dual of the disjoint union.}}
į:
{{defn|no=1|The '''product (category theory)|product''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the projective limit <math>\varprojlim</math> of the functor <math>I \to C, \, i \mapsto X_i</math>, where ''I'' is viewed as a discrete category. It is denoted by <math>\prod_i X_i</math> and is the dual of the coproduct of the family.}}
{{defn|no=2|The '''product of categories|product of a family of categories''' ''C''<sub>''i''</sub>'s indexed by a set ''I'' is the category denoted by <math>\prod_i C_i</math> whose class of objects is the product of the classes of objects of ''C''<sub>''i''</sub>'s and whose hom-sets are <math>\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)</math>; the morphisms are composed component-wise. It is the dual of the disjoint union.}}
Pakeistos 518-519 eilutės iš
{{defn|1=Given categories ''C'' and ''D'', a [[profunctor]] (or a distributor) from ''C'' to ''D'' is a functor of the form <math>D^{\text{op}} \times C \to \mathbf{Set}</math>.}}
į:
{{defn|1=Given categories ''C'' and ''D'', a '''profunctor''' (or a distributor) from ''C'' to ''D'' is a functor of the form <math>D^{\text{op}} \times C \to \mathbf{Set}</math>.}}
Pakeistos 521-522 eilutės iš
{{defn|1=An object ''A'' in an abelian category is [[projective object|projective]] if the functor <math>\operatorname{Hom}(A, -)</math> is exact. It is the dual of an injective object.}}
į:
{{defn|1=An object ''A'' in an abelian category is '''projective object|projective''' if the functor <math>\operatorname{Hom}(A, -)</math> is exact. It is the dual of an injective object.}}
Pakeista 524 eilutė iš:
{{defn|1=A [[PROP (category theory)|PROP]] is a symmetric strict monoidal category whose objects are natural numbers and whose tensor product [[addition]] of natural numbers.}}
į:
{{defn|1=A '''PROP (category theory)|PROP''' is a symmetric strict monoidal category whose objects are natural numbers and whose tensor product '''addition''' of natural numbers.}}
Pakeista 530 eilutė iš:
{{defn|1=[[Quillen’s theorem A]] provides a criterion for a functor to be a weak equivalence.}}
į:
{{defn|1='''Quillen’s theorem A''' provides a criterion for a functor to be a weak equivalence.}}
Pakeistos 540-541 eilutės iš
{{defn|1=A set-valued contravariant functor ''F'' on a category ''C'' is said to be [[representable functor|representable]] if it belongs to the essential image of the [[Yoneda embedding]] <math>C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})</math>; i.e., <math>F \simeq \operatorname{Hom}_C(-, Z)</math> for some object ''Z''. The object ''Z'' is said to be the representing object of ''F''.}}
į:
{{defn|1=A set-valued contravariant functor ''F'' on a category ''C'' is said to be '''representable functor|representable''' if it belongs to the essential image of the '''Yoneda embedding''' <math>C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})</math>; i.e., <math>F \simeq \operatorname{Hom}_C(-, Z)</math> for some object ''Z''. The object ''Z'' is said to be the representing object of ''F''.}}
Pakeista 543 eilutė iš:
{{defn|1=[[File:Section retract.svg|150px|thumb|''f'' is a retraction of ''g''. ''g'' is a section of ''f''.]]A morphism is a [[section (category theory)|retraction]] if it has a right inverse.}}
į:
{{defn|1='''File:Section retract.svg|150px|thumb|''f'' is a retraction of ''g''. ''g'' is a section of ''f''.'''A morphism is a '''section (category theory)|retraction''' if it has a right inverse.}}
Pakeistos 549-550 eilutės iš
{{defn|1=A morphism is a [[section (category theory)|section]] if it has a left inverse. For example, the [[axiom of choice]] says that any surjective function admits a section.}}
į:
{{defn|1=A morphism is a '''section (category theory)|section''' if it has a left inverse. For example, the '''axiom of choice''' says that any surjective function admits a section.}}
Pakeistos 552-553 eilutės iš
{{defn|1=[[Segal space]]s were certain simplicial spaces, introduced as models for [[(infinity,1)-category|(∞,&nbsp;1)-categories]].}}
į:
{{defn|1='''Segal space'''s were certain simplicial spaces, introduced as models for '''(infinity,1)-category|(∞,&nbsp;1)-categories'''.}}
Pakeistos 555-556 eilutės iš
{{defn|1=An abelian category is [[semisimple category|semisimple]] if every short exact sequence splits. For example, a ring is [[semisimple ring|semisimple]] if and only if the category of modules over it is semisimple.}}
į:
{{defn|1=An abelian category is '''semisimple category|semisimple''' if every short exact sequence splits. For example, a ring is '''semisimple ring|semisimple''' if and only if the category of modules over it is semisimple.}}
Pakeista 558 eilutė iš:
{{defn|1=Given a ''k''-linear category ''C'' over a field ''k'', a [[Serre functor]] <math>f: C \to C</math> is an auto-equivalence such that <math>\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*</math> for any objects ''A'', ''B''.}}
į:
{{defn|1=Given a ''k''-linear category ''C'' over a field ''k'', a '''Serre functor''' <math>f: C \to C</math> is an auto-equivalence such that <math>\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*</math> for any objects ''A'', ''B''.}}
Pakeista 560 eilutė iš:
{{defn|1=In a category, a [[sieve (category theory)|sieve]] is a set ''S'' of objects having the property: if ''f'' is a morphism with the codomain in ''S'', then the domain of ''f'' is in ''S''.}} conflict with the literature? -->
į:
{{defn|1=In a category, a '''sieve (category theory)|sieve''' is a set ''S'' of objects having the property: if ''f'' is a morphism with the codomain in ''S'', then the domain of ''f'' is in ''S''.}} conflict with the literature? -->
Pakeistos 562-563 eilutės iš
{{defn|1=A simple object in an abelian category is <span id="simple object"></span>an object ''A'' that is not isomorphic to the zero object and whose every [[subobject]] is isomorphic to zero or to ''A''. For example, a [[simple module]] is precisely a simple object in the category of (say left) modules.}}
į:
{{defn|1=A simple object in an abelian category is <span id="simple object"></span>an object ''A'' that is not isomorphic to the zero object and whose every '''subobject''' is isomorphic to zero or to ''A''. For example, a '''simple module''' is precisely a simple object in the category of (say left) modules.}}
Pakeistos 565-566 eilutės iš
{{defn|1=The [[simplex category]] &Delta; is the category where an object is a set [''n''] = { 0, 1, …, ''n'' }, n ≥ 0, totally ordered in the standard way and a morphism is an order-preserving function.}}
į:
{{defn|1=The '''simplex category''' &Delta; is the category where an object is a set [''n''] = { 0, 1, …, ''n'' }, n ≥ 0, totally ordered in the standard way and a morphism is an order-preserving function.}}
Pakeistos 571-572 eilutės iš
{{defn|1=[[Simplicial localization]] is a method of localizing a category.}}
į:
{{defn|1='''Simplicial localization''' is a method of localizing a category.}}
Pakeistos 574-575 eilutės iš
{{defn|1=A [[simplicial object]] in a category ''C'' is roughly a sequence of objects <math>X_0, X_1, X_2, \dots</math> in ''C'' that forms a simplicial set. In other words, it is a covariant or contravariant functor Δ → ''C''. For example, a [[simplicial presheaf]] is a simplicial object in the category of presheaves.}}
į:
{{defn|1=A '''simplicial object''' in a category ''C'' is roughly a sequence of objects <math>X_0, X_1, X_2, \dots</math> in ''C'' that forms a simplicial set. In other words, it is a covariant or contravariant functor Δ → ''C''. For example, a '''simplicial presheaf''' is a simplicial object in the category of presheaves.}}
Pakeistos 577-578 eilutės iš
{{defn|1=A [[simplicial set]] is a contravariant functor from &Delta; to '''Set''', where &Delta; is the [[simplex category]], a category whose objects are the sets [''n''] = { 0, 1, …, ''n'' } and whose morphisms are order-preserving functions. One writes <math>X_n = X([n])</math> and an element of the set <math>X_n</math> is called an ''n''-simplex. For example, <math>\Delta^n = \operatorname{Hom}_{\Delta}(-, [n])</math> is a simplicial set called the standard ''n''-simplex. By Yoneda's lemma, <math>X_n \simeq \operatorname{Nat}(\Delta^n, X)</math>.}}
į:
{{defn|1=A '''simplicial set''' is a contravariant functor from &Delta; to '''Set''', where &Delta; is the '''simplex category''', a category whose objects are the sets [''n''] = { 0, 1, …, ''n'' } and whose morphisms are order-preserving functions. One writes <math>X_n = X([n])</math> and an element of the set <math>X_n</math> is called an ''n''-simplex. For example, <math>\Delta^n = \operatorname{Hom}_{\Delta}(-, [n])</math> is a simplicial set called the standard ''n''-simplex. By Yoneda's lemma, <math>X_n \simeq \operatorname{Nat}(\Delta^n, X)</math>.}}
Pakeistos 580-581 eilutės iš
{{defn|1=A category equipped with a [[Grothendieck topology]].}}
į:
{{defn|1=A category equipped with a '''Grothendieck topology'''.}}
Pakeistos 583-584 eilutės iš
{{defn|1=A category is [[Skeleton (category theory)|skeletal]] if isomorphic objects are necessarily identical.}}
į:
{{defn|1=A category is '''Skeleton (category theory)|skeletal''' if isomorphic objects are necessarily identical.}}
Pakeistos 586-587 eilutės iš
{{defn|1=Given a category ''C'' and an object ''A'' in it, the [[slice category]] ''C''<sub>/''A''</sub> of ''C'' over ''A'' is the category whose objects are all the morphisms in ''C'' with codomain ''A'', whose morphisms are morphisms in ''C'' such that if ''f'' is a morphism from <math>p_X: X \to A</math> to <math>p_Y: Y \to A</math>, then <math>p_Y \circ f = p_X</math> in ''C'' and whose composition is that of ''C''.}}
į:
{{defn|1=Given a category ''C'' and an object ''A'' in it, the '''slice category''' ''C''<sub>/''A''</sub> of ''C'' over ''A'' is the category whose objects are all the morphisms in ''C'' with codomain ''A'', whose morphisms are morphisms in ''C'' such that if ''f'' is a morphism from <math>p_X: X \to A</math> to <math>p_Y: Y \to A</math>, then <math>p_Y \circ f = p_X</math> in ''C'' and whose composition is that of ''C''.}}
Pakeistos 589-591 eilutės iš
{{defn|no=1|1=A [[small category]] is a category in which the class of all morphisms is a [[Set (mathematics)|set]] (i.e., not a [[proper class]]); otherwise '''large'''. A category is '''locally small''' if the morphisms between every pair of objects ''A'' and ''B'' form a set. Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a '''quasicategory''' is a category whose objects and morphisms merely form a [[conglomerate (set theory)|conglomerate]].<ref>{{cite book |last=Adámek |first=Jiří |author2=Herrlich, Horst |author3=Strecker, George E |title=Abstract and Concrete Categories (The Joy of Cats) |origyear=1990 |url=http://katmat.math.uni-bremen.de/acc/ |format=PDF |year=2004 |publisher= Wiley & Sons |location=New York |isbn=0-471-60922-6 |page=40}}</ref> (NB: some authors use the term "quasicategory" with a different meaning.<ref>{{cite journal|doi=10.1016/S0022-4049(02)00135-4|last=Joyal|first=A.|title=Quasi-categories and Kan complexes|journal=Journal of Pure and Applied Algebra|volume=175|year=2002|issue=1–3|pages=207–222|ref=harv}}</ref>)}}
{{defn|no=2|An object in a category is said to be [[small object|small]] if it is κ-compact for some regular cardinal κ. The notion prominently appears in Quiilen's [[small object argument]] (cf. https://ncatlab.org/nlab/show/small+object+argument)}}
į:
{{defn|no=1|1=A '''small category''' is a category in which the class of all morphisms is a '''Set (mathematics)|set''' (i.e., not a '''proper class'''); otherwise '''large'''. A category is '''locally small''' if the morphisms between every pair of objects ''A'' and ''B'' form a set. Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a '''quasicategory''' is a category whose objects and morphisms merely form a '''conglomerate (set theory)|conglomerate'''.<ref>{{cite book |last=Adámek |first=Jiří |author2=Herrlich, Horst |author3=Strecker, George E |title=Abstract and Concrete Categories (The Joy of Cats) |origyear=1990 |url=http://katmat.math.uni-bremen.de/acc/ |format=PDF |year=2004 |publisher= Wiley & Sons |location=New York |isbn=0-471-60922-6 |page=40}}</ref> (NB: some authors use the term "quasicategory" with a different meaning.<ref>{{cite journal|doi=10.1016/S0022-4049(02)00135-4|last=Joyal|first=A.|title=Quasi-categories and Kan complexes|journal=Journal of Pure and Applied Algebra|volume=175|year=2002|issue=1–3|pages=207–222|ref=harv}}</ref>)}}
{{defn|no=2|An object in a category is said to be '''small object|small''' if it is κ-compact for some regular cardinal κ. The notion prominently appears in Quiilen's '''small object argument''' (cf. https://ncatlab.org/nlab/show/small+object+argument)}}
Pakeistos 593-594 eilutės iš
{{defn|1=A [[combinatorial species|(combinatorial) species]] is an endofunctor on the groupoid of finite sets with bijections. It is categorically equivalent to a [[symmetric sequence]].}}
į:
{{defn|1=A '''combinatorial species|(combinatorial) species''' is an endofunctor on the groupoid of finite sets with bijections. It is categorically equivalent to a '''symmetric sequence'''.}}
Pakeistos 596-597 eilutės iš
{{defn|1=An ∞-category is [[stable ∞-category|stable]] if (1) it has a zero object, (2) every morphism in it admits a fiber and a cofiber and (3) a triangle in it is a fiber sequence if and only if it is a cofiber sequence.}}
į:
{{defn|1=An ∞-category is '''stable ∞-category|stable''' if (1) it has a zero object, (2) every morphism in it admits a fiber and a cofiber and (3) a triangle in it is a fiber sequence if and only if it is a cofiber sequence.}}
Pakeistos 599-600 eilutės iš
{{defn|A morphism ''f'' in a category admitting finite limits and finite colimits is [[strict morphism|strict]] if the natural morphism <math>\operatorname{Coim}(f) \to \operatorname{Im}(f)</math> is an isomorphism.}}
į:
{{defn|A morphism ''f'' in a category admitting finite limits and finite colimits is '''strict morphism|strict''' if the natural morphism <math>\operatorname{Coim}(f) \to \operatorname{Im}(f)</math> is an isomorphism.}}
Pakeistos 602-603 eilutės iš
{{defn|A strict 0-category is a set and for any integer ''n'' > 0, a [[strict n-category|strict ''n''-category]] is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category. '''Note''': the term "''n''-category" typically refers to "[[weak n-category|weak ''n''-category]]"; not strict one.}}
į:
{{defn|A strict 0-category is a set and for any integer ''n'' > 0, a '''strict n-category|strict ''n''-category''' is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category. '''Note''': the term "''n''-category" typically refers to "'''weak n-category|weak ''n''-category'''"; not strict one.}}
Pakeistos 605-606 eilutės iš
{{defn|1=A topology on a category is [[subcanonical]] if every representable contravariant functor on ''C'' is a sheaf with respect to that topology.<ref>{{harvnb|Vistoli|2004|loc=Definition 2.57.}}</ref> Generally speaking, some [[flat topology]] may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical.}}
į:
{{defn|1=A topology on a category is '''subcanonical''' if every representable contravariant functor on ''C'' is a sheaf with respect to that topology.<ref>{{harvnb|Vistoli|2004|loc=Definition 2.57.}}</ref> Generally speaking, some '''flat topology''' may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical.}}
Pakeistos 608-609 eilutės iš
{{defn|1=A category ''A'' is a [[subcategory]] of a category ''B'' if there is an inclusion functor from ''A'' to ''B''.}}
į:
{{defn|1=A category ''A'' is a '''subcategory''' of a category ''B'' if there is an inclusion functor from ''A'' to ''B''.}}
Pakeistos 611-612 eilutės iš
{{defn|1=Given an object ''A'' in a category, a [[subobject]] of ''A'' is an equivalence class of monomorphisms to ''A''; two monomorphisms ''f'', ''g'' are considered equivalent if ''f'' factors through ''g'' and ''g'' factors through ''f''.}}
į:
{{defn|1=Given an object ''A'' in a category, a '''subobject''' of ''A'' is an equivalence class of monomorphisms to ''A''; two monomorphisms ''f'', ''g'' are considered equivalent if ''f'' factors through ''g'' and ''g'' factors through ''f''.}}
Pakeistos 614-615 eilutės iš
{{defn|1=A [[subquotient]] is a quotient of a subobject.}}
į:
{{defn|1=A '''subquotient''' is a quotient of a subobject.}}
Pakeistos 617-618 eilutės iš
{{defn|1=A [[subterminal object]] is an object ''X'' such that every object has at most one morphism into ''X''.}}
į:
{{defn|1=A '''subterminal object''' is an object ''X'' such that every object has at most one morphism into ''X''.}}
Pakeistos 620-621 eilutės iš
{{defn|1=A [[symmetric monoidal category]] is a [[monoidal category]] (i.e., a category with ⊗) that has maximally symmetric braiding.}}
į:
{{defn|1=A '''symmetric monoidal category''' is a '''monoidal category''' (i.e., a category with ⊗) that has maximally symmetric braiding.}}
Pakeistos 623-624 eilutės iš
{{defn|1=A [[symmetric sequence]] is a sequence of objects with actions of [[symmetric group]]s. It is categorically equivalent to a [[combinatorial species|(combinatorial) species]].}}
į:
{{defn|1=A '''symmetric sequence''' is a sequence of objects with actions of '''symmetric group'''s. It is categorically equivalent to a '''combinatorial species|(combinatorial) species'''.}}
Pakeistos 630-631 eilutės iš
{{defn|1=A [[t-structure]] is an additional structure on a [[triangulated category]] (more generally [[stable ∞-category]]) that axiomatizes the notions of complexes whose cohomology concentrated in non-negative degrees or non-positive degrees.}}
į:
{{defn|1=A '''t-structure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞-category''') that axiomatizes the notions of complexes whose cohomology concentrated in non-negative degrees or non-positive degrees.}}
Pakeistos 633-634 eilutės iš
{{defn|The [[Tannakian duality]] states that, in an appropriate setup, to give a morphism <math>f: X \to Y</math> is to give a pullback functor <math>f^*</math> along it. In other words, the Hom set <math>\operatorname{Hom}(X, Y)</math> can be identified with the functor category <math>\operatorname{Fct}(D(Y), D(X))</math>, perhaps in the [[derived algebraic geometry|derived sense]], where <math>D(X)</math> is the category associated to ''X'' (e.g., the derived category).<ref>Jacob Lurie. Tannaka duality for geometric stacks. http://math.harvard.edu/~lurie/, 2004.</ref><ref>{{cite arxiv|last=Bhatt|first=Bhargav|date=2014-04-29|title=Algebraization and Tannaka duality|eprint=1404.7483|class=math.AG}}</ref>}}
į:
{{defn|The '''Tannakian duality''' states that, in an appropriate setup, to give a morphism <math>f: X \to Y</math> is to give a pullback functor <math>f^*</math> along it. In other words, the Hom set <math>\operatorname{Hom}(X, Y)</math> can be identified with the functor category <math>\operatorname{Fct}(D(Y), D(X))</math>, perhaps in the '''derived algebraic geometry|derived sense''', where <math>D(X)</math> is the category associated to ''X'' (e.g., the derived category).<ref>Jacob Lurie. Tannaka duality for geometric stacks. http://math.harvard.edu/~lurie/, 2004.</ref><ref>{{cite arxiv|last=Bhatt|first=Bhargav|date=2014-04-29|title=Algebraization and Tannaka duality|eprint=1404.7483|class=math.AG}}</ref>}}
Pakeistos 636-637 eilutės iš
{{defn|1=Usually synonymous with [[monoidal category]] (though some authors distinguish between the two concepts.)}}
į:
{{defn|1=Usually synonymous with '''monoidal category''' (though some authors distinguish between the two concepts.)}}
Pakeistos 639-640 eilutės iš
{{defn|1=A [[tensor triangulated category]] is a category that carries the structure of a symmetric monoidal category and that of a triangulated category in a compatible way.}}
į:
{{defn|1=A '''tensor triangulated category''' is a category that carries the structure of a symmetric monoidal category and that of a triangulated category in a compatible way.}}
Pakeistos 646-648 eilutės iš
{{defn|no=1|An object ''A'' is [[terminal object|terminal]] (also called final) if there is exactly one morphism from each object to ''A''; e.g., [[singleton (mathematics)|singleton]]s in '''[[Category of sets|Set]]'''. It is the dual of an [[initial object]].}}
{{defn|no=2|An object ''A'' in an ∞-category ''C'' is terminal if <math>\operatorname{Map}_C(B, A)</math> is [[contractible space|contractible]] for every object ''B'' in ''C''.}}
į:
{{defn|no=1|An object ''A'' is '''terminal object|terminal''' (also called final) if there is exactly one morphism from each object to ''A''; e.g., '''singleton (mathematics)|singleton'''s in ''''''Category of sets|Set''''''. It is the dual of an '''initial object'''.}}
{{defn|no=2|An object ''A'' in an ∞-category ''C'' is terminal if <math>\operatorname{Map}_C(B, A)</math> is '''contractible space|contractible''' for every object ''B'' in ''C''.}}
Pakeistos 650-651 eilutės iš
{{defn|1=A full subcategory of an abelian category is [[thick subcategory|thick]] if it is closed under extensions.}}
į:
{{defn|1=A full subcategory of an abelian category is '''thick subcategory|thick''' if it is closed under extensions.}}
Pakeistos 653-654 eilutės iš
{{defn|1=A [[thin category|thin]] is a category where there is at most one morphism between any pair of objects.}}
į:
{{defn|1=A '''thin category|thin''' is a category where there is at most one morphism between any pair of objects.}}
Pakeistos 656-657 eilutės iš
{{defn|1=A [[triangulated category]] is a category where one can talk about distinguished triangles, generalization of exact sequences. An abelian category is a prototypical example of a triangulated category. A [[derived category]] is a triangulated category that is not necessary an abelian category.}}
į:
{{defn|1=A '''triangulated category''' is a category where one can talk about distinguished triangles, generalization of exact sequences. An abelian category is a prototypical example of a triangulated category. A '''derived category''' is a triangulated category that is not necessary an abelian category.}}
Pakeista 660 eilutė iš:
== U ==
į:
[+U+]
Pakeista 663 eilutė iš:
{{defn|no=1|Given a functor <math>f: C \to D</math> and an object ''X'' in ''D'', a [[universal morphism]] from ''X'' to ''f'' is an initial object in the [[comma category]] <math>(X \downarrow f)</math>. (Its dual is also called a universal morphism.) For example, take ''f'' to be the forgetful functor <math>\mathbf{Vec}_k \to \mathbf{Set}</math> and ''X'' a set. An initial object of <math>(X \downarrow f)</math> is a function <math>j: X \to f(V_X)</math>. That it is initial means that if <math>k: X \to f(W)</math> is another morphism, then there is a unique morphism from ''j'' to ''k'', which consists of a linear map <math>V_X \to W</math> that extends ''k'' via ''j''; that is to say, <math>V_X</math> is the [[free vector space]] generated by ''X''.}}
į:
{{defn|no=1|Given a functor <math>f: C \to D</math> and an object ''X'' in ''D'', a '''universal morphism''' from ''X'' to ''f'' is an initial object in the '''comma category''' <math>(X \downarrow f)</math>. (Its dual is also called a universal morphism.) For example, take ''f'' to be the forgetful functor <math>\mathbf{Vec}_k \to \mathbf{Set}</math> and ''X'' a set. An initial object of <math>(X \downarrow f)</math> is a function <math>j: X \to f(V_X)</math>. That it is initial means that if <math>k: X \to f(W)</math> is another morphism, then there is a unique morphism from ''j'' to ''k'', which consists of a linear map <math>V_X \to W</math> that extends ''k'' via ''j''; that is to say, <math>V_X</math> is the '''free vector space''' generated by ''X''.}}
Pakeistos 672-673 eilutės iš
{{defn|1=A [[Waldhausen category]] is, roughly, a category with families of cofibrations and weak equivalences.}}
į:
{{defn|1=A '''Waldhausen category''' is, roughly, a category with families of cofibrations and weak equivalences.}}
Pakeista 675 eilutė iš:
{{defn|1=A category is wellpowered if for each object there is only a set of pairwise non-isomorphic [[subobject]]s.}}
į:
{{defn|1=A category is wellpowered if for each object there is only a set of pairwise non-isomorphic '''subobject'''s.}}
Pakeista 683 eilutė iš:
|author=[[Barry Mazur]]
į:
|author='''Barry Mazur'''
Pakeista 685 eilutė iš:
|width=33%}}The [[Yoneda lemma]] says: for each set-valued contravariant functor ''F'' on ''C'' and an object ''X'' in ''C'', there is a natural bijection
į:
|width=33%}}The '''Yoneda lemma''' says: for each set-valued contravariant functor ''F'' on ''C'' and an object ''X'' in ''C'', there is a natural bijection
Pakeista 690 eilutė iš:
{{defn|no=2|1=If <math>F: C \to D</math> is a functor and ''y'' is the Yoneda embedding of ''C'', then the [[Yoneda extension]] of ''F'' is the left Kan extension of ''F'' along ''y''.}}
į:
{{defn|no=2|1=If <math>F: C \to D</math> is a functor and ''y'' is the Yoneda embedding of ''C'', then the '''Yoneda extension''' of ''F'' is the left Kan extension of ''F'' along ''y''.}}
Pakeista 693 eilutė iš:
== Z ==
į:
[+Z+]
Pakeista 696 eilutė iš:
{{defn|1=A [[zero object]] is an object that is both initial and terminal, such as a [[trivial group]] in '''[[Category of groups|Grp]]'''.}}
į:
{{defn|1=A '''zero object''' is an object that is both initial and terminal, such as a '''trivial group''' in ''''''Category of groups|Grp''''''.}}
Pakeista 707 eilutė iš:
|editor=[[Alexandre Grothendieck]] |editor2=[[Jean-Louis Verdier]]
į:
|editor='''Alexandre Grothendieck''' |editor2='''Jean-Louis Verdier'''
Pakeista 711 eilutė iš:
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
į:
| publisher = '''Springer Science+Business Media|Springer-Verlag'''
Pakeistos 728-732 eilutės iš
*[[André Joyal|A. Joyal]], [http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf The theory of quasi-categories II] (Volume I is missing??)
*[[Jacob Lurie|Lurie, J.]], ''[https://web.archive.org/web/20150206092410/http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf Higher Algebra]''
*Lurie, J., ''[[Higher Topos Theory]]''
* {{cite book | last=Mac Lane | first=Saunders | authorlink=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | edition=2nd | series=[[Graduate Texts in Mathematics]] | volume=5 | location=New York, NY | publisher=[[Springer-Verlag]] | year=1998 | isbn=0-387-98403-8 | zbl=0906.18001 |ref=harv}}
* {{cite book | editor1-last=Pedicchio | editor1-first=Maria Cristina | editor2-last=Tholen | editor2-first=Walter | title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory | series=Encyclopedia of Mathematics and Its Applications | volume=97 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2004 | isbn=0-521-83414-7 | zbl=1034.18001 }}
į:
*'''André Joyal|A. Joyal''', [http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf The theory of quasi-categories II] (Volume I is missing??)
*'''Jacob Lurie|Lurie, J.''', ''[https://web.archive.org/web/20150206092410/http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf Higher Algebra]''
*Lurie, J., '''''Higher Topos Theory'''''
* {{cite book | last=Mac Lane | first=Saunders | authorlink=Saunders Mac Lane | title='''Categories for the Working Mathematician''' | edition=2nd | series='''Graduate Texts in Mathematics''' | volume=5 | location=New York, NY | publisher='''Springer-Verlag''' | year=1998 | isbn=0-387-98403-8 | zbl=0906.18001 |ref=harv}}
* {{cite book | editor1-last=Pedicchio | editor1-first=Maria Cristina | editor2-last=Tholen | editor2-first=Walter | title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory | series=Encyclopedia of Mathematics and Its Applications | volume=97 | location=Cambridge | publisher='''Cambridge University Press''' | year=2004 | isbn=0-521-83414-7 | zbl=1034.18001 }}
Pakeista 735 eilutė iš:
== Further reading ==
į:
[+Further reading+]
Pakeista 738 eilutė iš:
*[[History of topos theory]]
į:
*'''History of topos theory'''
Pakeista 742 eilutė iš:
* [http://www.andrew.cmu.edu/user/awodey/catlog/ Categorical Logic] lecture notes by [[Steve Awodey]]
į:
* [http://www.andrew.cmu.edu/user/awodey/catlog/ Categorical Logic] lecture notes by '''Steve Awodey'''
2019 vasario 10 d., 09:04 atliko AndriusKulikauskas -
Pridėtos 1-742 eilutės:
See: [[Category theory]]

This is a glossary of properties and concepts in [[category theory]] in [[mathematics]].

*'''Notes on foundations''': In many expositions (e.g., Vistoli), the set-theoretic issues are ignored; this means, for instance, that one does not distinguish between small and large categories and that one can arbitrarily form a localization of a category.<ref>If one believes in the existence of [[strongly inaccessible cardinal]]s, then there can be a rigorous theory where statements and constructions have references to [[Grothendieck universe]]s.</ref> Like those expositions, this glossary also generally ignores the set-theoretic issues, except when they are relevant (e.g., the discussion on accessibility.)

Especially for higher categories, the concepts from algebraic topology are also used in the category theory. For that see also [[glossary of algebraic topology]].

The notations and the conventions used throughout the article are:
*[''n''] = { 0, 1, 2, …, ''n'' }, which is viewed as a category (by writing <math>i \to j \Leftrightarrow i \le j</math>.)
*'''Cat''', the [[category of categories|category of (small) categories]], where the objects are categories (which are small with respect to some universe) and the morphisms [[functor]]s.
*'''Fct'''(''C'', ''D''), the [[functor category]]: the category of [[functor]]s from a category ''C'' to a category ''D''.
*'''Set''', the category of (small) sets.
*''s'''''Set''', the category of [[simplicial set]]s.
*"weak" instead of "strict" is given the default status; e.g., "''n''-category" means "weak ''n''-category", not the strict one, by default.
*By an [[∞-category]], we mean a [[quasi-category]], the most popular model, unless other models are being discussed.
*The number [[zero]] 0 is a natural number.
{{Compact ToC|short1|sym=yes|x=[[#XYZ|XYZ]]|y=|z=|seealso=yes|refs=yes}}

==A==
{{glossary}}
{{term|1=abelian}}
{{defn|1=A category is [[abelian category|abelian]] if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.}}

{{term|1=accessible}}
{{defn|no=1|Given a [[cardinal number]] κ, an object ''X'' in a category is [[accessible object|κ-accessible]] (or κ-compact or κ-presentable) if <math>\operatorname{Hom}(X, -)</math> commutes with κ-filtered colimits.}}
{{defn|no=2|Given a [[regular cardinal]] κ, a category is [[accessible category|κ-accessible]] if it has κ-filtered colimits and there exists a small set ''S'' of κ-compact objects that generates the category under colimits, meaning every object can be written as a colimit of diagrams of objects in ''S''.}}

{{term|1=additive}}
{{defn|1=A category is [[additive category|additive]] if it is preadditive (to be precise, has some pre-additive structure) and admits all finite [[coproduct]]s. Although "preadditive" is an additional structure, one can show "additive" is a ''property'' of a category; i.e., one can ask whether a given category is additive or not.<ref>Remark 2.7. of https://ncatlab.org/nlab/show/additive+category</ref>}}

{{term|1=adjunction}}
{{defn|1=An [[adjoint functor|adjunction]] (also called an adjoint pair) is a pair of functors ''F'': ''C'' → ''D'', ''G'': ''D'' → ''C'' such that there is a "natural" bijection
:<math>\operatorname{Hom}_D (F(X), Y) \simeq \operatorname{Hom}_C (X, G(Y))</math>;
''F'' is said to be left adjoint to ''G'' and ''G'' to right adjoint to ''F''. Here, "natural" means there is a natural isomorphism <math>\operatorname{Hom}_D (F(-), -) \simeq \operatorname{Hom}_C (-, G(-))</math> of bifunctors (which are contravariant in the first variable.)}}

{{term|1=algebra for a monad}}
{{defn|1=Given a monad ''T'' in a category ''X'', an [[algebra for a monad|algebra for ''T'']] or a ''T''-algebra is an object in ''X'' with a [[monoid action]] of ''T'' ("algebra" is misleading and "''T''-object" is perhaps a better term.) For example, given a group ''G'' that determines a monad ''T'' in '''Set''' in the standard way, a ''T''-algebra is a set with an [[group action|action]] of ''G''.}}

{{term|1=amnestic}}
{{defn|1=A functor is amnestic if it has the property: if ''k'' is an isomorphism and ''F''(''k'') is an identity, then ''k'' is an identity.}}
{{glossary end}}

==B==
{{glossary}}
{{term|1=balanced}}
{{defn|1=A category is balanced if every bimorphism is an isomorphism.}}

{{term|1=Beck's theorem}}
{{defn|1=[[Beck's monadicity theorem|Beck's theorem]] characterizes the category of [[algebra for a monad|algebras for a given monad]].}}

{{term|1=bicategory}}
{{defn|1=A [[bicategory]] is a model of a weak [[2-category]].}}

{{term|1=bifunctor}}
{{defn|1=A [[bifunctor]] from a pair of categories ''C'' and ''D'' to a category ''E'' is a functor ''C'' × ''D'' → ''E''. For example, for any category ''C'', <math>\operatorname{Hom}(-, -)</math> is a bifunctor from ''C''<sup>op</sup> and ''C'' to '''Set'''.}}

{{term|1=bimorphism}}
{{defn|1=A [[bimorphism]] is a morphism that is both an epimorphism and a monomorphism.}}

{{term|1=Bousfield localization}}
{{defn|1=See [[Bousfield localization]].}}

{{glossary end}}

==C==
{{glossary}}
{{term|1=calculus of functors}}
{{defn|1=The [[calculus of functors]] is a technique of studying functors in the manner similar to the way a [[function (mathematics)|function]] is studied via its [[Taylor series]] expansion; whence, the term "calculus".}}

{{term|1=cartesian closed}}
{{defn|1=A category is [[Cartesian closed category|cartesian closed]] if it has a terminal object and that any two objects have a product and exponential.}}

{{term|1=cartesian functor}}
{{defn|1=Given relative categories <math>p: F \to C, q: G \to C</math> over the same base category ''C'', a functor <math>f: F \to G</math> over ''C'' is cartesian if it sends cartesian morphisms to cartesian morphisms.}}

{{term|1=cartesian morphism}}
{{defn|no=1|1=Given a functor π: ''C'' → ''D'' (e.g., a [[prestack]] over schemes), a morphism ''f'': ''x'' → ''y'' in ''C'' is [[cartesian morphism|π-cartesian]] if, for each object ''z'' in ''C'', each morphism ''g'': ''z'' → ''y'' in ''C'' and each morphism ''v'': π(''z'') → π(''x'') in ''D'' such that π(''g'') = π(''f'') ∘ ''v'', there exists a unique morphism ''u'': ''z'' → ''x'' such that π(''u'') = ''v'' and ''g'' = ''f'' ∘ ''u''.}}
{{defn|no=2|1=Given a functor π: ''C'' → ''D'' (e.g., a [[prestack]] over rings), a morphism ''f'': ''x'' → ''y'' in ''C'' is [[cartesian morphism|π-coCartesian]] if, for each object ''z'' in ''C'', each morphism ''g'': ''x'' → ''z'' in ''C'' and each morphism ''v'': π(''y'') → π(''z'') in ''D'' such that π(''g'') = ''v'' ∘ π(''f''), there exists a unique morphism ''u'': ''y'' → ''z'' such that π(''u'') = ''v'' and ''g'' = ''u'' ∘ ''f''. (In short, ''f'' is the dual of a π-cartesian morphism.)}}

{{term|1=Cartesian square}}
{{defn|1=A commutative diagram that is isomorphic to the diagram given as a fiber product.<!-- really need a diagram here -->}}

{{term|1=categorical logic}}
{{defn|1=[[Categorical logic]] is an approach to [[mathematical logic]] that uses category theory.}}

{{term|1=categorification}}
{{defn|1=[[Categorification]] is a process of replacing sets and set-theoretic concepts with categories and category-theoretic concepts in some nontrivial way to capture categoric flavors. Decategorification is the reverse of categorification.}}

{{term|1=category}}
{{defn|1=A [[category (mathematics)|category]] consists of the following data
#A class of objects,
#For each pair of objects ''X'', ''Y'', a set <math>\operatorname{Hom}(X, Y)</math>, whose elements are called morphisms from ''X'' to ''Y'',
#For each triple of objects ''X'', ''Y'', ''Z'', a map (called composition)
#:<math>\circ: \operatorname{Hom}(Y, Z) \times \operatorname{Hom}(X, Y) \to \operatorname{Hom}(X, Z), \, (g, f) \mapsto g \circ f</math>,
#For each object ''X'', an identity morphism <math>\operatorname{id}_X \in \operatorname{Hom}(X, X)</math>
subject to the conditions: for any morphisms <math>f: X \to Y</math>, <math>g: Y \to Z</math> and <math>h: Z \to W</math>,
*<math>(h \circ g) \circ f = h \circ (g \circ f)</math> and <math>\operatorname{id}_Y \circ f = f \circ \operatorname{id}_X = f</math>.
For example, a [[partially ordered set]] can be viewed as a category: the objects are the elements of the set and for each pair of objects ''x'', ''y'', there is a unique morphism <math>x \to y</math> if and only if <math>x \le y</math>; the associativity of composition means transitivity.}}

{{term|1=category of categories}}
{{defn|1=The [[category of categories|category of (small) categories]], denoted by '''Cat''', is a category where the objects are all the categories which are small with respect to some fixed universe and the morphisms are all the [[functor]]s.}}

{{term|1=classifying space}}
{{defn|1=The [[classifying space of a category]] ''C'' is the geometric realization of the nerve of ''C''.}}

{{term|1=co-}}
{{defn|1=Often used synonymous with op-; for example, a [[colimit]] refers to an op-limit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an op-fibration is not the same thing as a [[cofibration]].}}

{{term|1=coend}}
{{defn|1=The coend of a functor <math>F: C^{\text{op}} \times C \to X</math> is the dual of the [[end (category theory)|end]] of ''F'' and is denoted by
:<math>\int^{c \in C} F(c, c)</math>.
For example, if ''R'' is a ring, ''M'' a right ''R''-module and ''N'' a left ''R''-module, then the [[tensor product of modules|tensor product]] of ''M'' and ''N'' is
:<math>M \otimes_R N = \int^{R} M \otimes_{\mathbb{Z}} N</math>
where ''R'' is viewed as a category with one object whose morphisms are the elements of ''R''.}}

{{term|1=coequalizer}}
{{defn|1=The [[coequalizer]] of a pair of morphisms <math>f, g: A \to B</math> is the colimit of the pair. It is the dual of an equalizer.}}

{{term|1=coimage}}
{{defn|1=The [[coimage]] of a morphism ''f'': ''X'' → ''Y'' is the coequalizer of <math>X \times_Y X \rightrightarrows X</math>.}}

{{term|1=colored operad}}
{{defn|1=Another term for [[multicategory]], a generalized category where a morphism can have several domains. The notion of "colored operad" is more primitive than that of operad: in fact, an operad can be defined as a colored operad with a single object.}}

{{term|1=comma}}
{{defn|1=Given functors <math>f: C \to B, g: D \to B</math>, the [[comma category]] <math>(f \downarrow g)</math> is a category where (1) the objects are morphisms <math>f(c) \to g(d)</math> and (2) a morphism from <math>\alpha: f(c) \to g(d)</math> to <math>\beta: f(c') \to g(d')</math> consists of <math>c \to c'</math> and <math>d \to d'</math> such that <math>f(c) \to f(c') \overset{\beta}\to g(d')</math> is <math>f(c) \overset{\alpha}\to g(d) \to g(d').</math> For example, if ''f'' is the identity functor and ''g'' is the constant functor with a value ''b'', then it is the slice category of ''B'' over an object ''b''.
}}

{{term|1=comonad}}
{{defn|1=A [[comonad]] in a category ''X'' is a [[comonid]] in the monoidal category of endofunctors of ''X''.}}

{{term|1=compact}}
{{defn|1=Probably synonymous with [[#accessible]].}}

{{term|1=complete}}
{{defn|1=A category is [[complete category|complete]] if all small limits exist.}}

{{term|1=composition}}
{{defn|no=1|1=A composition of morphisms in a category is part of the datum defining the category.}}
{{defn|no=2|1=If <math>f: C \to D, \, g: D \to E</math> are functors, then the composition <math>g \circ f</math> or <math>gf</math> is the functor defined by: for an object ''x'' and a morphism ''u'' in ''C'', <math>(g \circ f)(x) = g(f(x)), \, (g \circ f)(u) = g(f(u))</math>.}}
{{defn|no=3|1=Natural transformations are composed pointwise: if <math>\varphi: f \to g, \, \psi: g \to h</math> are natural transformations, then <math>\psi \circ \varphi</math> is the natural transformation given by <math>(\psi \circ \varphi)_x = \psi_x \circ \varphi_x</math>.}}

{{term|1=concrete}}
{{defn|1=A [[concrete category]] ''C'' is a category such that there is a faithful functor from ''C'' to '''[[Category of sets|Set]]'''; e.g., '''[[category of vector spaces|Vec]]''', '''[[category of groups|Grp]]''' and '''[[category of topological spaces|Top]]'''.}}

{{term|1=cone}}
{{defn|1=A [[cone (category theory)|cone]] is a way to express the [[universal property]] of a colimit (or dually a limit). One can show<ref>{{harvnb|Kashiwara|Schapira|2006|loc=Ch. 2, Exercise 2.8.}}</ref> that the colimit <math>\varinjlim</math> is the left adjoint to the diagonal functor <math>\Delta: C \to \operatorname{Fct}(I, C)</math>, which sends an object ''X'' to the constant functor with value ''X''; that is, for any ''X'' and any functor <math>f: I \to C</math>,
:<math>\operatorname{Hom}(\varinjlim f, X) \simeq \operatorname{Hom}(f, \Delta_X),</math>
provided the colimit in question exists. The right-hand side is then the set of cones with vertex ''X''.<ref>{{harvnb|Mac Lane|1998|loc=Ch. III, § 3.}}.</ref><!--For example, let <math>f: \mathbb{N} \to \mathbf{Set}</math> be a functor that maps each <math>i \to j</math> to an inclusion. Then the cone is a map from the union of <math>f(i)</math> over all ''i'' to any -->}}

{{term|1=connected}}
{{defn|1=A category is [[connected category|connected]] if, for each pair of objects ''x'', ''y'', there exists a finite sequence of objects ''z''<sub>''i''</sub> such that <math>z_0 = x, z_n = y</math> and either <math>\operatorname{Hom}(z_i, z_{i+1})</math> or <math>\operatorname{Hom}(z_{i+1}, z_i)</math> is nonempty for any ''i''.}}

{{term|1=conservative functor}}
{{defn|1=A [[conservative functor]] is a functor that reflects isomorphisms. Many forgetful functors are conservative, but the forgetful functor from '''Top''' to '''Set''' is not conservative.}}

{{term|1=constant}}
{{defn|1=A functor is [[constant functor|constant]] if it maps every object in a category to the same object ''A'' and every morphism to the identity on ''A''. Put in another way, a functor <math>f: C \to D</math> is constant if it factors as: <math>C \to \{ A \} \overset{i}\to D</math> for some object ''A'' in ''D'', where ''i'' is the inclusion of the discrete category { ''A'' }.}}

{{term|1=contravariant functor}}
{{defn|1=A [[contravariant functor]] ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a [[presheaf (category theory)|presheaf]] especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let <math>\mathfrak{P}(S)</math> be the power set of ''S'' and for each function <math>f: S \to T</math>, define
:<math>\mathfrak{P}(f): \mathfrak{P}(T) \to \mathfrak{P}(S)</math>
by sending a subset ''A'' of ''T'' to the pre-image <math>f^{-1}(A)</math>. With this, <math>\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}</math> is a contravariant functor.}}

{{term|1=coproduct}}
{{defn|1=The [[coproduct]] of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the inductive limit <math>\varinjlim</math> of the functor <math>I \to C, \, i \mapsto X_i</math>, where ''I'' is viewed as a discrete category. It is the dual of the product of the family. For example, a coproduct in '''[[category of groups|Grp]]''' is a [[free product]].}}

{{term|1=core}}
{{defn|1=The [[core (category theory)|core]] of a category is the maximal groupoid contained in the category.}}

{{glossary end}}

==D==
{{glossary}}
{{term|1=Day convolution}}
{{defn|Given a group or monoid ''M'', the [[Day convolution]] is the tensor product in <math>\mathbf{Fct}(M, \mathbf{Set})</math>.<ref>http://ncatlab.org/nlab/show/Day+convolution</ref>}}

{{term|1=density theorem}}
{{defn|1=The [[density theorem (category theory)|density theorem]] states that every presheaf (a set-valued contravariant functor) is a colimit of representable presheaves. Yoneda's lemma embeds a category ''C'' into the category of presheaves on ''C''. The density theorem then says the image is "dense", so to say. The name "density" is because of the analogy with the [[Jacobson density theorem]] (or other variants) in abstract algebra.}}

{{term|1=diagonal functor}}
{{defn|1=Given categories ''I'', ''C'', the [[diagonal functor]] is the functor
:<math>\Delta: C \to \mathbf{Fct}(I, C), \, A \mapsto \Delta_A</math>
that sends each object ''A'' to the constant functor with value ''A'' and each morphism <math>f: A \to B</math> to the natural transformation <math>\Delta_{f, i}: \Delta_A(i) = A \to \Delta_B(i) =B</math> that is ''f'' at each ''i''.}}

{{term|1=diagram}}
{{defn|1=Given a category ''C'', a [[diagram (category theory)|diagram]] in ''C'' is a functor <math>f: I \to C</math> from a small category ''I''.}}

{{term|1=differential graded category}}
{{defn|1=A [[differential graded category]] is a category whose Hom sets are equipped with structures of [[differential graded module]]s. In particular, if the category has only one object, it is the same as a differential graded module.}}

{{term|1=discrete}}
{{defn|1=A category is [[discrete category|discrete]] if each morphism is an identity morphism (of some object). For example, a set can be viewed as a discrete category.}}

{{term|1=distributor}}
{{defn|1=Another term for "profunctor".}}

{{term|1=Dwyer–Kan equivalence}}
{{defn|1=A [[Dwyer–Kan equivalence]] is a generalization of an equivalence of categories to the simplicial context.<ref>{{cite arxiv|last=Hinich|first=V.|date=2013-11-17|title=Dwyer-Kan localization revisited|eprint=1311.4128|class=math.QA}}</ref>}}
{{glossary end}}

==E==
{{glossary}}
{{term|1=Eilenberg–Moore category}}
{{defn|1=Another name for the category of [[algebra for a monad|algebras for a given monad]].}}

{{term|1=end}}
{{defn|1=The [[end (category theory)|end]] of a functor <math>F: C^{\text{op}} \times C \to X</math> is the limit
:<math>\int_{c \in C} F(c, c) = \varprojlim (F^{\#}: C^{\#} \to X)</math>
where <math>C^{\#}</math> is the category (called the [[subdivision category]] of ''C'') whose objects are symbols <math>c^{\#}, u^{\#}</math> for all objects ''c'' and all morphisms ''u'' in ''C'' and whose morphisms are <math>b^{\#} \to u^{\#}</math> and <math>u^{\#} \to c^{\#}</math> if <math>u: b \to c</math> and where <math>F^{\#}</math> is induced by ''F'' so that <math>c^{\#}</math> would go to <math>F(c, c)</math> and <math>u^{\#}, u: b \to c</math> would go to <math>F(b, c)</math>. For example, for functors <math>F, G: C \to X</math>,
:<math>\int_{c \in C} \operatorname{Hom}(F(c), G(c))</math>
is the set of natural transformations from ''F'' to ''G''. For more examples, see [http://mathoverflow.net/questions/78471/intuition-for-coends this mathoverflow thread]. The dual of an end is a coend.}}

{{term|1=endofunctor}}
{{defn|1=A functor between the same category.}}

{{term|1=enriched category}}
{{defn|1=Given a monoidal category (''C'', ⊗, 1), a [[category enriched]] over ''C'' is, informally, a category whose Hom sets are in ''C''. More precisely, a category ''D'' enriched over ''C'' is a data consisting of
# A class of objects,
# For each pair of objects ''X'', ''Y'' in ''D'', an object <math>\operatorname{Map}_D(X, Y)</math> in ''C'', called the [[mapping object]] from ''X'' to ''Y'',
# For each triple of objects ''X'', ''Y'', ''Z'' in ''D'', a morphism in ''C'',
#:<math>\circ: \operatorname{Map}_D(Y, Z) \otimes \operatorname{Map}_D(X, Y) \to \operatorname{Map}_D(X, Z)</math>,
#:called the composition,
#For each object ''X'' in ''D'', a morphism <math>1_X: 1 \to \operatorname{Map}_D(X, X)</math> in ''C'', called the unit morphism of ''X''
subject to the conditions that (roughly) the compositions are associative and the unit morphisms act as the multiplicative identity.

For example, a category enriched over sets is an ordinary category.}}

{{term|1=empty}}
{{defn|The [[empty category (category theory)|empty category]] is a category with no object. It is the same thing as the [[empty set]] when the empty set is viewed as a discrete category.}}

{{term|1=epimorphism}}
{{defn|1=A morphism ''f'' is an [[epimorphism]] if <math>g=h</math> whenever <math>g\circ f=h\circ f</math>. In other words, ''f'' is the dual of a monomorphism.}}

{{term|1=equalizer}}
{{defn|1=The [[equalizer (mathematics)|equalizer]] of a pair of morphisms <math>f, g: A \to B</math> is the limit of the pair. It is the dual of a coequalizer.}}

{{term|1=equivalence}}
{{defn|no=1|A functor is an [[equivalence of categories|equivalence]] if it is faithful, full and essentially surjective.}}
{{defn|no=2|A morphism in an ∞-category ''C'' is an equivalence if it gives an isomorphism in the homotopy category of ''C''.}}

{{term|1=equivalent}}
{{defn|1=A category is equivalent to another category if there is an [[equivalence of categories|equivalence]] between them.}}

{{term|1=essentially surjective}}
{{defn|1=A functor ''F'' is called [[essentially surjective]] (or isomorphism-dense) if for every object ''B'' there exists an object ''A'' such that ''F''(''A'') is isomorphic to ''B''.}}

{{term|1=evaluation}}
{{defn|1=Given categories ''C'', ''D'' and an object ''A'' in ''C'', the [[evaluation (category theory)|evaluation]] at ''A'' is the functor
:<math>\mathbf{Fct}(C, D) \to D, \,\, F \mapsto F(A).</math>
For example, the [[Eilenberg–Steenrod axioms]] give an instance when the functor is an equivalence.}}
{{glossary end}}

==F==
{{glossary}}
{{term|1=faithful}}
{{defn|1=A functor is [[faithful functor|faithful]] if it is injective when restricted to each [[hom-set]].}}

{{term|1=fundamental category}}
{{defn|1=The [[fundamental category functor]] <math>\tau_1: s\mathbf{Set} \to \mathbf{Cat}</math> is the left adjoint to the nerve functor ''N''. For every category ''C'', <math>\tau_1 NC = C</math>.}}

{{term|1=fundamental groupoid}}
{{defn|1=The [[fundamental groupoid]] <math>\Pi_1 X</math> of a Kan complex ''X'' is the category where an object is a 0-simplex (vertex) <math>\Delta^0 \to X</math>, a morphism is a homotopy class of a 1-simplex (path) <math>\Delta^1 \to X</math> and a composition is determined by the Kan property.<!-- check this: Equivalently, it is the groupoid completion of the fundamental category <math>\tau_1 X</math> of ''X''.-->}}

{{term|1=fibered category}}
{{defn|1=A functor π: ''C'' → ''D'' is said to exhibit ''C'' as a [[fibered category|category fibered over]] ''D'' if, for each morphism ''g'': ''x'' → π(''y'') in ''D'', there exists a π-cartesian morphism ''f'': ''x<nowiki>'</nowiki>'' → ''y'' in ''C'' such that π(''f'') = ''g''. If ''D'' is the category of affine schemes (say of finite type over some field), then π is more commonly called a [[prestack]]. '''Note''': π is often a forgetful functor and in fact the [[Grothendieck construction]] implies that every fibered category can be taken to be that form (up to equivalences in a suitable sense).}}

{{term|1=fiber product}}
{{defn|1=Given a category ''C'' and a set ''I'', the [[fiber product]] over an object ''S'' of a family of objects ''X''<sub>''i''</sub> in ''C'' indexed by ''I'' is the product of the family in the [[slice category]] <math>C_{/S}</math> of ''C'' over ''S'' (provided there are <math>X_i \to S</math>). The fiber product of two objects ''X'' and ''Y'' over an object ''S'' is denoted by <math>X \times_S Y</math> and is also called a [[Cartesian square]].}}

{{term|1=filtered}}
{{defn|no=1|1=A [[filtered category]] (also called a filtrant category) is a nonempty category with the properties (1) given objects ''i'' and ''j'', there are an object ''k'' and morphisms ''i'' → ''k'' and ''j'' → ''k'' and (2) given morphisms ''u'', ''v'': ''i'' → ''j'', there are an object ''k'' and a morphism ''w'': ''j'' → ''k'' such that ''w'' ∘ ''u'' = ''w'' ∘ ''v''. A category ''I'' is filtered if and only if, for each finite category ''J'' and functor ''f'': ''J'' → ''I'', the set <math>\varprojlim \operatorname{Hom}(f(j), i)</math> is nonempty for some object ''i'' in ''I''.}}
{{defn|no=2|1=Given a cardinal number π, a category is said to be π-filtrant if, for each category ''J'' whose set of morphisms has cardinal number strictly less than π, the set <math>\varprojlim \operatorname{Hom}(f(j), i)</math> is nonempty for some object ''i'' in ''I''.}}

{{term|1=finitary monad}}
{{defn|1=A [[finitary monad]] or an algebraic monad is a monad on '''Set''' whose underlying endofunctor commutes with filtered colimits.}}

{{term|1=finite}}
{{defn|1=A category is finite if it has only finitely many morphisms.}}

{{term|1=forgetful functor}}
{{defn|The [[forgetful functor]] is, roughly, a functor that loses some of data of the objects; for example, the functor <math>\mathbf{Grp} \to \mathbf{Set}</math> that sends a group to its underlying set and a group homomorphism to itself is a forgetful functor.}}

{{term|1=free functor}}
{{defn|1=A [[free functor]] is a left adjoint to a forgetful functor. For example, for a ring ''R'', the functor that sends a set ''X'' to the [[free module|free ''R''-module]] generated by ''X'' is a free functor (whence the name).}}

{{term|1=Frobenius category}}
{{defn|1=A [[Frobenius category]] is an [[exact category]] that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects.}}

{{term|1=Fukaya category}}
{{defn|1=See [[Fukaya category]].}}

{{term|1=full}}
{{defn|no=1|1=A functor is [[full functor|full]] if it is surjective when restricted to each [[hom-set]].}}
{{defn|no=2|1=A category ''A'' is a [[full subcategory]] of a category ''B'' if the inclusion functor from ''A'' to ''B'' is full.}}

{{term|1=functor}}
{{defn|1=Given categories ''C'', ''D'', a [[functor]] ''F'' from ''C'' to ''D'' is a structure-preserving map from ''C'' to ''D''; i.e., it consists of an object ''F''(''x'') in ''D'' for each object ''x'' in ''C'' and a morphism ''F''(''f'') in ''D'' for each morphism ''f'' in ''C'' satisfying the conditions: (1) <math>F(f \circ g) = F(f) \circ F(g)</math> whenever <math>f \circ g</math> is defined and (2) <math>F(\operatorname{id}_x) = \operatorname{id}_{F(x)}</math>. For example,
:<math>\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}, \, S \mapsto \mathfrak{P}(S)</math>,
where <math>\mathfrak{P}(S)</math> is the [[power set]] of ''S'' is a functor if we define: for each function <math>f: S \to T</math>, <math>\mathfrak{P}(f): \mathfrak{P}(S) \to \mathfrak{P}(T)</math> by <math>\mathfrak{P}(f)(A) = f(A)</math>.}}

{{term|1=functor category}}
{{defn|1=The [[functor category]] '''Fct'''(''C'', ''D'') from a category ''C'' to a category ''D'' is the category where the objects are all the functors from ''C'' to ''D'' and the morphisms are all the natural transformations between the functors.}}

{{glossary end}}

==G==
{{glossary}}
{{term|1=Gabriel–Popescu theorem}}
{{defn|1=The [[Gabriel–Popescu theorem]] says an abelian category is a [[Serre quotient category|quotient]] of the category of modules.}}

{{term|1=generator}}
{{defn|1=In a category ''C'', a family of objects <math>G_i, i \in I</math> is a [[generator (category theory)|system of generators]] of ''C'' if the functor <math>X \mapsto \prod_{i \in I} \operatorname{Hom}(G_i, X)</math> is conservative. Its dual is called a system of cogenerators.}}

{{term|1=Grothendieck's Galois theory}}
{{defn|1=A category-theoretic generalization of [[Galois theory]]; see [[Grothendieck's Galois theory]].}}

{{term|1=Grothendieck category}}
{{defn|1=A [[Grothendieck category]] is a certain well-behaved kind of an abelian category.}}

{{term|1=Grothendieck construction}}
{{defn|1=Given a functor <math>U: C \to \mathbf{Cat}</math>, let ''D''<sub>''U''</sub> be the category where the objects are pairs (''x'', ''u'') consisting of an object ''x'' in ''C'' and an object ''u'' in the category ''U''(''x'') and a morphism from (''x'', ''u'') to (''y'', ''v'') is a pair consisting of a morphism ''f'': ''x'' → ''y'' in ''C'' and a morphism ''U''(''f'')(''u'') → ''v'' in ''U''(''y''). The passage from ''U'' to ''D''<sub>''U''</sub> is then called the [[Grothendieck construction]].}}

{{term|1=Grothendieck fibration}}
{{defn|1=A [[fibered category]].}}

{{term|1=groupoid}}
{{defn|no=1|1=A category is called a [[groupoid]] if every morphism in it is an isomorphism.}}
{{defn|no=2|1=An ∞-category is called an [[∞-groupoid]] if every morphism in it is an equivalence (or equivalently if it is a [[Kan complex]].)}}

{{glossary end}}

==H==
{{glossary}}
{{term|1=Hall algebra of a category}}
{{defn|1=See [[Ringel–Hall algebra]].}}

{{term|1=heart}}
{{defn|1=The [[heart (category theory)|heart]] of a [[t-structure]] (<math>D^{\ge 0}</math>, <math>D^{\le 0}</math>) on a triangulated category is the intersection <math>D^{\ge 0} \cap D^{\le 0}</math>. It is an abelian category.}}

{{term|1=Higher category theory}}
{{defn|1=[[Higher category theory]] is a subfield of category theory that concerns the study of [[n-category|''n''-categories]] and [[∞-categories]].}}

{{term|1=homological dimension}}
{{defn|1=The [[homological dimension]] of an abelian category with enough injectives is the least non-negative intege ''n'' such that every object in the category admits an injective resolution of length at most ''n''. The dimension is ∞ if no such integer exists. For example, the homological dimension of [[category of modules|Mod<sub>''R''</sub>]] with a principal ideal domain ''R'' is at most one.}}

{{term|1=homotopy category}}
{{defn|1=See<!-- for now --> [[homotopy category]]. It is closely related to a [[localization of a category]].}}

{{term|1=homotopy hypothesis}}
{{defn|1=The [[homotopy hypothesis]] states an [[∞-groupoid]] is a space (less equivocally, an ''n''-groupoid can be used as a homotopy ''n''-type.)}}
{{glossary end}}

==I==
{{glossary}}
{{term|1=identity}}
{{defn|no=1|1=The [[identity morphism]] ''f'' of an object ''A'' is a morphism from ''A'' to ''A'' such that for any morphisms ''g'' with domain ''A'' and ''h'' with codomain ''A'', <math>g\circ f=g</math> and <math>f\circ h=h</math>.}}
{{defn|no=2|The [[identity functor]] on a category ''C'' is a functor from ''C'' to ''C'' that sends objects and morphisms to themselves.}}
{{defn|no=3|Given a functor ''F'': ''C'' → ''D'', the [[identity natural transformation]] from ''F'' to ''F'' is a natural transformation consisting of the identity morphisms of ''F''(''X'') in ''D'' for the objects ''X'' in ''C''.}}

{{term|1=image}}
{{defn|1=The [[image of a morphism|image]] of a morphism ''f'': ''X'' → ''Y'' is the equalizer of <math>Y \rightrightarrows Y \sqcup_X Y</math>.}}

{{term|1=ind-limit}}
{{defn|1=A colimit (or inductive limit) in <math>\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})</math>.}}

{{term|1=∞-category}}
{{defn|1=An [[∞-category]] ''C'' is a [[simplicial set]] satisfying the following condition: for each 0 < ''i'' < ''n'',
*every map of simplicial sets <math>f: \Lambda^n_i \to C</math> extends to an ''n''-simplex <math>f: \Delta^n \to C</math>
where &Delta;<sup>''n''</sup> is the standard ''n''-simplex and <math>\Lambda^n_i</math> is obtained from &Delta;<sup>''n''</sup> by removing the ''i''-th face and the interior (see [[Kan fibration#Definition]]). For example, the [[nerve of a category]] satisfies the condition and thus can be considered as an ∞-category.}}

{{term|1=initial}}
{{defn|no=1|1=An object ''A'' is [[initial object|initial]] if there is exactly one morphism from ''A'' to each object; e.g., [[empty set]] in '''[[Category of sets|Set]]'''.}}
{{defn|no=2|1=An object ''A'' in an ∞-category ''C'' is initial if <math>\operatorname{Map}_C(A, B)</math> is [[contractible space|contractible]] for each object ''B'' in ''C''.}}

{{term|1=injective}}
{{defn|1=An object ''A'' in an abelian category is [[injective object|injective]] if the functor <math>\operatorname{Hom}(-, A)</math> is exact. It is the dual of a projective object.}}

{{term|1=internal Hom}}
{{defn|1=Given a [[monoidal category]] (''C'', ⊗), the [[internal Hom]] is a functor <math>[-, -]: C^{\text{op}} \times C \to C</math> such that <math>[Y, -]</math> is the right adjoint to <math>- \otimes Y</math> for each object ''Y'' in ''C''. For example, the [[category of modules]] over a commutative ring ''R'' has the internal Hom given as <math>[M, N] = \operatorname{Hom}_R(M, N)</math>, the set of ''R''-linear maps.}}

{{term|1=inverse}}
{{defn|1=A morphism ''f'' is an [[inverse function|inverse]] to a morphism ''g'' if <math>g\circ f</math> is defined and is equal to the identity morphism on the codomain of ''g'', and <math>f\circ g</math> is defined and equal to the identity morphism on the domain of ''g''. The inverse of ''g'' is unique and is denoted by ''g''<sup>−1</sup>. ''f'' is a left inverse to ''g'' if <math>f\circ g</math> is defined and is equal to the identity morphism on the domain of ''g'', and similarly for a right inverse.}}

{{term|1=isomorphic}}
{{defn|no=1|1=An object is [[isomorphic]] to another object if there is an isomorphism between them.}}
{{defn|no=2|1=A category is isomorphic to another category if there is an isomorphism between them.}}

{{term|1=isomorphism}}
{{defn|1=A morphism ''f'' is an [[isomorphism]] if there exists an ''inverse'' of ''f''.}}
{{glossary end}}

==K==
{{glossary}}
{{term|1=Kan complex}}
{{defn|1=A [[Kan complex]] is a [[fibrant object]] in the category of simplicial sets.}}

{{term|1=Kan extension}}
{{defn|no=1|Given a category ''C'', the left [[Kan extension]] functor along a functor <math>f: I \to J</math> is the left adjoint (if it exists) to <math>f^* = - \circ f: \operatorname{Fct}(J, C) \to \operatorname{Fct}(I, C)</math> and is denoted by <math>f_!</math>. For any <math>\alpha: I \to C</math>, the functor <math>f_! \alpha: J \to C</math> is called the left Kan extension of α along ''f''.<ref>http://www.math.harvard.edu/~lurie/282ynotes/LectureXI-Homological.pdf</ref> One can show:
:<math>(f_! \alpha)(j) = \varinjlim_{f(i) \to j} \alpha(i)</math>
where the colimit runs over all objects <math>f(i) \to j</math> in the comma category.}}
{{defn|no=2|The right Kan extension functor is the right adjoint (if it exists) to <math>f^*</math>.}}

{{term|1=Kleisli category}}
{{defn|1=Given a monad ''T'', the [[Kleisli category]] of ''T'' is the full subcategory of the category of ''T''-algebras (called Eilenberg–Moore category) that consists of free ''T''-algebras.}}
{{glossary end}}

==L==
{{glossary}}
{{term|1=lax}}
{{defn|1=The term "[[lax functor]]" is essentially synonymous with "[[pseudo-functor]]".}}

{{term|1=length}}
{{defn|1=An object in an abelian category is said to have <span id="finite length"></span>finite length if it has a [[composition series]]. The maximum number of proper subobjects in any such composition series is called the '''length''' of ''A''.<ref>{{harvnb|Kashiwara|Schapira|2006|loc=exercise 8.20}}</ref>}}

{{term|1=limit}}
{{defn|no=1|The [[limit (category theory)|limit]] (or [[projective limit]]) of a functor <math>f: I^{\text{op}} \to \mathbf{Set}</math> is
::<math>\varprojlim_{i \in I} f(i) = \{ (x_i|i) \in \prod_{i} f(i) | f(s)(x_j) = x_i \text{ for any } s: i \to j \}.</math>}}
{{defn|no=2|The limit <math>\varprojlim_{i \in I} f(i)</math> of a functor <math>f: I^{\text{op}} \to C</math> is an object, if any, in ''C'' that satisfies: for any object ''X'' in ''C'', <math>\operatorname{Hom}(X, \varprojlim_{i \in I} f(i)) = \varprojlim_{i \in I} \operatorname{Hom}(X, f(i))</math>; i.e., it is an object representing the functor <math>X \mapsto \varprojlim_i \operatorname{Hom}(X, f(i)).</math>}}
{{defn|no=3|The [[colimit]] (or [[inductive limit]]) <math>\varinjlim_{i \in I} f(i)</math> is the dual of a limit; i.e., given a functor <math>f: I \to C</math>, it satisfies: for any ''X'', <math>\operatorname{Hom}(\varinjlim f(i), X) = \varprojlim \operatorname{Hom}(f(i), X)</math>. Explicitly, to give <math>\varinjlim f(i) \to X</math> is to give a family of morphisms <math>f(i) \to X</math> such that for any <math>i \to j</math>, <math>f(i) \to X</math> is <math>f(i) \to f(j) \to X</math>. Perhaps the simplest example of a colimit is a [[coequalizer]]. For another example, take ''f'' to be the identity functor on ''C'' and suppose <math>L = \varinjlim_{X \in C} f(X)</math> exists; then the identity morphism on ''L'' corresponds to a compatible family of morphisms <math>\alpha_X: X \to L</math> such that <math>\alpha_L</math> is the identity. If <math>f: X \to L</math> is any morphism, then <math>f = \alpha_L \circ f = \alpha_X</math>; i.e., ''L'' is a final object of ''C''.
}}

{{term|1=localization of a category}}
{{defn|1=See [[localization of a category]].}}
{{glossary end}}

==M==
{{glossary}}
{{term|1=monad}}
{{defn|1=A [[monad (category theory)|monad]] in a category ''X'' is a [[monoid object]] in the monoidal category of endofunctors of ''X'' with the monoidal structure given by composition. For example, given a group ''G'', define an endofunctor ''T'' on '''Set''' by <math>T(X) = G \times X</math>. Then define the multiplication ''μ'' on ''T'' as the natural transformation <math>\mu: T \circ T \to T</math> given by
:<math>\mu_X: G \times (G \times X) \to G \times X, \,\, (g, (h, x)) \mapsto (gh, x)</math>
and also define the identity map ''η'' in the analogous fashion. Then (''T'', ''μ'', ''η'') constitutes a monad in '''Set'''. More substantially, an adjunction between functors <math>F: X \rightleftarrows A : G</math> determines a monad in ''X''; namely, one takes <math>T = G \circ F</math>, the identity map ''η'' on ''T'' to be a unit of the adjunction and also defines ''μ'' using the adjunction.}}

{{term|1=monadic}}
{{defn|no=1|1=An adjunction is said to be [[monadic adjunction|monadic]] if it comes from the monad that it determines by means of the [[Eilenberg–Moore category]] (the category of algebras for the monad).}}
{{defn|no=2|1=A functor is said to be [[monadic functor|monadic]] if it is a constituent of a monadic adjunction.}}

{{term|1=monoidal category}}
{{defn|1=A [[monoidal category]], also called a tensor category, is a category ''C'' equipped with (1) a [[bifunctor]] <math>\otimes: C \times C \to C</math>, (2) an identity object and (3) natural isomorphisms that make ⊗ associative and the identity object an identity for ⊗, subject to certain coherence conditions.}}

{{term|1=monoid object}}
{{defn|1=A [[monoid object]] in a monoidal category is an object together with the multiplication map and the identity map that satisfy the expected conditions like associativity. For example, a monoid object in '''Set''' is a usual monoid (unital semigroup) and a monoid object in '''[[category of modules|''R''-mod]]''' is an [[associative algebra]] over a commutative ring ''R''.}}

{{term|1=monomorphism}}
{{defn|1=A morphism ''f'' is a [[monomorphism]] (also called monic) if <math>g=h</math> whenever <math>f\circ g=f\circ h</math>; e.g., an [[Injective function|injection]] in '''[[Category of sets|Set]]'''. In other words, ''f'' is the dual of an epimorphism.}}

{{term|1=multicategory}}
{{defn|1=A [[multicategory]] is a generalization of a category in which a morphism is allowed to have more than one domain. It is the same thing as a [[colored operad]].<ref>https://ncatlab.org/nlab/show/multicategory</ref>}}
{{glossary end}}

==N==
{{glossary}}

{{term|1=''n''-category}}
{{quote box
|quote=[T]he issue of comparing definitions of weak ''n''-category is a slippery one, as it is hard to say what it even ''means'' for two such definitions to be equivalent. [...] It is widely held that the structure formed by weak ''n''-categories and the functors, transformations, ... between them should be a weak (''n'' + 1)-category; and if this is the case then the question is whether your weak (''n'' + 1)-category of weak ''n-''categories is equivalent to mine—but whose definition of weak (''n'' + 1)-category are we using here... ?
|source=[http://www.tac.mta.ca/tac/volumes/10/1/10-01abs.html A survey of definitions of ''n''-category]
|author=Tom Leinster
|align=right
|width=33%
}}{{defn|no=1|1=A [[strict n-category|strict ''n''-category]] is defined inductively: a strict 0-category is a set and a strict ''n''-category is a category whose Hom sets are strict (''n''-1)-categories. Precisely, a strict ''n''-category is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category.}}
{{defn|no=2|1=The notion of a [[weak n-category|weak ''n''-category]] is obtained from the strict one by weakening the conditions like associativity of composition to hold only up to [[coherent isomorphism]]s in the weak sense.}}
{{defn|no=3|1=One can define an ∞-category as a kind of a colim of ''n''-categories. Conversely, if one has the notion of a (weak) ∞-category (say a [[quasi-category]]) in the beginning, then a weak ''n''-category can be defined as a type of a truncated ∞-category.}}

{{term|1=natural}}
{{defn|no=1|A natural transformation is, roughly, a map between functors. Precisely, given a pair of functors ''F'', ''G'' from a category ''C'' to category ''D'', a [[natural transformation]] φ from ''F'' to ''G'' is a set of morphisms in ''D''
:<math>\{ \phi_x: F(x) \to G(x) \mid x \in \operatorname{Ob}(C) \}</math>
satisfying the condition: for each morphism ''f'': ''x'' → ''y'' in ''C'', <math>\phi_y \circ F(f) = G(f) \circ \phi_x</math>. For example, writing <math>GL_n(R)</math> for the group of invertible ''n''-by-''n'' matrices with coefficients in a commutative ring ''R'', we can view <math>GL_n</math> as a functor from the category '''CRing''' of commutative rings to the category '''Grp''' of groups. Similarly, <math>R \mapsto R^*</math> is a functor from '''CRing''' to '''Grp'''. Then the [[determinant]] det is a natural transformation from <math>GL_n</math> to -<sup>*</sup>.}}
{{defn|no=2|A [[natural isomorphism]] is a natural transformation that is an isomorphism (i.e., admits the inverse).}}

[[Image:Nerve-2-simplex.png|thumb|right|The composition is encoded as a 2-simplex.]]
{{term|1=nerve}}
{{defn|1=The [[nerve functor]] ''N'' is the functor from '''Cat''' to ''s'''''Set''' given by <math>N(C)_n = \operatorname{Hom}_{\mathbf{Cat}}([n], C)</math>. For example, if <math>\varphi</math> is a functor in <math>N(C)_2</math> (called a 2-simplex), let <math>x_i = \varphi(i), \, 0 \le i \le 2</math>. Then <math>\varphi(0 \to 1)</math> is a morphism <math>f: x_0 \to x_1</math> in ''C'' and also <math>\varphi(1 \to 2) = g: x_1 \to x_2</math> for some ''g'' in ''C''. Since <math>0 \to 2</math> is <math>0 \to 1</math> followed by <math>1 \to 2</math> and since <math>\varphi</math> is a functor, <math>\varphi(0 \to 2) = g \circ f</math>. In other words, <math>\varphi</math> encodes ''f'', ''g'' and their compositions.}}

{{term|1=normal}}
{{defn|1=A category is [[normal category|normal]] if every monic is normal.{{citation needed|date=October 2015}}}}
{{glossary end}}

==O==
{{glossary}}
{{term|1=object}}
{{defn|no=1|1=An object is part of a data defining a category.}}
{{defn|no=2|1=An [adjective] object in a category ''C'' is a contravariant functor (or presheaf) from some fixed category corresponding to the "adjective" to ''C''. For example, a [[simplicial object]] in ''C'' is a contravariant functor from the simplicial category to ''C'' and a [[Γ-object]] is a pointed contravariant functor from [[Γ (category theory)|Γ]] (roughly the pointed category of pointed finite sets) to ''C'' provided ''C'' is pointed.}}

{{term|1=op-fibration}}
{{defn|1=A functor π:''C'' → ''D'' is an [[op-fibration]] if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one π-coCartesian morphism ''f'': ''x'' → ''y<nowiki>'</nowiki>'' in ''C'' such that π(''f'') = ''g''. In other words, π is the dual of a [[Grothendieck fibration]].}}

{{term|1=opposite}}
{{defn|1=The [[opposite category]] of a category is obtained by reversing the arrows. For example, if a partially ordered set is viewed as a category, taking its opposite amounts to reversing the ordering.}}
{{glossary end}}

==P==
{{glossary}}
{{term|1=perfect}}
{{defn|Sometimes synonymous with "compact". See [[perfect complex]].}}

{{term|1=pointed}}
{{defn|1=A category (or ∞-category) is called pointed if it has a zero object.}}

{{term|1=polynomial}}
{{defn|1=A functor from the category of finite-dimensional vector spaces to itself is called a [[polynomial functor]] if, for each pair of vector spaces ''V'', ''W'', {{nowrap|''F'': Hom(''V'', ''W'') → Hom(''F''(''V''), ''F''(''W''))}} is a polynomial map between the vector spaces. A [[Schur functor]] is a basic example.}}

{{term|1=preadditive}}
{{defn|1=A category is [[preadditive category|preadditive]] if it is [[enriched category|enriched]] over the [[monoidal category]] of [[abelian group]]s. More generally, it is [[preadditive category#R-linear categories|''R''-linear]] if it is enriched over the monoidal category of [[module (mathematics)|''R''-modules]], for ''R'' a [[commutative ring]].}}

{{term|1=presentable}}
{{defn|Given a [[regular cardinal]] κ, a category is [[presentabl category|κ-presentable]] if it admits all small colimits and is [[#accessible|κ-accessible]]. A category is presentable if it is κ-presentable for some regular cardinal κ (hence presentable for any larger cardinal). '''Note''': Some authors call a presentable category a [[locally presentable category]].}}

{{term|1=presheaf}}
{{defn|1=Another term for a contravariant functor: a functor from a category ''C''<sup>op</sup> to '''Set''' is a presheaf of sets on ''C'' and a functor from ''C''<sup>op</sup> to ''s'''''Set''' is a presheaf of simplicial sets or [[simplicial presheaf]], etc. A [[Grothendieck topology|topology]] on ''C'', if any, tells which presheaf is a sheaf (with respect to that topology).}}

{{term|1=product}}
{{defn|no=1|The [[product (category theory)|product]] of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the projective limit <math>\varprojlim</math> of the functor <math>I \to C, \, i \mapsto X_i</math>, where ''I'' is viewed as a discrete category. It is denoted by <math>\prod_i X_i</math> and is the dual of the coproduct of the family.}}
{{defn|no=2|The [[product of categories|product of a family of categories]] ''C''<sub>''i''</sub>'s indexed by a set ''I'' is the category denoted by <math>\prod_i C_i</math> whose class of objects is the product of the classes of objects of ''C''<sub>''i''</sub>'s and whose hom-sets are <math>\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)</math>; the morphisms are composed component-wise. It is the dual of the disjoint union.}}

{{term|1=profunctor}}
{{defn|1=Given categories ''C'' and ''D'', a [[profunctor]] (or a distributor) from ''C'' to ''D'' is a functor of the form <math>D^{\text{op}} \times C \to \mathbf{Set}</math>.}}

{{term|1=projective}}
{{defn|1=An object ''A'' in an abelian category is [[projective object|projective]] if the functor <math>\operatorname{Hom}(A, -)</math> is exact. It is the dual of an injective object.}}

{{term|1=PROP}}
{{defn|1=A [[PROP (category theory)|PROP]] is a symmetric strict monoidal category whose objects are natural numbers and whose tensor product [[addition]] of natural numbers.}}
{{glossary end}}

==Q==
{{glossary}}
{{term|1=Quillen}}
{{defn|1=[[Quillen’s theorem A]] provides a criterion for a functor to be a weak equivalence.}}
{{glossary end}}

==R==
{{glossary}}
{{term|1=reflect}}
{{defn|no=1|1=A functor is said to reflect identities if it has the property: if ''F''(''k'') is an identity then ''k'' is an identity as well.}}
{{defn|no=2|1=A functor is said to reflect isomorphisms if it has the property: ''F''(''k'') is an isomorphism then ''k'' is an isomorphism as well.}}

{{term|1=representable}}
{{defn|1=A set-valued contravariant functor ''F'' on a category ''C'' is said to be [[representable functor|representable]] if it belongs to the essential image of the [[Yoneda embedding]] <math>C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})</math>; i.e., <math>F \simeq \operatorname{Hom}_C(-, Z)</math> for some object ''Z''. The object ''Z'' is said to be the representing object of ''F''.}}

{{term|1=retraction}}
{{defn|1=[[File:Section retract.svg|150px|thumb|''f'' is a retraction of ''g''. ''g'' is a section of ''f''.]]A morphism is a [[section (category theory)|retraction]] if it has a right inverse.}}
{{glossary end}}

==S==
{{glossary}}
{{term|1=section}}
{{defn|1=A morphism is a [[section (category theory)|section]] if it has a left inverse. For example, the [[axiom of choice]] says that any surjective function admits a section.}}

{{term|1=Segal space}}
{{defn|1=[[Segal space]]s were certain simplicial spaces, introduced as models for [[(infinity,1)-category|(∞,&nbsp;1)-categories]].}}

{{term|1=semisimple}}
{{defn|1=An abelian category is [[semisimple category|semisimple]] if every short exact sequence splits. For example, a ring is [[semisimple ring|semisimple]] if and only if the category of modules over it is semisimple.}}

{{term|1=Serre functor}}
{{defn|1=Given a ''k''-linear category ''C'' over a field ''k'', a [[Serre functor]] <math>f: C \to C</math> is an auto-equivalence such that <math>\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*</math> for any objects ''A'', ''B''.}}
<!--{{term|1=sieve}}
{{defn|1=In a category, a [[sieve (category theory)|sieve]] is a set ''S'' of objects having the property: if ''f'' is a morphism with the codomain in ''S'', then the domain of ''f'' is in ''S''.}} conflict with the literature? -->
{{term|1=simple object}}
{{defn|1=A simple object in an abelian category is <span id="simple object"></span>an object ''A'' that is not isomorphic to the zero object and whose every [[subobject]] is isomorphic to zero or to ''A''. For example, a [[simple module]] is precisely a simple object in the category of (say left) modules.}}

{{term|1=simplex category}}
{{defn|1=The [[simplex category]] &Delta; is the category where an object is a set [''n''] = { 0, 1, …, ''n'' }, n ≥ 0, totally ordered in the standard way and a morphism is an order-preserving function.}}

{{term|1=simplicial category}}
{{defn|1=A category enriched over simplicial sets.}}

{{term|1=Simplicial localization}}
{{defn|1=[[Simplicial localization]] is a method of localizing a category.}}

{{term|1=simplicial object}}
{{defn|1=A [[simplicial object]] in a category ''C'' is roughly a sequence of objects <math>X_0, X_1, X_2, \dots</math> in ''C'' that forms a simplicial set. In other words, it is a covariant or contravariant functor Δ → ''C''. For example, a [[simplicial presheaf]] is a simplicial object in the category of presheaves.}}

{{term|1=simplicial set}}
{{defn|1=A [[simplicial set]] is a contravariant functor from &Delta; to '''Set''', where &Delta; is the [[simplex category]], a category whose objects are the sets [''n''] = { 0, 1, …, ''n'' } and whose morphisms are order-preserving functions. One writes <math>X_n = X([n])</math> and an element of the set <math>X_n</math> is called an ''n''-simplex. For example, <math>\Delta^n = \operatorname{Hom}_{\Delta}(-, [n])</math> is a simplicial set called the standard ''n''-simplex. By Yoneda's lemma, <math>X_n \simeq \operatorname{Nat}(\Delta^n, X)</math>.}}

{{term|1=site}}
{{defn|1=A category equipped with a [[Grothendieck topology]].}}

{{term|1=skeletal}}
{{defn|1=A category is [[Skeleton (category theory)|skeletal]] if isomorphic objects are necessarily identical.}}

{{term|1=slice}}
{{defn|1=Given a category ''C'' and an object ''A'' in it, the [[slice category]] ''C''<sub>/''A''</sub> of ''C'' over ''A'' is the category whose objects are all the morphisms in ''C'' with codomain ''A'', whose morphisms are morphisms in ''C'' such that if ''f'' is a morphism from <math>p_X: X \to A</math> to <math>p_Y: Y \to A</math>, then <math>p_Y \circ f = p_X</math> in ''C'' and whose composition is that of ''C''.}}

{{term|1=small}}
{{defn|no=1|1=A [[small category]] is a category in which the class of all morphisms is a [[Set (mathematics)|set]] (i.e., not a [[proper class]]); otherwise '''large'''. A category is '''locally small''' if the morphisms between every pair of objects ''A'' and ''B'' form a set. Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a '''quasicategory''' is a category whose objects and morphisms merely form a [[conglomerate (set theory)|conglomerate]].<ref>{{cite book |last=Adámek |first=Jiří |author2=Herrlich, Horst |author3=Strecker, George E |title=Abstract and Concrete Categories (The Joy of Cats) |origyear=1990 |url=http://katmat.math.uni-bremen.de/acc/ |format=PDF |year=2004 |publisher= Wiley & Sons |location=New York |isbn=0-471-60922-6 |page=40}}</ref> (NB: some authors use the term "quasicategory" with a different meaning.<ref>{{cite journal|doi=10.1016/S0022-4049(02)00135-4|last=Joyal|first=A.|title=Quasi-categories and Kan complexes|journal=Journal of Pure and Applied Algebra|volume=175|year=2002|issue=1–3|pages=207–222|ref=harv}}</ref>)}}
{{defn|no=2|An object in a category is said to be [[small object|small]] if it is κ-compact for some regular cardinal κ. The notion prominently appears in Quiilen's [[small object argument]] (cf. https://ncatlab.org/nlab/show/small+object+argument)}}

{{term|species}}
{{defn|1=A [[combinatorial species|(combinatorial) species]] is an endofunctor on the groupoid of finite sets with bijections. It is categorically equivalent to a [[symmetric sequence]].}}

{{term|1=stable}}
{{defn|1=An ∞-category is [[stable ∞-category|stable]] if (1) it has a zero object, (2) every morphism in it admits a fiber and a cofiber and (3) a triangle in it is a fiber sequence if and only if it is a cofiber sequence.}}

{{term|strict}}
{{defn|A morphism ''f'' in a category admitting finite limits and finite colimits is [[strict morphism|strict]] if the natural morphism <math>\operatorname{Coim}(f) \to \operatorname{Im}(f)</math> is an isomorphism.}}

{{term|strict ''n''-category}}
{{defn|A strict 0-category is a set and for any integer ''n'' > 0, a [[strict n-category|strict ''n''-category]] is a category enriched over strict (''n''-1)-categories. For example, a strict 1-category is an ordinary category. '''Note''': the term "''n''-category" typically refers to "[[weak n-category|weak ''n''-category]]"; not strict one.}}

{{term|1=subcanonical}}
{{defn|1=A topology on a category is [[subcanonical]] if every representable contravariant functor on ''C'' is a sheaf with respect to that topology.<ref>{{harvnb|Vistoli|2004|loc=Definition 2.57.}}</ref> Generally speaking, some [[flat topology]] may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical.}}

{{term|1=subcategory}}
{{defn|1=A category ''A'' is a [[subcategory]] of a category ''B'' if there is an inclusion functor from ''A'' to ''B''.}}

{{term|1=subobject}}
{{defn|1=Given an object ''A'' in a category, a [[subobject]] of ''A'' is an equivalence class of monomorphisms to ''A''; two monomorphisms ''f'', ''g'' are considered equivalent if ''f'' factors through ''g'' and ''g'' factors through ''f''.}}

{{term|1=subquotient}}
{{defn|1=A [[subquotient]] is a quotient of a subobject.}}

{{term|1=subterminal object}}
{{defn|1=A [[subterminal object]] is an object ''X'' such that every object has at most one morphism into ''X''.}}

{{term|1=symmetric monoidal category}}
{{defn|1=A [[symmetric monoidal category]] is a [[monoidal category]] (i.e., a category with ⊗) that has maximally symmetric braiding.}}

{{term|1=symmetric sequence}}
{{defn|1=A [[symmetric sequence]] is a sequence of objects with actions of [[symmetric group]]s. It is categorically equivalent to a [[combinatorial species|(combinatorial) species]].}}

{{glossary end}}

==T==
{{glossary}}
{{term|1=t-structure}}
{{defn|1=A [[t-structure]] is an additional structure on a [[triangulated category]] (more generally [[stable ∞-category]]) that axiomatizes the notions of complexes whose cohomology concentrated in non-negative degrees or non-positive degrees.}}

{{term|1=Tannakian duality}}
{{defn|The [[Tannakian duality]] states that, in an appropriate setup, to give a morphism <math>f: X \to Y</math> is to give a pullback functor <math>f^*</math> along it. In other words, the Hom set <math>\operatorname{Hom}(X, Y)</math> can be identified with the functor category <math>\operatorname{Fct}(D(Y), D(X))</math>, perhaps in the [[derived algebraic geometry|derived sense]], where <math>D(X)</math> is the category associated to ''X'' (e.g., the derived category).<ref>Jacob Lurie. Tannaka duality for geometric stacks. http://math.harvard.edu/~lurie/, 2004.</ref><ref>{{cite arxiv|last=Bhatt|first=Bhargav|date=2014-04-29|title=Algebraization and Tannaka duality|eprint=1404.7483|class=math.AG}}</ref>}}

{{term|1=tensor category}}
{{defn|1=Usually synonymous with [[monoidal category]] (though some authors distinguish between the two concepts.)}}

{{term|1=tensor triangulated category}}
{{defn|1=A [[tensor triangulated category]] is a category that carries the structure of a symmetric monoidal category and that of a triangulated category in a compatible way.}}

{{term|tensor product}}
{{defn|Given a monoidal category ''B'', the '''tensor product of functors''' <math>F: C^{\text{op}} \to B</math> and <math>G: C \to B</math> is the coend:
:<math>F \otimes_C G = \int^{c \in C} F(c) \otimes G(c).</math>}}

{{term|1=terminal}}
{{defn|no=1|An object ''A'' is [[terminal object|terminal]] (also called final) if there is exactly one morphism from each object to ''A''; e.g., [[singleton (mathematics)|singleton]]s in '''[[Category of sets|Set]]'''. It is the dual of an [[initial object]].}}
{{defn|no=2|An object ''A'' in an ∞-category ''C'' is terminal if <math>\operatorname{Map}_C(B, A)</math> is [[contractible space|contractible]] for every object ''B'' in ''C''.}}

{{term|1=thick subcategory}}
{{defn|1=A full subcategory of an abelian category is [[thick subcategory|thick]] if it is closed under extensions.}}

{{term|1=thin}}
{{defn|1=A [[thin category|thin]] is a category where there is at most one morphism between any pair of objects.}}

{{term|1=triangulated category}}
{{defn|1=A [[triangulated category]] is a category where one can talk about distinguished triangles, generalization of exact sequences. An abelian category is a prototypical example of a triangulated category. A [[derived category]] is a triangulated category that is not necessary an abelian category.}}

{{glossary end}}

== U ==
{{glossary}}
{{term|1=universal}}
{{defn|no=1|Given a functor <math>f: C \to D</math> and an object ''X'' in ''D'', a [[universal morphism]] from ''X'' to ''f'' is an initial object in the [[comma category]] <math>(X \downarrow f)</math>. (Its dual is also called a universal morphism.) For example, take ''f'' to be the forgetful functor <math>\mathbf{Vec}_k \to \mathbf{Set}</math> and ''X'' a set. An initial object of <math>(X \downarrow f)</math> is a function <math>j: X \to f(V_X)</math>. That it is initial means that if <math>k: X \to f(W)</math> is another morphism, then there is a unique morphism from ''j'' to ''k'', which consists of a linear map <math>V_X \to W</math> that extends ''k'' via ''j''; that is to say, <math>V_X</math> is the [[free vector space]] generated by ''X''.}}
{{defn|no=2|Stated more explicitly, given ''f'' as above, a morphism <math>X \to f(u_X)</math> in ''D'' is universal if and only if the natural map
:<math>\operatorname{Hom}_C(u_X, c) \to \operatorname{Hom}_D(X, f(c)), \, \alpha \mapsto (X \to f(u_x) \overset{f(\alpha)}\to f(c))</math>
is bijective. In particular, if <math>\operatorname{Hom}_C(u_X, -) \simeq \operatorname{Hom}_D(X, f(-))</math>, then taking ''c'' to be ''u''<sub>''X''</sub> one gets a universal morphism by sending the identity morphism. In other words, having a universal morphism is equivalent to the representability of the functor <math>\operatorname{Hom}_D(X, f(-))</math>.}}
{{glossary end}}

==W==
{{glossary}}
{{term|1=Waldhausen category}}
{{defn|1=A [[Waldhausen category]] is, roughly, a category with families of cofibrations and weak equivalences.}}

{{term|1=wellpowered}}
{{defn|1=A category is wellpowered if for each object there is only a set of pairwise non-isomorphic [[subobject]]s.}}
{{glossary end}}

==Y==
{{glossary}}
{{term|1=Yoneda}}
{{defn|no=1|1={{quote box
|quote=Yoneda’s Lemma asserts ... in more evocative terms, a mathematical object X is best thought of in the context of a category surrounding it, and is determined by the network of relations it enjoys with all the objects of that category. Moreover, to understand X it might be more germane to deal directly with the functor representing it. This is reminiscent of Wittgenstein’s ’language game’; i.e., that the meaning of a word is—in essence—determined by, in fact is nothing more than, its relations to all the utterances in a language.
|author=[[Barry Mazur]]
|source=[http://www.math.harvard.edu/~mazur/papers/Thinking.about.Grothendieck%285%29.pdf Thinking about Grothendieck]
|width=33%}}The [[Yoneda lemma]] says: for each set-valued contravariant functor ''F'' on ''C'' and an object ''X'' in ''C'', there is a natural bijection
:<math>F(X) \simeq \operatorname{Nat}(\operatorname{Hom}_C(-, X), F)</math>
where Nat means the set of natural transformations. In particular, the functor
:<math>y: C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(-, X)</math>
is fully faithful and is called the Yoneda embedding.<ref>Technical note: the lemma implicitly involves a choice of '''Set'''; i.e., a choice of universe.</ref> }}
{{defn|no=2|1=If <math>F: C \to D</math> is a functor and ''y'' is the Yoneda embedding of ''C'', then the [[Yoneda extension]] of ''F'' is the left Kan extension of ''F'' along ''y''.}}
{{glossary end}}

== Z ==
{{glossary}}
{{term|1=zero}}
{{defn|1=A [[zero object]] is an object that is both initial and terminal, such as a [[trivial group]] in '''[[Category of groups|Grp]]'''.}}
{{glossary end}}

==Notes==
{{reflist}}

==References==
*{{cite book
| first = Michael
| last = Artin
| authorlink = Michael Artin
|editor=[[Alexandre Grothendieck]] |editor2=[[Jean-Louis Verdier]]
| title = Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1
|series=Lecture Notes in Mathematics |volume=269
| year = 1972
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| location = Berlin; New York
| language = French
| pages = xix+525
| nopp = true
|doi= 10.1007/BFb0081551
|isbn= 978-3-540-05896-0
}}
*{{Cite book
| last=Kashiwara
| first=Masaki
| last2=Schapira
| first2=Pierre
| title=Categories and sheaves
| year=2006
| ref=harv
|author-link = Masaki Kashiwara|author-link2 = Pierre Schapira (mathematician)}}
*[[André Joyal|A. Joyal]], [http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf The theory of quasi-categories II] (Volume I is missing??)
*[[Jacob Lurie|Lurie, J.]], ''[https://web.archive.org/web/20150206092410/http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf Higher Algebra]''
*Lurie, J., ''[[Higher Topos Theory]]''
* {{cite book | last=Mac Lane | first=Saunders | authorlink=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | edition=2nd | series=[[Graduate Texts in Mathematics]] | volume=5 | location=New York, NY | publisher=[[Springer-Verlag]] | year=1998 | isbn=0-387-98403-8 | zbl=0906.18001 |ref=harv}}
* {{cite book | editor1-last=Pedicchio | editor1-first=Maria Cristina | editor2-last=Tholen | editor2-first=Walter | title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory | series=Encyclopedia of Mathematics and Its Applications | volume=97 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2004 | isbn=0-521-83414-7 | zbl=1034.18001 }}
* {{Cite arxiv |title = Notes on Grothendieck topologies, fibered categories and descent theory |eprint = math/0412512|date = 2004-12-28|first = Angelo|last = Vistoli |ref=harv}}

== Further reading ==
* Groth, M., [http://www.math.uni-bonn.de/~mgroth/InfinityCategories.pdf A Short Course on ∞-categories]
* [http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf Cisinski's notes]
*[[History of topos theory]]
*http://plato.stanford.edu/entries/category-theory/
*{{Cite book |last=Leinster|first=Tom|date=2014|title=Basic Category Theory|series=Cambridge Studies in Advanced Mathematics |publisher=Cambridge University Press|volume=143 |arxiv=1612.09375|bibcode=2016arXiv161209375L}}
*Emily Riehl, [http://www.math.jhu.edu/~eriehl/ssets.pdf A leisurely introduction to simplicial sets]
* [http://www.andrew.cmu.edu/user/awodey/catlog/ Categorical Logic] lecture notes by [[Steve Awodey]]

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Puslapis paskutinį kartą pakeistas 2019 balandžio 05 d., 10:25
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