Introduction E9F5FC Understandable FFFFFF Questions FFFFC0 Notes EEEEEE Software 
Book.CategoryTheoryGlossary istorijaPaslėpti nežymius pakeitimus  Rodyti galutinio teksto pakeitimus 2019 balandžio 05 d., 10:25
atliko 
Pridėta 10 eilutė:
* Think through in what sense Product means "and" and Coproduct means "or". 2019 kovo 14 d., 00:03
atliko 
Pakeista 106 eilutė iš:
* A category is wellpowered if for each object there is only a set of pairwise nonisomorphic '''subobject'''s. į:
* A category is''' wellpowered''' if for each object there is only a set of pairwise nonisomorphic '''subobject'''s. 2019 kovo 06 d., 09:09
atliko 
Pakeistos 146151 eilutės iš
* A category is '''complete''' if all small limits 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 '''prelogos''') 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. * Wellpowered. 2019 kovo 03 d., 22:19
atliko 
Pridėtos 316320 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 
Pakeista 137 eilutė iš:
* A category is '' 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 
Pakeistos 1516 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 
Pridėta 6 eilutė:
 Ištrinta 7 eilutė:
2019 vasario 13 d., 14:51
atliko 
Pridėta 6 eilutė:
>>bgcolor=#FFFFC0<< Ištrintos 78 eilutės:
2019 vasario 13 d., 14:49
atliko 
Pridėtos 712 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.  2019 vasario 13 d., 14:46
atliko 
Pridėtos 12 eilutės:
>>bgcolor=#E9F5FC<<  Pridėtos 48 eilutės:
'''Investigation: Organize the concepts in category theory to reveal underlying themes'''  >><< 2019 vasario 13 d., 14:09
atliko 
Pridėta 122 eilutė:
* A category is '''closed''' if it has an internal Hom functor. Pridėtos 124125 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 
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''. 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/LectureXIHomological.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 
Pakeistos 258259 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 
Pakeistos 258259 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 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 
Pakeistos 201203 eilutės iš
* Given categories ''C'', ''D'', a '''functor''' ''F'' from ''C'' to ''D'' is a structurepreserving 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, 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 structurepreserving 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 
Pridėta 218 eilutė:
Pakeistos 220221 eilutės iš
į:
* A functor π:''C'' → ''D'' is an '''opfibration''' if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one πcoCartesian morphism ''f'': ''x'' → ''y'' in ''C'' such that π(''f'') = ''g''. In other words, π is the dual of a '''Grothendieck fibration'''. Ištrintos 224225 eilutės:
* 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 '''Γobject''' is a pointed contravariant functor from '''Γ''' (roughly the pointed category of pointed finite sets) to ''C'' provided ''C'' is pointed. Pridėtos 227241 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 Δ to '''Set''', where Δ is the '''simplex category''', a category whose objects are the sets [''n''] = { 0, 1, …, ''n'' } and whose morphisms are orderpreserving 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 setvalued 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 '''Γobject''' 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 finitedimensional 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 264279 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 '''opfibration''' if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one πcoCartesian 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 finitedimensional 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 265271 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 Δ to '''Set''', where Δ is the '''simplex category''', a category whose objects are the sets [''n''] = { 0, 1, …, ''n'' } and whose morphisms are orderpreserving 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 
Pridėtos 9697 eilutės:
* The '''simplex category''' Δ 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 orderpreserving function. Pakeistos 165166 eilutės iš
į:
* '''simplicial category''' A category enriched over simplicial sets. Pridėtos 222223 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 setvalued 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 241243 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 2simplex), 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 259268 eilutės iš
* 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 2simplex), 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:
į:
Pakeista 265 eilutė iš:
į:
Pakeistos 267268 eilutės iš
* '''simplicial category''' A category enriched over simplicial sets. į:
2019 vasario 12 d., 09:05
atliko 
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 204205 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 215216 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 '''Γobject''' is a pointed contravariant functor from '''Γ''' (roughly the pointed category of pointed finite sets) to ''C'' provided ''C'' is pointed. Pakeistos 224236 eilutės iš
* 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 πcartesian 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 228229 eilutės iš
* 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 πcartesian 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 241258 eilutės iš
* 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 260261 eilutės iš
* * A functor π:''C'' → ''D'' is an '''opfibration''' if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one πcoCartesian morphism ''f'': ''x'' → ''y<nowiki>'</nowiki> į:
* A functor π:''C'' → ''D'' is an '''opfibration''' if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one πcoCartesian 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 
Pridėtos 7576 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 8788 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 {$h: Z \to W$}, {$(h \circ g) \circ f = h \circ (g \circ f)$} and {$\operatorname{id}_Y \circ f = f \circ \operatorname{id}_X = f$}. Pakeistos 101102 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 201206 eilutės:
* 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 203213 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 '''homset'''. * A functor is '''full''' if it is surjective when restricted to each '''homset'''. * A functor ''F'' is called '''essentially surjective''' (or isomorphismdense) 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 215224 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 227230 eilutės iš
* 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 isomorphismdense) if for every object ''B'' there exists an object ''A'' such that ''F''(''A'') is isomorphic to ''B''. į:
Pakeista 229 eilutė iš:
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Ištrinta 233 eilutė:
Pakeista 243 eilutė iš:
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Pakeistos 252253 eilutės iš
* 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:
į:
2019 vasario 11 d., 14:25
atliko 
Pakeistos 8586 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 
Pridėtos 123126 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 147151 eilutės iš
į:
* The '''homological dimension''' of an abelian category with enough injectives is the least nonnegative 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 162175 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 165174 eilutės iš
* Given a category ''C'' and an object * A '''Waldhausen category''' is, roughly, a category with families of cofibrations and weak equivalences. * The 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 Kspectrum of C. * '''homotopy category'''. It is closely related to a '''localization of a category'''. 2019 vasario 11 d., 14:08
atliko 
Pridėta 93 eilutė:
* A category is wellpowered if for each object there is only a set of pairwise nonisomorphic '''subobject'''s. Pakeistos 123124 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 139141 eilutės iš
* A * 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'' 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 141142 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 164166 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. į:
Pakeistos 184185 eilutės iš
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Pridėta 190 eilutė:
* Given a cardinal number π, a category is said to be πfiltrant 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 
Pakeista 139 eilutė iš:
* A category is '''preadditive''' if it is '''enriched''' over the '''monoidal category''' of '''abelian group'''s. More generally, it is ''''' 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  2019 vasario 11 d., 13:41
atliko 
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 9597 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 0category 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 1category is an ordinary category. Pakeistos 105106 eilutės iš
į:
* A category is '''skeletal''' if isomorphic objects are necessarily identical. Ištrintos 121122 eilutės:
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 135136 eilutės iš
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* A '''derived category''' is a triangulated category that is not necessary an abelian category. Pakeistos 176178 eilutės iš
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Pakeista 178 eilutė iš:
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Pakeistos 180185 eilutės iš
* 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 
Pakeista 80 eilutė iš:
* A ''' 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 ''' 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 176178 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 ''' 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 ''' 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 preimage {$f^{1}(A)$}. 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 ''' 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 '''Γobject''' is a pointed contravariant functor from ''' 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 '''Γobject''' 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 ''' 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 ''' 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 
Pakeistos 6365 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 171173 eilutės iš
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Pridėtos 296297 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 
Pakeistos 106107 eilutės iš
į:
* The '''fundamental groupoid''' {$\Pi_1 X$} of a Kan complex ''X'' is the category where an object is a 0simplex (vertex) {$\Delta^0 \to X$}, a morphism is a homotopy class of a 1simplex (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 112115 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 homsets are {$\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)$}; the morphisms are composed componentwise. 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 118121 eilutės iš
* * The '''functor category''' 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 150151 eilutės iš
į:
Pakeistos 168175 eilutės iš
* A category is '''normal''' if every * 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 homsets are {$\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)$}; the morphisms are composed componentwise. It is the dual of the disjoint union į:
* A category is '''normal''' if every monomorphism is normal. Ištrintos 174178 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 
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 į:
* A '''thin''' is a category where there is at most one morphism between any pair of objects. Pakeistos 105107 eilutės iš
į:
* The '''core''' of a category is the maximal groupoid contained in the category. Pridėtos 114115 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 119122 eilutės iš
į:
* The '''heart''' of a '''tstructure''' ({$D^{\ge 0}$}, {$D^{\le 0}$}) 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 150151 eilutės iš
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Pakeista 153 eilutė iš:
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Pakeistos 166167 eilutės iš
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Pakeistos 172173 eilutės iš
* į:
* '''tensor category''' Usually synonymous with '''monoidal category''' (though some authors distinguish between the two concepts.) Ištrinta 174 eilutė:
Pridėtos 176178 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:
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Pakeista 184 eilutė iš:
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Pridėta 188 eilutė:
* Given a '''regular cardinal''' κ, a category is '''κpresentable''' if it admits all small colimits and is '''κaccessible'''. A category is presentable if it is κpresentable 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 
Pakeistos 8994 eilutės iš
* * A category is '''additive''' if it is preadditive (to be precise, has some preadditive 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 9195 eilutės iš
* * 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 setsSet''''''; e.g., ''''''category of vector spacesVec'''''', ''''''category of groupsGrp'''''' and ''''''category of topological spacesTop''''''. * 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 98110 eilutės:
* '''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 0simplex (vertex) {$\Delta^0 \to X$}, a morphism is a homotopy class of a 1simplex (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 πfiltrant 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 100103 eilutės iš
* A * * A category ''A'' is * The '''functor category 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 112141 eilutės:
* A category is '''preadditive''' if it is '''enriched''' over the '''monoidal category''' of '''abelian group'''s. More generally, it is '''preadditive category#Rlinear 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 preadditive 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 0simplex (vertex) {$\Delta^0 \to X$}, a morphism is a homotopy class of a 1simplex (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 πfiltrant 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:
Pakeista 145 eilutė iš:
į:
Pakeista 149 eilutė iš:
* '''''n''category''' A ''' į:
* '''''n''category''' A '''strict ''n''category''' is defined inductively: a strict 0category 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 1category is an ordinary category. Pakeistos 151153 eilutės iš
* A category (or ∞category) is called '''pointed''' if it has a zero object. * A category is '''preadditive''' if it is '''enriched categoryenriched''' over the '''monoidal category''' of '''abelian group'''s. More generally, it is '''preadditive category#Rlinear 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:
į:
Pridėtos 187188 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 Δ<sup>''n''</sup> is the standard ''n''simplex and {$\Lambda^n_i$} is obtained from Δ<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 200201 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 
Pakeista 135 eilutė iš:
* The '''product of a family of categories''' 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 homsets are {$\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)$}; the morphisms are composed componentwise. 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''' 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 259260 eilutės iš
* The '''product''' of a family of objects * The '''coproduct''' of a family of objects 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 groupsGrp'''''' is a '''free product'''. Pakeistos 262263 eilutės iš
* Given a category ''C'' and a set ''I'', the '''fiber product''' over an object ''S'' of a family of objects 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 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 
Pakeistos 7475 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 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:
į:
Pakeista 99 eilutė iš:
* A category is '''connected''' if, for each pair of objects ''x'', ''y'', there exists a finite sequence of objects 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 152154 eilutės iš
* The '''homological dimension''' of an abelian category with enough injectives is the least nonnegative 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 į:
* The '''homological dimension''' of an abelian category with enough injectives is the least nonnegative 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 '''κaccessible''' if it has κfiltered colimits and there exists a small set ''S'' of κcompact 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 
Pakeista 51 eilutė iš:
* An object in an abelian category is said to have 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 
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 preimage {$f^{1}(A)$}. 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 preimage {$f^{1}(A)$}. 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/LectureXIHomological.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. 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/LectureXIHomological.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 autoequivalence 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 autoequivalence such that {$\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*$} for any objects ''A'', ''B''. Pakeistos 225227 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. 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 {$\varprojlim_{i \in I} f(i) = \{ (x_ii) \in \prod_{i} f(i)  f(s)(x_j) = x_i \text{ for any } s: i \to j \}.$ į:
* The '''limit''' (or '''projective limit''') of a functor {$f: I^{\text{op}} \to \mathbf{Set}$} is {$\varprojlim_{i \in I} f(i) = \{ (x_ii) \in \prod_{i} f(i)  f(s)(x_j) = x_i \text{ for any } s: i \to j \}.$} 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 
Pakeistos 4344 eilutės iš
* An object in a category is said to be '''small''' if it is κcompact for some regular cardinal κ. The notion prominently appears in Quiilen's '''small object argument''' į:
* An object in a category is said to be '''small''' if it is κcompact for some regular cardinal κ. The notion prominently appears in Quiilen's '''small object argument''' [[https://ncatlab.org/nlab/show/small+object+argument  nlab: small object argument]]. Ištrintos 7172 eilutės:
2019 vasario 11 d., 12:36
atliko 
Pridėtos 5861 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 6369 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 7278 eilutės iš
* 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 functioninjection''' in ''''''Category of setsSet''''''. 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 
Ištrintos 3134 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 3435 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 3738 eilutės:
* A '''Kan complex''' is a '''fibrant object''' in the category of simplicial sets. 2019 vasario 11 d., 12:28
atliko 
Ištrintos 2026 eilutės:
* '''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 setsSet''''''. * 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 2227 eilutės iš
* * A * 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 κcompact 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 2526 eilutės iš
* * A ''' 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 4044 eilutės:
* Given a '''cardinal number''' κ, an object ''X'' in a category is '''κaccessible''' (or κcompact or κpresentable) if {$\operatorname{Hom}(X, )$} commutes with κfiltered 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 κcompact 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 4748 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 
Pakeista 222 eilutė iš:
* 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 ''' 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 
Pakeistos 233234 eilutės iš
* A '''cone''' is a way to express the '''universal property''' of a colimit (or dually a limit). One can show 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 righthand 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/intuitionforcoends 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/intuitionforcoends  this mathoverflow thread]]. The dual of an end is a coend. 2019 vasario 11 d., 11:33
atliko 
Pridėta 230 eilutė:
* Given a category ''C'', a '''diagram''' in ''C'' is a functor {$f: I \to C$} from a small category ''I''. Pridėtos 232240 eilutės:
* A '''cone''' is a way to express the '''universal property''' of a colimit (or dually a limit). One can show<ref>{{harvnbKashiwaraSchapira2006loc=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 righthand 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 {$\varprojlim_{i \in I} f(i) = \{ (x_ii) \in \prod_{i} f(i)  f(s)(x_j) = x_i \text{ for any } s: i \to j \}.$}}} * 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''. * '''indlimit''' 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 244246 eilutės iš
* :{$\operatorname{Hom}(\varinjlim f, X) \simeq \operatorname{Hom}(f, \Delta_X),$} provided the colimit in question exists. The righthand 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 247253 eilutės iš
* Given a category '' * The '''end''' of a functor :{$\int_{c \in C} F(c, c) = \varprojlim (F^{\#}: C^{\#} \to X)$} 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/intuitionforcoends 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 249255 eilutės iš
* The ''' * '''indlimit''' A colimit * The * The limit * 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/intuitionforcoends this mathoverflow thread]. The dual of an end is a coend. Pridėta 252 eilutė:
2019 vasario 11 d., 00:02
atliko 
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:
į:
2019 vasario 11 d., 00:00
atliko 
Pakeistos 3848 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 140158 eilutės iš
į:
* The '''homological dimension''' of an abelian category with enough injectives is the least nonnegative 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 '''tstructure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞category''') that axiomatizes the notions of complexes whose cohomology concentrated in nonnegative degrees or nonpositive degrees. [+Generalizations of a category+] * A '''bicategory''' is a model of a weak '''2category'''. * '''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 0category 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 1category 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 '''quasicategory''') 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 215226 eilutės:
[+Generalizations of a functor+] * The term "'''lax functor'''" is essentially synonymous with "'''pseudofunctor'''". [+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 253266 eilutės iš
[+ * * '''colored operad''' Another term for * 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 0category 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 1category 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 '''quasicategory''') 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 "'''pseudofunctor'''" 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 '''(∞, 1)categories'''. Pridėtos 277281 eilutės:
[+Dualities+] * '''co''' Often used synonymous with op; for example, a '''colimit''' refers to an oplimit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an opfibration 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 286329 eilutės iš
* Higher category theory is a subfield of category theory that concerns the study of ncategories 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 '''tstructure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞category''') that axiomatizes the notions of complexes whose cohomology concentrated in nonnegative degrees or nonpositive degrees. [+Dualities+] * '''co''' Often used synonymous with op; for example, a '''colimit''' refers to an oplimit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an opfibration 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 nonnegative 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 '''(∞, 1)categories'''. į:
* Higher category theory is a subfield of category theory that concerns the study of ncategories and ∞categories. 2019 vasario 10 d., 22:03
atliko 
Pridėtos 162163 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 257258 eilutės iš
į:
* '''Hall algebra of a category'''. See '''Ringel–Hall algebra'''. (based on equivalence classes of objects) Pakeistos 261262 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 '''tstructure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞category''') that axiomatizes the notions of complexes whose cohomology concentrated in nonnegative degrees or nonpositive degrees. Ištrintos 275277 eilutės:
* A '''comonad''' in a category ''X'' is a '''comonid''' in the monoidal category of endofunctors of ''X''. Pakeistos 279282 eilutės iš
* '''Hall algebra of a category'''. See '''Ringel–Hall algebra'''. į:
Ištrintos 291293 eilutės:
* A '''tstructure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞category''') that axiomatizes the notions of complexes whose cohomology concentrated in nonnegative degrees or nonpositive degrees. 2019 vasario 10 d., 21:51
atliko 
Pakeistos 129130 eilutės iš
į:
* The '''core''' of a category is the maximal groupoid contained in the category. Ištrinta 243 eilutė:
Pakeistos 246265 eilutės iš
į:
* Higher category theory is a subfield of category theory that concerns the study of ncategories 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 oplimit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an opfibration 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 271272 eilutės iš
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Ištrinta 274 eilutė:
Ištrinta 279 eilutė:
Ištrintos 283286 eilutės:
* 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 289290 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 292294 eilutės iš
* 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 295301 eilutės iš
* A ''' * A * A '''tstructure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞category''') that axiomatizes the notions of complexes whose cohomology concentrated in nonnegative degrees or nonpositive 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 '''tstructure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞category''') that axiomatizes the notions of complexes whose cohomology concentrated in nonnegative degrees or nonpositive degrees. 2019 vasario 10 d., 21:42
atliko 
Pakeistos 5153 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 237241 eilutės iš
į:
* '''localization of a category''' Pridėtos 239243 eilutės:
* '''Simplicial localization''' is a method of localizing a category. [+Areas of math+] Pakeistos 245250 eilutės iš
į:
* '''Grothendieck's Galois theory''' A categorytheoretic 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 261262 eilutės iš
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Pakeista 264 eilutė iš:
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Pakeista 270 eilutė iš:
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Pakeistos 278279 eilutės iš
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Pakeistos 286287 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:
2019 vasario 10 d., 21:36
atliko 
Pridėtos 2223 eilutės:
* '''compact''' Probably synonymous with '''#accessible'''. * '''perfect''' Sometimes synonymous with "compact". See '''perfect complex'''. Pakeistos 177179 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> }} {{defnno=21=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 213219 eilutės iš
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* 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 '''quasicategory''') 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 "'''pseudofunctor'''". Pakeistos 227229 eilutės iš
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* 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 setvalued 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 247248 eilutės iš
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Pakeista 261 eilutė iš:
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Ištrintos 265266 eilutės:
* 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 '''quasicategory''') in the beginning, then a weak ''n''category can be defined as a type of a truncated ∞category. Ištrinta 268 eilutė:
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''. 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 281285 eilutės iš
* 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 setvalued 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> }} {{defnno=21=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 
Pakeistos 6162 eilutės iš
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* '''composition''' A composition of morphisms in a category is part of the datum defining the category. Pakeistos 174175 eilutės iš
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* 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 202205 eilutės iš
[+ * 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 205219 eilutės iš
* ''' 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 0category 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 1category 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 setvalued 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:
Pridėtos 222230 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:
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Ištrintos 236238 eilutės:
# 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:
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Ištrinta 241 eilutė:
Pakeista 243 eilutė iš:
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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:
Pakeistos 263264 eilutės iš
* A strict 0category 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 1category 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 
Pakeistos 259294 eilutės iš
{{defnno=21=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='''JeanLouis Verdier''' title = Séminaire de Géométrie Algébrique du Bois Marie  196364  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 authorlink = Masaki Kashiwaraauthorlink2 = Pierre Schapira (mathematician)}} *'''André JoyalA. Joyal''', [http://mat.uab.cat/~kock/crm/hocat/advancedcourse/Quadern452.pdf The theory of quasicategories II] (Volume I is missing??) *'''Jacob LurieLurie, 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='''SpringerVerlag'''  year=1998  isbn=0387984038  zbl=0906.18001 ref=harv}} * {{cite book  editor1last=Pedicchio  editor1first=Maria Cristina  editor2last=Tholen  editor2first=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=0521834147  zbl=1034.18001 }} * {{Cite arxiv title = Notes on Grothendieck topologies, fibered categories and descent theory eprint = math/0412512date = 20041228first = Angelolast = Vistoli ref=harv}} [+Further reading+] * Groth, M., [http://www.math.unibonn.de/~mgroth/InfinityCategories.pdf A Short Course on ∞categories] * [http://www.math.univtoulouse.fr/~dcisinsk/1097.pdf Cisinski's notes] *'''History of topos theory''' *http://plato.stanford.edu/entries/categorytheory/ *{{Cite book last=Leinsterfirst=Tomdate=2014title=Basic Category Theoryseries=Cambridge Studies in Advanced Mathematics publisher=Cambridge University Pressvolume=143 arxiv=1612.09375bibcode=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:
{{defnno=21=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 
Pakeistos 3536 eilutės iš
į:
* A '''zero object''' is an object that is both initial and terminal, such as a '''trivial group''' in ''''''Category of groupsGrp''''''. Pakeistos 257277 eilutės iš
{{defnno=11={{quote box quote= width=33%}}The : {{defnno=21=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}} {{term1=zero}} {{defn1=A '''zero object''' is an object that is both initial and terminal, such as a '''trivial group''' in ''''''Category of groupsGrp''''''.}} {{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 setvalued 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> }} {{defnno=21=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 
Pakeistos 122124 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 nonisomorphic '''subobject'''s. Ištrintos 252253 eilutės:
Pakeistos 254269 eilutės iš
{{defnno=2Stated more explicitly, given ''f'' as above, a morphism {$X \to f(u_X)$} in ''D'' is universal if and only if the natural map {{glossary end}} ==W== {{glossary}} {{term1=Waldhausen category}} {{defn1=A '''Waldhausen category''' is, roughly, a category with families of cofibrations and weak equivalences.}} {{term1=wellpowered}} {{defn1=A category is wellpowered if for each object there is only a set of pairwise nonisomorphic '''subobject'''s.}} {{glossary end}} ==Y== {{glossary}} į:
{{defnno=2Stated 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 
Pakeistos 3235 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 spacecontractible''' for every object ''B'' in ''C''. Pakeistos 115122 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 170171 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 195197 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 208212 eilutės:
:{$\int^{c \in C} F(c, c)$}. For example, if ''R'' is a ring, ''M'' a right ''R''module and ''N'' a left ''R''module, then the '''tensor product of modulestensor 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 242247 eilutės iš
* 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 243244 eilutės:
Pakeistos 245304 eilutės iš
{{defn1=A category ''A'' is {{term1=subobject}} {{defn1=Given an object ''A'' in a category, a '''subobject''' of ''A'' is an equivalence class of monomorphisms to {{term1=subquotient}} {{defn1=A '''subquotient''' {{term1=subterminal object}} {{defn1= {{term1=symmetric monoidal category}} {{defn1=A '''symmetric monoidal category''' is a '''monoidal category {{defn1=A '''symmetric sequence''' is {{glossary end ==T== {{term1=tstructure}} {{defn1=A {{term1=Tannakian duality}} {{defnThe {{term1=tensor category}} {{defn1=Usually synonymous with {{term1=tensor triangulated category}} {{defn1=A {{termtensor product}} : {{term1=terminal}} {{defnno=1An object ''A'' is '''terminal objectterminal''' (also called final) if there is exactly one morphism from each object to ''A''; e.g., '''singleton (mathematics)singleton'''s in ''''''Category of setsSet''''''. It is the dual of an '''initial object'''.}} {{defnno=2An object ''A'' in an ∞category ''C'' is terminal if {$\operatorname{Map}_C(B, A)$} is '''contractible spacecontractible''' for every object ''B'' in ''C''.}} {{term1=thick subcategory}} {{defn1=A full subcategory of an abelian category is '''thick subcategorythick''' if it is closed under extensions.}} {{term1=thin}} {{defn1=A '''thin categorythin''' is a category where there is at most one morphism between any pair of objects.}} {{term1=triangulated category}} {{defn1=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}} {{term1=universal}} {{defnno=1Given 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 '''tstructure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞category''') that axiomatizes the notions of complexes whose cohomology concentrated in nonnegative degrees or nonpositive 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''.}} {{defnno=2Stated 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 
Pakeistos 108112 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 159160 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 237242 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 242244 eilutės iš
* A strict 0category 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 1category is an ordinary category. '''Note''': the term "''n''category" typically refers to "'''weak ''n''category'''"; not strict one. į:
* A strict 0category 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 1category 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 
Pakeistos 3132 eilutės iš
į:
* An object in a category is said to be '''small''' if it is κcompact 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 153155 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 Δ to '''Set''', where Δ is the '''simplex category''', a category whose objects are the sets [''n''] = { 0, 1, …, ''n'' } and whose morphisms are orderpreserving 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 232264 eilutės iš
{{defn1=A {{term1=simplicial set}} {{defn1=A {{term1=site}} {{defn1=A category equipped with a {{term1=skeletal}} {{defn1=A category is '''Skeleton (category theory)skeletal''' if isomorphic objects are necessarily identical.}} {{term1=slice}} {{defn1=Given a category ''C'' and an object {{term1=small}} {{defnno=11=A {{defnno=2An object in a category is said to be '''small objectsmall''' if it is κcompact for some regular cardinal κ. The notion prominently appears in Quiilen's '''small object argument''' (cf. https://ncatlab.org/nlab/show/small+object+argument)}} {{termspecies}} {{defn1=A '''combinatorial species(combinatorial) species''' is an endofunctor on the groupoid of finite sets with bijections. It is categorically equivalent to a '''symmetric sequence'''.}} {{term1=stable}} {{defn1=An ∞category is '''stable ∞categorystable''' 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.}} {{termstrict}} {{defnA morphism ''f'' in a category admitting finite limits and finite colimits is '''strict morphismstrict''' if the natural morphism {$\operatorname{Coim}(f) \to \operatorname{Im}(f)$} is an isomorphism.}} {{termstrict ''n''category}} {{defnA strict 0category is a set and for any integer ''n'' > 0, a '''strict ncategorystrict ''n''category''' is a category enriched over strict (''n''1)categories. For example, a strict 1category is an ordinary category. '''Note''': the term "''n''category" typically refers to "'''weak ncategoryweak ''n''category'''"; not strict one.}} {{term1=subcanonical}} {{defn1=A topology on a category is '''subcanonical''' if every representable contravariant functor on ''C'' is a sheaf with respect to that topology.<ref>{{harvnbVistoli2004loc=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 0category 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 1category 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>{{harvnbVistoli2004loc=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 
Pakeistos 3031 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 4344 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 106107 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 146152 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 setvalued 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 autoequivalence such that {$\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*$} for any objects ''A'', ''B''.}} * The '''simplex category''' Δ 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 orderpreserving function. * '''simplicial category''' A category enriched over simplicial sets. Pakeistos 224257 eilutės iš
* A functor is said to ==S== {{glossary}} {{term1=section}} {{defn1=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.}} {{term1=Segal space}} {{defn1='''Segal space'''s were certain simplicial spaces, introduced as models for '''(infinity,1)category(∞, 1)categories'''.}} {{term1=semisimple}} {{defn1=An abelian category is '''semisimple categorysemisimple''' if every short exact sequence splits. For example, a ring is '''semisimple ringsemisimple''' if and only if the category of modules over it is semisimple.}} {{term1=Serre functor}} {{defn1=Given a ''k''linear category ''C'' over a field ''k'', a '''Serre functor''' {$f: C \to C$} is an autoequivalence such that {$\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*$} for any objects ''A'', ''B''.}} <!{{term1=sieve}} {{defn1=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? > {{term1=simple object}} {{defn1=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.}} {{term1=simplex category}} {{defn1=The '''simplex category''' Δ 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 orderpreserving function.}} {{term1=simplicial category}} {{defn1=A category enriched over simplicial sets.}} {{term1=Simplicial localization}} {{defn1='''Simplicial localization''' is a method of localizing a category.}} į:
* '''Segal space'''s were certain simplicial spaces, introduced as models for '''(∞, 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 
Pakeistos 4142 eilutės iš
į:
* A morphism is a '''retraction''' if it has a right inverse. Pakeistos 218233 eilutės iš
==R== {{glossary}} {{term1= {{defnno=11=A functor is said to reflect identities {{defnno=21=A {{term1=representable}} {{defn1=A {{term1=retraction}} {{defn1='''File:Section retract.svg150pxthumb''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 setvalued 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 
Pakeistos 2930 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 99100 eilutės iš
į:
* A category is '''preadditive''' if it is '''enriched categoryenriched''' over the '''monoidal category''' of '''abelian group'''s. More generally, it is '''preadditive category#Rlinear 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 '''κpresentable''' if it admits all small colimits and is '''κaccessible'''. A category is presentable if it is κpresentable 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 homsets are {$\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)$}; the morphisms are composed componentwise. 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 139140 eilutės iš
į:
* A functor from the category of finitedimensional 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 165166 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 213245 eilutės iš
{{term1=polynomial}} {{defn1=A functor from the category of finitedimensional vector spaces {{term1 {{defn1 {{term1 {{defnGiven a '''regular cardinal''' κ, a category is '''presentabl categoryκpresentable''' if it admits all small colimits and is '''#accessibleκaccessible'''. A category is presentable if it is κpresentable for some regular cardinal κ (hence presentable for any larger cardinal). '''Note''': Some authors call a presentable category a '''locally presentable category'''.}} {{term1 {{defn1=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 topologytopology''' on ''C'', if any, tells which presheaf is a sheaf (with respect to that topology).}} {{term1=product}} {{defnno=1The '''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.}} {{defnno=2The '''product of categoriesproduct 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 homsets are {$\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)$}; the morphisms are composed componentwise. It is the dual of the disjoint union.}} {{term1=profunctor}} {{defn1=Given categories ''C'' and ''D'', a '''profunctor''' (or a distributor) from ''C'' to ''D'' is a functor of the form {$D^{\text{op}} \times C \to \mathbf{Set}$}.}} {{term1=projective}} {{defn1=An object ''A'' in an abelian category is '''projective objectprojective''' if the functor {$\operatorname{Hom}(A, )$} is exact. It is the dual of an injective object.}} {{term1=PROP}} {{defn1=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}} {{term1=Quillen}} {{defn1='''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 
Pridėtos 1920 eilutės:
* An '''object''' is part of a data defining a category. Pakeistos 9599 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 133134 eilutės iš
* 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 '''Γobject''' 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 '''opfibration''' if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one πcoCartesian 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 203229 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'':{$\{ \phi_x: F(x) \to G(x) \mid x \in \operatorname{Ob}(C) \}$} ==O== {{glossary}} {{term1=object}} {{defnno=11=An object is part of a data defining a category {{defnno=21=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 '''Γobject''' is a pointed contravariant functor from '''Γ (category theory)Γ''' (roughly the pointed category of pointed finite sets) to ''C'' provided ''C'' is pointed.}} {{term1=opfibration}} {{defn1=A functor π:''C'' → ''D'' is an '''opfibration''' if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one πcoCartesian morphism ''f'': ''x'' → ''y<nowiki>'</nowiki>'' in ''C'' such that π(''f'') = ''g''. In other words, π is the dual of a '''Grothendieck fibration'''.}} {{term1=opposite}} {{defn1=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}} {{term1=perfect}} {{defnSometimes synonymous with "compact". See '''perfect complex'''.}} {{term1=pointed}} {{defn1=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 
Pakeistos 2627 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 3738 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 functioninjection''' in ''''''Category of setsSet''''''. In other words, ''f'' is the dual of an epimorphism. Pakeistos 9193 eilutės iš
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* 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 ncategorystrict ''n''category''' is defined inductively: a strict 0category 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 1category is an ordinary category. Pakeistos 122128 eilutės iš
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* 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 2simplex), 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 190242 eilutės iš
:{$\mu_X: G \times (G \times X) \to G \times X, \,\, (g, (h, x)) \mapsto (gh, x)$} and also define the identity map {{term1=monadic}} {{defnno=11=An adjunction is said to be {{defnno=21=A functor is said to be {{term1=monoidal category {{defn1=A '''monoidal category {{term1=monoid object}} {{defn1=A '''monoid object''' in a monoidal {{defn1 {{term1=multicategory}} {{defn1=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}} {{term1=''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/10/1/1001abs.html A survey of definitions of ''n''category] author=Tom Leinster align=right width=33% }}{{defnno=11=A '''strict ncategorystrict ''n''category''' is defined inductively: a strict 0category 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 1category is an ordinary category.}} {{defnno=21=The notion of a '''weak ncategoryweak ''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.}} {{defnno=31=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 '''quasicategory''') in the beginning, then a weak ''n''category can be defined as a type of a truncated ∞category.}} {{term1=natural}} {{defnno=1A 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>.}} {{defnno=2A '''natural isomorphism''' is a natural transformation that is an isomorphism (i.e., admits the inverse).}} '''Image:Nerve2simplex.pngthumbrightThe composition is encoded as a 2simplex.''' {{term1=nerve}} {{defn1=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 2simplex), 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.}} {{term1=normal}} {{defn1=A category is '''normal categorynormal''' if every monic is normal.{{citation neededdate=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 '''quasicategory''') 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>. {{defnno=2A '''natural isomorphism''' is a natural transformation that is an isomorphism (i.e., admits the inverse).}} ==O== 2019 vasario 10 d., 11:35
atliko 
Pakeistos 2426 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 8789 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 117118 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/LectureXIHomological.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 137138 eilutės iš
* '''indlimit''' A colimit (or inductive limit) in {$\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}.}} į:
* '''indlimit''' 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: {$\varprojlim_{i \in I} f(i) = \{ (x_ii) \in \prod_{i} f(i)  f(s)(x_j) = x_i \text{ for any } s: i \to j \}.$}}} * 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 178220 eilutės iš
* A category is '''isomorphic''' to another category {{term1=Kan complex}} {{defn1=A '''Kan complex''' is a '''fibrant object''' in the category of simplicial sets.}} {{term1=Kan extension}} {{defnno=1Given a category :{$(f_! \alpha)(j) = \varinjlim_{f(i) \to j} \alpha(i)$} where the colimit runs over all objects {$f(i) \to j$} in the comma category.}} {{defnno=2The right Kan extension functor is the right adjoint (if it exists) to {$f^*$}.}} {{term1=Kleisli category}} {{defn1=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}} {{term1=lax}} {{defn1=The term "'''lax functor'''" is essentially synonymous with "'''pseudofunctor'''".}} {{term1=length}} {{defn1=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>{{harvnbKashiwaraSchapira2006loc=exercise 8.20}}</ref>}} {{term1=limit}} {{defnno=1The '''limit (category theory)limit''' (or '''projective limit''') of a functor {$f: I^{\text{op}} \to \mathbf{Set}$} is ::{$\varprojlim_{i \in I} f(i) = \{ (x_ii) \in \prod_{i} f(i)  f(s)(x_j) = x_i \text{ for any } s: i \to j \}.$}}} {{defnno=2The 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 {$X \mapsto \varprojlim_i \operatorname{Hom}(X, f(i)).$}}} {{defnno=3The '''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''. }} {{term1=localization of a category}} {{defn1=See '''localization of a category'''.}} {{glossary end}} ==M== {{glossary}} {{term1=monad}} {{defn1=A '''monad (category theory) į:
* The term "'''lax functor'''" is essentially synonymous with "'''pseudofunctor'''". * 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 
Pakeista 7 eilutė iš:
*'''Cat''', the ''' 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 2124 eilutės iš
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* An object ''A'' is '''initial''' if there is exactly one morphism from ''A'' to each object; e.g., '''empty set''' in ''''''Category of setsSet''''''. * 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 3134 eilutės iš
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* 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 8385 eilutės iš
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* '''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 Δ<sup>''n''</sup> is the standard ''n''simplex and {$\Lambda^n_i$} is obtained from Δ<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 111113 eilutės iš
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* 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)$}, the set of ''R''linear maps. Pakeistos 131133 eilutės iš
į:
* The '''image''' of a morphism ''f'': ''X'' → ''Y'' is the equalizer of {$Y \rightrightarrows Y \sqcup_X Y$}. * '''indlimit''' A colimit (or inductive limit) in {$\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}.}} Pakeistos 165227 eilutės iš
{{term1=Higher {{defn1= {{term1=homological {{defn1=The {{term1=homotopy category}} {{defn1=See<! for now > {{defn1=The {{glossary end}} ==I== {{glossary}} {{term1=identity}} {{defnno=11=The {{defnno=2The '''identity functor''' on a category ''C'' is a functor from ''C'' to ''C'' that sends objects and morphisms to themselves.}} {{defnno=3Given a functor ''F'': ''C'' → ''D'', the '''identity natural transformation''' from ''F'' to ''F'' is a natural transformation consisting of the identity morphisms of ''F''(''X'') in ''D'' for the objects ''X'' in ''C''.}} {{term1=image}} {{defn1=The '''image of a morphismimage''' of a morphism ''f'': ''X'' → ''Y'' is the equalizer of {$Y \rightrightarrows Y \sqcup_X Y$}.}} {{term1=indlimit}} {{defn1=A colimit (or inductive limit) in {$\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}.}} {{term1=∞category}} {{defn1=An '''∞category''' ''C'' is a '''simplicial set''' satisfying the following condition: for each 0 < ''i'' < ''n'', *every map of simplicial sets {$f: \Lambda^n_i \to C$} extends to an ''n''simplex {$f: \Delta^n \to C$} where Δ<sup>''n''</sup> is the standard ''n''simplex and {$\Lambda^n_i$} is obtained from Δ<sup>''n''</sup> by removing the ''i''th face and the interior (see '''Kan fibration#Definition'''). For example, the '''nerve of a category''' satisfies the condition and thus can be considered as an ∞category.}} {{term1=initial}} {{defnno=11=An object ''A'' is '''initial objectinitial''' if there is exactly one morphism from ''A'' to each object; e.g., '''empty set''' in ''''''Category of setsSet''''''.}} {{defnno=21=An object ''A'' in an ∞category ''C'' is initial if {$\operatorname{Map}_C(A, B)$} is '''contractible spacecontractible''' for each object ''B'' in ''C''.}} {{term1=injective}} {{defn1=An object ''A'' in an abelian category is '''injective objectinjective''' if the functor {$\operatorname{Hom}(, A)$} is exact. It is the dual of a projective object.}} {{term1=internal Hom}} {{defn1=Given a '''monoidal category''' (''C'', ⊗), the '''internal Hom''' is a functor {$[, ]: 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)$}, the set of ''R''linear maps.}} {{term1=inverse}} {{defn1=A morphism ''f'' is an '''inverse functioninverse''' to a morphism ''g'' if {$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$} is defined and is equal to the identity morphism on the domain of ''g'', and similarly for a right inverse.}} {{term1=isomorphic}} {{defnno=11=An object is '''isomorphic''' to another object if there is an isomorphism between them.}} {{defnno=21=A category is isomorphic to another category if there is an isomorphism between them.}} {{term1=isomorphism}} {{defn1=A morphism ''f'' is an '''isomorphism''' if there exists an ''inverse'' of ''f''.}} {{glossary end}} ==K== {{glossary}} į:
* '''Hall algebra of a category'''. See '''Ringel–Hall algebra'''. * '''Higher category theory''' is a subfield of category theory that concerns the study of '''''n''categories''' and '''∞categories'''. * The '''homological dimension''' of an abelian category with enough injectives is the least nonnegative integer ''n'' such that every object in the category admits an injective resolution of length at most ''n''. The dimension is ∞ if no such integer exists. For example, the homological dimension of '''Mod<sub>''R''</sub>''' with a principal ideal domain ''R'' is at most one. * The '''homotopy hypothesis''' states an '''∞groupoid''' is a space (less equivocally, an ''n''groupoid can be used as a homotopy ''n''type.)}} * Given a functor ''F'': ''C'' → ''D'', the '''identity natural transformation''' from ''F'' to ''F'' is a natural transformation consisting of the identity morphisms of ''F''(''X'') in ''D'' for the objects ''X'' in ''C''. * An object is '''isomorphic''' to another object if there is an isomorphism between them. * A category is '''isomorphic''' to another category if there is an isomorphism between them. 2019 vasario 10 d., 11:09
atliko 
Pakeistos 7677 eilutės iš
į:
* The '''heart''' of a '''tstructure''' ({$D^{\ge 0}$}, {$D^{\le 0}$}) on a triangulated category is the intersection {$D^{\ge 0} \cap D^{\le 0}$}. It is an abelian category. Pakeistos 158166 eilutės iš
==H== {{glossary}} {{term1= {{defn1=See {{term1=heart}} {{defn1=The '''heart (category theory)heart''' of a '''tstructure''' ({$D^{\ge 0}$}, {$D^{\le 0}$}) on a triangulated category is the intersection {$D^{\ge 0} \cap D^{\le 0}$}. It is an abelian category.}} į:
* Hall algebra of a category. See '''Ringel–Hall algebra'''. 2019 vasario 10 d., 11:06
atliko 
Pakeistos 2021 eilutės iš
į:
* In a category ''C'', a family of objects {$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)$} is conservative. Its dual is called a system of cogenerators. Pakeistos 7374 eilutės iš
į:
* A '''Grothendieck category''' is a certain wellbehaved kind of an abelian category. * '''Grothendieck fibration''' A '''fibered category'''. * A category is called a '''groupoid''' if every morphism in it is an isomorphism. An ∞category is called an '''∞groupoid''' if every morphism in it is an equivalence (or equivalently if it is a '''Kan complex'''. Pakeistos 149176 eilutės iš
{{glossary end}} ==G== {{glossary}} {{term1=Gabriel–Popescu {{defn1=The {{term1=generator}} {{defn1=In a category {{term1=Grothendieck {{defn1=A categorytheoretic generalization of {{term1=Grothendieck {{defn1=A {{term1=Grothendieck construction}} {{defn1=Given {{term1=Grothendieck fibration}} {{defn1=A '''fibered category'''.}} {{term1=groupoid}} {{defnno=11=A category is called a '''groupoid''' if every morphism in it is an isomorphism.}} {{defnno=21=An ∞category is called an '''∞groupoid''' if every morphism in it is an equivalence (or equivalently if it is a '''Kan complex'''.)}} į:
* The '''Gabriel–Popescu theorem''' says an abelian category is a '''quotient''' of the category of modules. * '''Grothendieck's Galois theory''' A categorytheoretic generalization of '''Galois theory'''; see '''Grothendieck's Galois theory'''. * '''Grothendieck construction''' Given a functor {$U: C \to \mathbf{Cat}$}, let ''D''<sub>''U''</sub> be the category where the objects are pairs (''x'', ''u'') consisting of an object ''x'' in ''C'' and an object ''u'' in the category ''U''(''x'') and a morphism from (''x'', ''u'') to (''y'', ''v'') is a pair consisting of a morphism ''f'': ''x'' → ''y'' in ''C'' and a morphism ''U''(''f'')(''u'') → ''v'' in ''U''(''y''). The passage from ''U'' to ''D''<sub>''U''</sub> is then called the '''Grothendieck construction'''. 2019 vasario 10 d., 11:01
atliko 
Pakeistos 6873 eilutės iš
į:
* A '''Frobenius category''' is an '''exact category''' that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects. * '''Fukaya category'''. * A category ''A'' is a '''full subcategory''' of a category ''B'' if the inclusion functor from ''A'' to ''B'' is full. * The '''functor category''' '''Fct'''(''C'', ''D'') from a category ''C'' to a category ''D'' is the category where the objects are all the functors from ''C'' to ''D'' and the morphisms are all the natural transformations between the functors. Ištrintos 147155 eilutės:
* A '''Frobenius category''' is an '''exact category''' that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects. * '''Fukaya category'''. * A category ''A'' is a '''full subcategory''' of a category ''B'' if the inclusion functor from ''A'' to ''B'' is full. {{term1=functor category}} {{defn1=The '''functor category''' '''Fct'''(''C'', ''D'') from a category ''C'' to a category ''D'' is the category where the objects are all the functors from ''C'' to ''D'' and the morphisms are all the natural transformations between the functors.}} 2019 vasario 10 d., 10:59
atliko 
Pakeistos 6468 eilutės iš
į:
* The '''fundamental groupoid''' {$\Pi_1 X$} of a Kan complex ''X'' is the category where an object is a 0simplex (vertex) {$\Delta^0 \to X$}, a morphism is a homotopy class of a 1simplex (path) {$\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)$} is nonempty for some object ''i'' in ''I''. * Given a cardinal number π, a category is said to be πfiltrant if, for each category ''J'' whose set of morphisms has cardinal number strictly less than π, the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''. * A category is '''finite''' if it has only finitely many morphisms. Pridėtos 7174 eilutės:
* Given categories ''C'', ''D'', a '''functor''' ''F'' from ''C'' to ''D'' is a structurepreserving map from ''C'' to ''D''; i.e., it consists of an object ''F''(''x'') in ''D'' for each object ''x'' in ''C'' and a morphism ''F''(''f'') in ''D'' for each morphism ''f'' in ''C'' satisfying the conditions: (1) {$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)$}. Pakeistos 8081 eilutės iš
* A '''contravariant functor''' ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a '''presheaf (category theory)presheaf''' especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let {$\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)$} by sending a subset ''A'' of ''T'' to the preimage {$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)$} by sending a subset ''A'' of ''T'' to the preimage {$f^{1}(A)$}. With this, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}$} is a contravariant functor.}} Pakeistos 8990 eilutės iš
į:
* The '''fundamental category functor''' {$\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$}. * A functor π: ''C'' → ''D'' is said to exhibit ''C'' as a '''category fibered over''' ''D'' if, for each morphism ''g'': ''x'' → π(''y'') in ''D'', there exists a πcartesian morphism ''f'': ''x<nowiki>'</nowiki>'' → ''y'' in ''C'' such that π(''f'') = ''g''. If ''D'' is the category of affine schemes (say of finite type over some field), then π is more commonly called a '''prestack'''. '''Note''': π is often a forgetful functor and in fact the '''Grothendieck construction''' implies that every fibered category can be taken to be that form (up to equivalences in a suitable sense). * The '''forgetful functor''' is, roughly, a functor that loses some of data of the objects; for example, the functor {$\mathbf{Grp} \to \mathbf{Set}$} that sends a group to its underlying set and a group homomorphism to itself is a forgetful functor. * A '''free functor''' is a left adjoint to a forgetful functor. For example, for a ring ''R'', the functor that sends a set ''X'' to the '''free modulefree ''R''module''' generated by ''X'' is a free functor (whence the name). * A functor is '''full''' if it is surjective when restricted to each '''homset'''. Pakeistos 111112 eilutės iš
į:
* Given a category ''C'' and a set ''I'', the '''fiber product''' over an object ''S'' of a family of objects ''X''<sub>''i''</sub> in ''C'' indexed by ''I'' is the product of the family in the '''slice category''' {$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$} and is also called a '''Cartesian square'''. Pakeistos 140184 eilutės iš
{{term1=fundamental category}} {{defn1=The {{term1=fundamental groupoid}} {{defn1=The {{term1=fibered category}} {{defn1=A functor π: ''C'' → ''D'' is said to exhibit ''C'' as a '''fibered categorycategory fibered over''' ''D'' if, for each morphism ''g'': ''x'' → π(''y'') in ''D'', there exists a πcartesian morphism ''f'': ''x<nowiki>'</nowiki>'' → ''y'' in ''C'' such that π(''f'') = ''g''. If ''D'' is the category of affine schemes (say of finite type over some field), then π is more commonly called a '''prestack'''. '''Note''': π is often a forgetful functor and in fact the '''Grothendieck construction''' implies that every fibered category can be taken to be that form (up to equivalences in a suitable sense).}} {{term1=fiber product}} {{defn1=Given a category ''C'' and a set ''I'', the '''fiber product''' over an object ''S'' of a family of objects ''X''<sub>''i''</sub> in ''C'' indexed by ''I'' is the product of the family in the '''slice category''' {$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$} and is also called a '''Cartesian square'''.}} {{term1=filtered}} {{defnno=11=A '''filtered category''' (also called a filtrant category) is a nonempty category with the properties (1) given objects ''i'' and ''j'', there are an object ''k'' and morphisms ''i'' → ''k'' and ''j'' → ''k'' and (2) given morphisms ''u'', ''v'': ''i'' → ''j'', there are an object ''k'' and a morphism ''w'': ''j'' → ''k'' such that ''w'' ∘ ''u'' = ''w'' ∘ ''v''. A category ''I'' is filtered if and only if, for each finite category ''J'' and functor ''f'': ''J'' → ''I'', the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''.}} {{defnno=21=Given a cardinal number π, a category is said to be πfiltrant if, for each category ''J'' whose set of morphisms has cardinal number strictly less than π, the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object ''i'' in ''I''.}} {{term1=finitary monad}} {{defn1=A '''finitary monad''' or an algebraic monad is a monad on '''Set''' whose underlying endofunctor commutes with filtered colimits.}} {{term1=finite}} {{defn1=A category is finite if it has only finitely many morphisms.}} {{term1=forgetful functor}} {{defnThe '''forgetful functor''' is, roughly, a functor that loses some of data of the objects; for example, the functor {$\mathbf{Grp} \to \mathbf{Set}$} that sends a group to its underlying set and a group homomorphism to itself is a forgetful functor.}} {{term1=free functor}} {{defn1=A '''free functor''' is a left adjoint to a forgetful functor. For example, for a ring ''R'', the functor that sends a set ''X'' to the '''free modulefree ''R''module''' generated by ''X'' is a free functor (whence the name).}} {{term1=Frobenius category}} {{defn1=A '''Frobenius category''' is an '''exact category''' that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects.}} {{term1=Fukaya category}} {{defn1=See '''Fukaya category'''.}} {{term1=full}} {{defnno=11=A functor is '''full functorfull''' if it is surjective when restricted to each '''homset'''.}} {{defnno=21=A category ''A'' is a '''full subcategory''' of a category ''B'' if the inclusion functor from ''A'' to ''B'' is full.}} {{term1=functor}} {{defn1=Given categories ''C'', ''D'', a '''functor''' ''F'' from ''C'' to ''D'' is a structurepreserving map from ''C'' to ''D''; i.e., it consists of an object ''F''(''x'') in ''D'' for each object ''x'' in ''C'' and a morphism ''F''(''f'') in ''D'' for each morphism ''f'' in ''C'' satisfying the conditions: (1) {$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)$}.}} į:
* A '''finitary monad''' or an algebraic monad is a monad on '''Set''' whose underlying endofunctor commutes with filtered colimits. * A '''Frobenius category''' is an '''exact category''' that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects. * '''Fukaya category'''. * A category ''A'' is a '''full subcategory''' of a category ''B'' if the inclusion functor from ''A'' to ''B'' is full. 2019 vasario 10 d., 10:44
atliko 
Pakeistos 2627 eilutės iš
į:
* A morphism ''f'' is an '''epimorphism''' if {$g=h$} whenever {$g\circ f=h\circ f$}. In other words, ''f'' is the dual of a monomorphism. Pakeistos 6364 eilutės iš
į:
* The '''empty category''' is a category with no object. It is the same thing as the '''empty set''' when the empty set is viewed as a discrete category. Pakeistos 7783 eilutės iš
į:
* A functor is an '''equivalence''' if it is faithful, full and essentially surjective. A morphism in an ∞category ''C'' is an equivalence if it gives an isomorphism in the homotopy category of ''C''. * A category is equivalent to another category if there is an '''equivalence''' between them. * A functor ''F'' is called '''essentially surjective''' (or isomorphismdense) if for every object ''B'' there exists an object ''A'' such that ''F''(''A'') is isomorphic to ''B''. * Given categories ''C'', ''D'' and an object ''A'' in ''C'', the '''evaluation''' at ''A'' is the functor:{$\mathbf{Fct}(C, D) \to D, \,\, F \mapsto F(A).$} For example, the '''Eilenberg–Steenrod axioms''' give an instance when the functor is an equivalence. * A functor is '''faithful''' if it is injective when restricted to each '''homset'''. Pakeistos 99100 eilutės iš
į:
* The '''equalizer (mathematics)equalizer''' of a pair of morphisms {$f, g: A \to B$} is the limit of the pair. It is the dual of a coequalizer. Pakeistos 126128 eilutės iš
* The '''density theorem į:
* The '''density theorem''' states that every presheaf (a setvalued contravariant functor) is a colimit of representable presheaves. Yoneda's lemma embeds a category ''C'' into the category of presheaves on ''C''. The density theorem then says the image is "dense", so to say. The name "density" is because of the analogy with the '''Jacobson density theorem''' (or other variants) in abstract algebra. Ištrintos 129159 eilutės:
{{term1=empty}} {{defnThe '''empty category (category theory)empty category''' is a category with no object. It is the same thing as the '''empty set''' when the empty set is viewed as a discrete category.}} {{term1=epimorphism}} {{defn1=A morphism ''f'' is an '''epimorphism''' if {$g=h$} whenever {$g\circ f=h\circ f$}. In other words, ''f'' is the dual of a monomorphism.}} {{term1=equalizer}} {{defn1=The '''equalizer (mathematics)equalizer''' of a pair of morphisms {$f, g: A \to B$} is the limit of the pair. It is the dual of a coequalizer.}} {{term1=equivalence}} {{defnno=1A functor is an '''equivalence of categoriesequivalence''' if it is faithful, full and essentially surjective.}} {{defnno=2A morphism in an ∞category ''C'' is an equivalence if it gives an isomorphism in the homotopy category of ''C''.}} {{term1=equivalent}} {{defn1=A category is equivalent to another category if there is an '''equivalence of categoriesequivalence''' between them.}} {{term1=essentially surjective}} {{defn1=A functor ''F'' is called '''essentially surjective''' (or isomorphismdense) if for every object ''B'' there exists an object ''A'' such that ''F''(''A'') is isomorphic to ''B''.}} {{term1=evaluation}} {{defn1=Given categories ''C'', ''D'' and an object ''A'' in ''C'', the '''evaluation (category theory)evaluation''' at ''A'' is the functor :{$\mathbf{Fct}(C, D) \to D, \,\, F \mapsto F(A).$} For example, the '''Eilenberg–Steenrod axioms''' give an instance when the functor is an equivalence.}} {{glossary end}} ==F== {{glossary}} {{term1=faithful}} {{defn1=A functor is '''faithful functorfaithful''' if it is injective when restricted to each '''homset'''.}} 2019 vasario 10 d., 10:30
atliko 
Pakeistos 5362 eilutės iš
į:
* Eilenberg–Moore category. Another name for the category of '''algebras for a given monad'''. * '''enriched category''' Given a monoidal category (''C'', ⊗, 1), a '''category enriched''' over ''C'' is, informally, a category whose Hom sets are in ''C''. More precisely, a category ''D'' enriched over ''C'' is a data consisting of # A class of objects, # For each pair of objects ''X'', ''Y'' in ''D'', an object {$\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 7475 eilutės iš
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* '''endofunctor.''' A functor between the same category. Pakeistos 8691 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/intuitionforcoends this mathoverflow thread]. The dual of an end is a coend. Pakeistos 122144 eilutės iš
{{term1=end}} {{defn1=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/intuitionforcoends this mathoverflow thread]. The dual of an end is a coend.}} {{term1=endofunctor}} {{defn1=A functor between the same category.}} {{term1=enriched category}} {{defn1=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 
Pakeistos 6465 eilutės iš
į:
* '''distributor'''. Another term for "profunctor". Pakeistos 105119 eilutės iš
{{term1=distributor}} {{defn1=Another term for "profunctor".}} {{term1=Dwyer–Kan {{defn1=A {{glossary end}} ==E== {{glossary}} {{term1=Eilenberg–Moore {{defn1=Another name 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 
Pakeistos 5153 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 7475 eilutės iš
į:
* Given a category ''C'', a '''diagram''' in ''C'' is a functor {$f: I \to C$} from a small category ''I''. Pakeistos 105112 eilutės iš
{{defn1=Given a category ''C'', a '''diagram (category theory)diagram''' in ''C'' is a functor {$f: I \to C$} from a small category ''I''.}} {{term1=differential graded category}} {{defn1=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.}} {{term1=discrete}} {{defn1=A category is '''discrete categorydiscrete''' 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 
Pakeistos 4651 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 setsSet''''''; e.g., ''''''category of vector spacesVec'''''', ''''''category of groupsGrp'''''' and ''''''category of topological spacesTop''''''. * 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 5764 eilutės iš
* '''Beck's theorem''' characterizes the category of * A * 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 preimage {$f^{1}(A)$}. 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 6679 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>{{harvnbKashiwaraSchapira2006loc=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 righthand 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 groupsGrp'''''' 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 '''2category'''. * '''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 8294 eilutės iš
{{term1=category of categories}} {{defn1=The {{term1=classifying space}} {{defn1=The '''classifying space of {{defn1=Often used synonymous with op; for example, a '''colimit''' refers to an oplimit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an opfibration is not the same thing as a '''cofibration'''.}} {{term1=coend}} {{defn1= 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 oplimit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an opfibration 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 88158 eilutės iš
where ''R'' is viewed as a category with one object whose morphisms are the elements of ''R''. {{term1=coequalizer}} {{defn1=The {{term1=coimage}} {{defn1=The '''coimage''' {{defn1=Another term for {{defn1=Given functors {$f: C }} {{term1=comonad}} {{defn1=A {{term1=compact}} {{defn1=Probably synonymous with '''#accessible''' {{term1=complete}} {{defn1=A category is '''complete categorycomplete''' if all small limits exist.}} {{term1=composition}} {{defnno=11=A composition of morphisms in a category is part of the datum defining the category {{defnno=21=If {$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))$}.}} {{defnno=31=Natural transformations are composed pointwise: if {$\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$}.}} {{term1=concrete}} {{defn1=A '''concrete category''' ''C'' is a category such that there is a faithful functor from ''C'' to ''''''Category of setsSet''''''; e.g., ''''''category of vector spacesVec'''''', ''''''category of groupsGrp'''''' and ''''''category of topological spacesTop''''''.}} {{term1=cone}} {{defn1=A '''cone (category theory)cone''' is a way to express the '''universal property''' of a colimit (or dually a limit). One can show<ref>{{harvnbKashiwaraSchapira2006loc=Ch. 2, Exercise 2.8.}}</ref> that the colimit {$\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),$} provided the colimit in question exists. The righthand side is then the set of cones with vertex ''X''.<ref>{{harvnbMac Lane1998loc=Ch. III, § 3.}}.</ref><!For example, let {$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)$} over all ''i'' to any >}} {{term1=connected}} {{defn1=A category is '''connected categoryconnected''' if, for each pair of objects ''x'', ''y'', there exists a finite sequence of objects ''z''<sub>''i''</sub> such that {$z_0 = x, z_n = y$} and either {$\operatorname{Hom}(z_i, z_{i+1})$} or {$\operatorname{Hom}(z_{i+1}, z_i)$} is nonempty for any ''i''.}} {{term1=conservative functor}} {{defn1=A '''conservative functor''' is a functor that reflects isomorphisms. Many forgetful functors are conservative, but the forgetful functor from '''Top''' to '''Set''' is not conservative.}} {{term1=constant}} {{defn1=A functor is '''constant functorconstant''' if it maps every object in a category to the same object ''A'' and every morphism to the identity on ''A''. Put in another way, a functor {$f: C \to D$} is constant if it factors as: {$C \to \{ A \} \overset{i}\to D$} for some object ''A'' in ''D'', where ''i'' is the inclusion of the discrete category { ''A'' }.}} {{term1=contravariant functor}} {{defn1=A '''contravariant functor''' ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a '''presheaf (category theory)presheaf''' especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let {$\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)$} by sending a subset ''A'' of ''T'' to the preimage {$f^{1}(A)$}. With this, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}$} is a contravariant functor.}} {{term1=coproduct}} {{defn1=The '''coproduct''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the inductive limit {$\varinjlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where ''I'' is viewed as a discrete category. It is the dual of the product of the family. For example, a coproduct in ''''''category of groupsGrp'''''' is a '''free product'''.}} {{term1=core}} {{defn1=The '''core (category theory)core''' of a category is the maximal groupoid contained in the category.}} {{glossary end}} ==D== {{glossary}} {{term1=Day convolution}} {{defnGiven a group or monoid ''M'', the '''Day convolution''' is the tensor product in {$\mathbf{Fct}(M, \mathbf{Set})$}.<ref>http://ncatlab.org/nlab/show/Day+convolution</ref>}} {{term1=density theorem}} {{defn1=The '''density theorem (category theory)density theorem''' states that every presheaf (a setvalued contravariant functor) is a colimit of representable presheaves. Yoneda's lemma embeds a category ''C'' into the category of presheaves on ''C''. The density theorem then says the image is "dense", so to say. The name "density" is because of the analogy with the '''Jacobson density theorem''' (or other variants) in abstract algebra.}} {{term1=diagonal functor}} {{defn1=Given categories ''I'', ''C'', the '''diagonal functor''' is the functor :{$\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})$}. * The '''density theorem (category theory)density theorem''' states that every presheaf (a setvalued contravariant functor) is a colimit of representable presheaves. Yoneda's lemma embeds a category ''C'' into the category of presheaves on ''C''. The density theorem then says the image is "dense", so to say. The name "density" is because of the analogy with the '''Jacobson density theorem''' (or other variants) in abstract algebra. 2019 vasario 10 d., 09:38
atliko 
Pakeistos 2426 eilutės iš
į:
* Given a functor π: ''C'' → ''D'' (e.g., a '''prestack''' over schemes), a morphism ''f'': ''x'' → ''y'' in ''C'' is '''πcartesian''' if, for each object ''z'' in ''C'', each morphism ''g'': ''z'' → ''y'' in ''C'' and each morphism ''v'': π(''z'') → π(''x'') in ''D'' such that π(''g'') = π(''f'') ∘ ''v'', there exists a unique morphism ''u'': ''z'' → ''x'' such that π(''u'') = ''v'' and ''g'' = ''f'' ∘ ''u''. * Given a functor π: ''C'' → ''D'' (e.g., a '''prestack''' over rings), a morphism ''f'': ''x'' → ''y'' in ''C'' is '''πcoCartesian''' if, for each object ''z'' in ''C'', each morphism ''g'': ''x'' → ''z'' in ''C'' and each morphism ''v'': π(''y'') → π(''z'') in ''D'' such that π(''g'') = ''v'' ∘ π(''f''), there exists a unique morphism ''u'': ''y'' → ''z'' such that π(''u'') = ''v'' and ''g'' = ''u'' ∘ ''f''. (In short, ''f'' is the dual of a πcartesian morphism.) Pridėtos 2939 eilutės:
* A '''category (mathematics)category''' consists of the following data #A class of objects, #For each pair of objects ''X'', ''Y'', a set {$\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$}; the associativity of composition means transitivity.}} Pakeistos 5152 eilutės iš
į:
* Given relative categories {$p: F \to C, q: G \to C$} over the same base category ''C'', a functor {$f: F \to G$} over ''C'' is cartesian if it sends cartesian morphisms to cartesian morphisms. Pakeistos 6087 eilutės iš
{{term1=cartesian functor}} {{defn1=Given relative categories {$p: F \to C, q: G \to C$} over the same base category {{term1=cartesian morphism}} {{defnno=11=Given a functor π: ''C'' → ''D'' (e.g., a '''prestack''' over schemes), a morphism ''f'': ''x'' → ''y'' in ''C'' is '''cartesian morphismπcartesian''' if, for each object ''z'' in ''C'', each morphism ''g'': ''z'' → ''y'' in ''C'' and each morphism ''v'': π(''z'') → π(''x'') in ''D'' such that π(''g'') = π(''f'') ∘ ''v'', there exists a unique morphism ''u'': ''z'' → ''x'' such that π(''u'') = ''v'' and ''g'' = ''f'' ∘ ''u'' {{defnno=21=Given a functor π: ''C'' → ''D'' (e.g., a '''prestack''' over rings), a morphism ''f'': ''x'' → ''y'' in ''C'' is '''cartesian morphismπcoCartesian''' if, for each object ''z'' in ''C'', each morphism ''g'': ''x'' → ''z'' in ''C'' and each morphism ''v'': π(''y'') → π(''z'') in ''D'' such that π(''g'') = ''v'' ∘ π(''f''), there exists a unique morphism ''u'': ''y'' → ''z'' such that π(''u'') = ''v'' and ''g'' = ''u'' ∘ ''f''. (In short, ''f'' is the dual of a πcartesian morphism.)}} {{term1=Cartesian square}} {{defn1=A commutative diagram that is isomorphic to the diagram given as a fiber product.<! really need a diagram here >}} {{term1=categorical logic}} {{defn1='''Categorical logic''' is an approach to '''mathematical logic''' that uses category theory.}} {{term1=categorification}} {{defn1='''Categorification''' is a process of replacing sets and settheoretic concepts with categories and categorytheoretic concepts in some nontrivial way to capture categoric flavors. Decategorification is the reverse of categorification.}} {{term1=category}} {{defn1=A '''category (mathematics)category''' consists of the following data #A class of objects, #For each pair of objects ''X'', ''Y'', a set {$\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$}; the associativity of composition means transitivity.}} į:
* '''Cartesian square''' A commutative diagram that is isomorphic to the diagram given as a fiber product. * '''Categorical logic''' is an approach to '''mathematical logic''' that uses category theory. * '''Categorification''' is a process of replacing sets and settheoretic concepts with categories and categorytheoretic concepts in some nontrivial way to capture categoric flavors. Decategorification is the reverse of categorification. 2019 vasario 10 d., 09:33
atliko 
Pakeistos 1719 eilutės iš
[+Types of * A category is '''abelian''' if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. į:
[+Types of object+] Pridėtos 2027 eilutės:
[+Types of morphism+] * A '''bimorphism''' is a morphism that is both an epimorphism and a monomorphism. [+Types of category+] * A category is '''abelian''' if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. Pakeistos 2973 eilutės iš
{{defn1=A category is {{term1=adjunction}} {{defn1=An :{$\operatorname{Hom}_D (F(X), Y) \simeq \operatorname{Hom}_C (X, G(Y))$}; {{term1=algebra for a monad}} {{defn1=Given {{term1=amnestic}} {{defn1=A functor is amnestic if it has the property: if {{glossary end}} ==B== {{glossary}} {{term1=balanced}} {{defn1=A category is balanced if every bimorphism is {{term1=Beck {{defn1='''Beck's monadicity theoremBeck's theorem {{term1=bicategory}} {{defn1=A {{term1=bifunctor}} {{defn1=A '''bifunctor''' from {{term1=bimorphism}} {{defn1=A {{term1=Bousfield localization}} {{defn1=See '''Bousfield localization {{glossary end}} ==C== {{glossary}} {{term {{defn1=The {{term1=cartesian closed}} {{defn1=A category is '''Cartesian closed categorycartesian closed''' if it has a terminal object and that any two objects have a product and exponential.}} į:
* A category is '''additive''' if it is preadditive (to be precise, has some preadditive structure) and admits all finite '''coproduct'''s. Although "preadditive" is an additional structure, one can show "additive" is a ''property'' of a category; i.e., one can ask whether a given category is additive or not. * An '''adjunction''' (also called an adjoint pair) is a pair of functors ''F'': ''C'' → ''D'', ''G'': ''D'' → ''C'' such that there is a "natural" bijection:{$\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}(, )$} is a bifunctor from ''C''<sup>op</sup> and ''C'' to '''Set'''. [+Other+] * Given a monad ''T'' in a category ''X'', an '''algebra for a algebra for ''T''''' or a ''T''algebra is an object in ''X'' with a '''monoid action''' of ''T'' ("algebra" is misleading and "''T''object" is perhaps a better term.) For example, given a group ''G'' that determines a monad ''T'' in '''Set''' in the standard way, a ''T''algebra is a set with an '''action''' of ''G''. * '''Beck's theorem''' characterizes the category of '''algebras for a given monad'''. * A '''bicategory''' is a model of a weak '''2category'''. * '''Bousfield localization''' * The '''calculus of functors''' is a technique of studying functors in the manner similar to the way a '''function (mathematics)function''' is studied via its '''Taylor series''' expansion; whence, the term "calculus". 2019 vasario 10 d., 09:18
atliko 
Ištrintos 47 eilutės:
Especially for higher categories, the concepts from algebraic topology are also used in the category theory. For that see also '''glossary of algebraic topology'''. Pakeista 6 eilutė iš:
*[''n''] = { 0, 1, 2, …, ''n'' }, which is viewed as a category (by writing į:
*[''n''] = { 0, 1, 2, …, ''n'' }, which is viewed as a category (by writing {$i \to j \Leftrightarrow i \le j$}.) Pakeistos 1424 eilutės iš
==A== {{glossary}} {{term1=abelian}} {{defn1=A category is '''abelian categoryabelian''' if {{term1=accessible}} {{defnno=1Given a {{defnno=2Given į:
 [+Types of category+] * A category is '''abelian''' if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. * Given a '''cardinal number''' κ, an object ''X'' in a category is '''κaccessible''' (or κcompact or κpresentable) if {$\operatorname{Hom}(X, )$} commutes with κfiltered colimits. * Given a '''regular cardinal''' κ, a category is '''κaccessible''' if it has κfiltered colimits and there exists a small set ''S'' of κcompact objects that generates the category under colimits, meaning every object can be written as a colimit of diagrams of objects in ''S''. Pakeistos 2830 eilutės iš
: ''F'' is said to be left adjoint to ''G'' and ''G'' to right adjoint to ''F''. Here, "natural" means there is a natural isomorphism į:
:{$\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())$} of bifunctors (which are contravariant in the first variable.)}} Pakeistos 5051 eilutės iš
{{defn1=A '''bifunctor''' from a pair of categories ''C'' and ''D'' to a category ''E'' is a functor ''C'' × ''D'' → ''E''. For example, for any category ''C'', į:
{{defn1=A '''bifunctor''' from a pair of categories ''C'' and ''D'' to a category ''E'' is a functor ''C'' × ''D'' → ''E''. For example, for any category ''C'', {$\operatorname{Hom}(, )$} is a bifunctor from ''C''<sup>op</sup> and ''C'' to '''Set'''.}} Pakeistos 6970 eilutės iš
{{defn1=Given relative categories į:
{{defn1=Given relative categories {$p: F \to C, q: G \to C$} over the same base category ''C'', a functor {$f: F \to G$} over ''C'' is cartesian if it sends cartesian morphisms to cartesian morphisms.}} Pakeista 87 eilutė iš:
#For each pair of objects ''X'', ''Y'', a set į:
#For each pair of objects ''X'', ''Y'', a set {$\operatorname{Hom}(X, Y)$}, whose elements are called morphisms from ''X'' to ''Y'', Pakeistos 8994 eilutės iš
#: #For each object ''X'', an identity morphism subject to the conditions: for any morphisms * For example, a '''partially ordered set''' can be viewed as a category: the objects are the elements of the set and for each pair of objects ''x'', ''y'', there is a unique morphism į:
#:{$\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$}; the associativity of composition means transitivity.}} Pakeistos 105106 eilutės iš
{{defn1=The coend of a functor : į:
{{defn1=The coend of a functor {$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)$}. Pakeista 108 eilutė iš:
: į:
:{$M \otimes_R N = \int^{R} M \otimes_{\mathbb{Z}} N$} Pakeistos 112113 eilutės iš
{{defn1=The '''coequalizer''' of a pair of morphisms į:
{{defn1=The '''coequalizer''' of a pair of morphisms {$f, g: A \to B$} is the colimit of the pair. It is the dual of an equalizer.}} Pakeistos 115116 eilutės iš
{{defn1=The '''coimage''' of a morphism ''f'': ''X'' → ''Y'' is the coequalizer of į:
{{defn1=The '''coimage''' of a morphism ''f'': ''X'' → ''Y'' is the coequalizer of {$X \times_Y X \rightrightarrows X$}.}} Pakeista 121 eilutė iš:
{{defn1=Given functors į:
{{defn1=Given functors {$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').$} For example, if ''f'' is the identity functor and ''g'' is the constant functor with a value ''b'', then it is the slice category of ''B'' over an object ''b''. Pakeistos 135137 eilutės iš
{{defnno=21=If į:
{{defnno=21=If {$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))$}.}} {{defnno=31=Natural transformations are composed pointwise: if {$\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$}.}} Pakeistos 142145 eilutės iš
{{defn1=A '''cone (category theory)cone''' is a way to express the '''universal property''' of a colimit (or dually a limit). One can show<ref>{{harvnbKashiwaraSchapira2006loc=Ch. 2, Exercise 2.8.}}</ref> that the colimit : provided the colimit in question exists. The righthand side is then the set of cones with vertex ''X''.<ref>{{harvnbMac Lane1998loc=Ch. III, § 3.}}.</ref><!For example, let į:
{{defn1=A '''cone (category theory)cone''' is a way to express the '''universal property''' of a colimit (or dually a limit). One can show<ref>{{harvnbKashiwaraSchapira2006loc=Ch. 2, Exercise 2.8.}}</ref> that the colimit {$\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),$} provided the colimit in question exists. The righthand side is then the set of cones with vertex ''X''.<ref>{{harvnbMac Lane1998loc=Ch. III, § 3.}}.</ref><!For example, let {$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)$} over all ''i'' to any >}} Pakeistos 147148 eilutės iš
{{defn1=A category is '''connected categoryconnected''' if, for each pair of objects ''x'', ''y'', there exists a finite sequence of objects ''z''<sub>''i''</sub> such that į:
{{defn1=A category is '''connected categoryconnected''' if, for each pair of objects ''x'', ''y'', there exists a finite sequence of objects ''z''<sub>''i''</sub> such that {$z_0 = x, z_n = y$} and either {$\operatorname{Hom}(z_i, z_{i+1})$} or {$\operatorname{Hom}(z_{i+1}, z_i)$} is nonempty for any ''i''.}} Pakeistos 153154 eilutės iš
{{defn1=A functor is '''constant functorconstant''' if it maps every object in a category to the same object ''A'' and every morphism to the identity on ''A''. Put in another way, a functor į:
{{defn1=A functor is '''constant functorconstant''' if it maps every object in a category to the same object ''A'' and every morphism to the identity on ''A''. Put in another way, a functor {$f: C \to D$} is constant if it factors as: {$C \to \{ A \} \overset{i}\to D$} for some object ''A'' in ''D'', where ''i'' is the inclusion of the discrete category { ''A'' }.}} Pakeistos 156159 eilutės iš
{{defn1=A '''contravariant functor''' ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a '''presheaf (category theory)presheaf''' especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let : by sending a subset ''A'' of ''T'' to the preimage į:
{{defn1=A '''contravariant functor''' ''F'' from a category ''C'' to a category ''D'' is a (covariant) functor from ''C''<sup>op</sup> to ''D''. It is sometimes also called a '''presheaf (category theory)presheaf''' especially when ''D'' is '''Set''' or the variants. For example, for each set ''S'', let {$\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)$} by sending a subset ''A'' of ''T'' to the preimage {$f^{1}(A)$}. With this, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}$} is a contravariant functor.}} Pakeistos 161162 eilutės iš
{{defn1=The '''coproduct''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the inductive limit į:
{{defn1=The '''coproduct''' of a family of objects ''X''<sub>''i''</sub> in a category ''C'' indexed by a set ''I'' is the inductive limit {$\varinjlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where ''I'' is viewed as a discrete category. It is the dual of the product of the family. For example, a coproduct in ''''''category of groupsGrp'''''' is a '''free product'''.}} Pakeistos 171172 eilutės iš
{{defnGiven a group or monoid ''M'', the '''Day convolution''' is the tensor product in į:
{{defnGiven a group or monoid ''M'', the '''Day convolution''' is the tensor product in {$\mathbf{Fct}(M, \mathbf{Set})$}.<ref>http://ncatlab.org/nlab/show/Day+convolution</ref>}} Pakeistos 178180 eilutės iš
: that sends each object ''A'' to the constant functor with value ''A'' and each morphism į:
:{$\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$} that is ''f'' at each ''i''.}} Pakeistos 182183 eilutės iš
{{defn1=Given a category ''C'', a '''diagram (category theory)diagram''' in ''C'' is a functor į:
{{defn1=Given a category ''C'', a '''diagram (category theory)diagram''' in ''C'' is a functor {$f: I \to C$} from a small category ''I''.}} Pakeistos 203206 eilutės iš
{{defn1=The '''end (category theory)end''' of a functor : where : į:
{{defn1=The '''end (category theory)end''' of a functor {$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))$} Pakeista 215 eilutė iš:
# For each pair of objects ''X'', ''Y'' in ''D'', an object į:
# For each pair of objects ''X'', ''Y'' in ''D'', an object {$\operatorname{Map}_D(X, Y)$} in ''C'', called the '''mapping object''' from ''X'' to ''Y'', Pakeista 217 eilutė iš:
#: į:
#:{$\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 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 228229 eilutės iš
{{defn1=A morphism ''f'' is an '''epimorphism''' if į:
{{defn1=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 231232 eilutės iš
{{defn1=The '''equalizer (mathematics)equalizer''' of a pair of morphisms į:
{{defn1=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:
: į:
:{$\mathbf{Fct}(C, D) \to D, \,\, F \mapsto F(A).$} Pakeistos 255256 eilutės iš
{{defn1=The '''fundamental category functor''' į:
{{defn1=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 258259 eilutės iš
{{defn1=The '''fundamental groupoid''' į:
{{defn1=The '''fundamental groupoid''' {$\Pi_1 X$} of a Kan complex ''X'' is the category where an object is a 0simplex (vertex) {$\Delta^0 \to X$}, a morphism is a homotopy class of a 1simplex (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 264265 eilutės iš
{{defn1=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''' į:
{{defn1=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 267269 eilutės iš
{{defnno=11=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 {{defnno=21=Given a cardinal number π, a category is said to be πfiltrant if, for each category ''J'' whose set of morphisms has cardinal number strictly less than π, the set į:
{{defnno=11=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''.}} {{defnno=21=Given a cardinal number π, a category is said to be πfiltrant 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 277278 eilutės iš
{{defnThe '''forgetful functor''' is, roughly, a functor that loses some of data of the objects; for example, the functor į:
{{defnThe '''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 293296 eilutės iš
{{defn1=Given categories ''C'', ''D'', a '''functor''' ''F'' from ''C'' to ''D'' is a structurepreserving 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) : where į:
{{defn1=Given categories ''C'', ''D'', a '''functor''' ''F'' from ''C'' to ''D'' is a structurepreserving 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 308309 eilutės iš
{{defn1=In a category ''C'', a family of objects į:
{{defn1=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 317318 eilutės iš
{{defn1=Given a functor į:
{{defn1=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 334335 eilutės iš
{{defn1=The '''heart (category theory)heart''' of a '''tstructure''' ( į:
{{defn1=The '''heart (category theory)heart''' of a '''tstructure''' ({$D^{\ge 0}$}, {$D^{\le 0}$}) on a triangulated category is the intersection {$D^{\ge 0} \cap D^{\le 0}$}. It is an abelian category.}} Pakeista 352 eilutė iš:
{{defnno=11=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'', į:
{{defnno=11=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 357358 eilutės iš
{{defn1=The '''image of a morphismimage''' of a morphism ''f'': ''X'' → ''Y'' is the equalizer of į:
{{defn1=The '''image of a morphismimage''' of a morphism ''f'': ''X'' → ''Y'' is the equalizer of {$Y \rightrightarrows Y \sqcup_X Y$}.}} Pakeistos 360361 eilutės iš
{{defn1=A colimit (or inductive limit) in į:
{{defn1=A colimit (or inductive limit) in {$\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}.}} Pakeistos 364366 eilutės iš
*every map of simplicial sets where Δ<sup>''n''</sup> is the standard ''n''simplex and į:
*every map of simplicial sets {$f: \Lambda^n_i \to C$} extends to an ''n''simplex {$f: \Delta^n \to C$} where Δ<sup>''n''</sup> is the standard ''n''simplex and {$\Lambda^n_i$} is obtained from Δ<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 369370 eilutės iš
{{defnno=21=An object ''A'' in an ∞category ''C'' is initial if į:
{{defnno=21=An object ''A'' in an ∞category ''C'' is initial if {$\operatorname{Map}_C(A, B)$} is '''contractible spacecontractible''' for each object ''B'' in ''C''.}} Pakeistos 372373 eilutės iš
{{defn1=An object ''A'' in an abelian category is '''injective objectinjective''' if the functor į:
{{defn1=An object ''A'' in an abelian category is '''injective objectinjective''' if the functor {$\operatorname{Hom}(, A)$} is exact. It is the dual of a projective object.}} Pakeistos 375376 eilutės iš
{{defn1=Given a '''monoidal category''' (''C'', ⊗), the '''internal Hom''' is a functor į:
{{defn1=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)$}, the set of ''R''linear maps.}} Pakeistos 378379 eilutės iš
{{defn1=A morphism ''f'' is an '''inverse functioninverse''' to a morphism ''g'' if į:
{{defn1=A morphism ''f'' is an '''inverse functioninverse''' to a morphism ''g'' if {$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$} is defined and is equal to the identity morphism on the domain of ''g'', and similarly for a right inverse.}} Pakeistos 394398 eilutės iš
{{defnno=1Given a category ''C'', the left '''Kan extension''' functor along a functor : where the colimit runs over all objects {{defnno=2The right Kan extension functor is the right adjoint (if it exists) to į:
{{defnno=1Given a category ''C'', the left '''Kan extension''' functor along a functor {$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$} is called the left Kan extension of α along ''f''.<ref>http://www.math.harvard.edu/~lurie/282ynotes/LectureXIHomological.pdf</ref> One can show: :{$(f_! \alpha)(j) = \varinjlim_{f(i) \to j} \alpha(i)$} where the colimit runs over all objects {$f(i) \to j$} in the comma category.}} {{defnno=2The right Kan extension functor is the right adjoint (if it exists) to {$f^*$}.}} Pakeistos 412415 eilutės iš
{{defnno=1The '''limit (category theory)limit''' (or '''projective limit''') of a functor :: į:
{{defnno=1The '''limit (category theory)limit''' (or '''projective limit''') of a functor {$f: I^{\text{op}} \to \mathbf{Set}$} is ::{$\varprojlim_{i \in I} f(i) = \{ (x_ii) \in \prod_{i} f(i)  f(s)(x_j) = x_i \text{ for any } s: i \to j \}.$}}} {{defnno=2The limit {$\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)).$}}} {{defnno=3The '''colimit''' (or '''inductive limit''') {$\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$}; i.e., ''L'' is a final object of ''C''. Pakeistos 425428 eilutės iš
{{defn1=A '''monad (category theory)monad''' in a category ''X'' is a '''monoid object''' in the monoidal category of endofunctors of ''X'' with the monoidal structure given by composition. For example, given a group ''G'', define an endofunctor ''T'' on '''Set''' by : and also define the identity map ''η'' in the analogous fashion. Then (''T'', ''μ'', ''η'') constitutes a monad in '''Set'''. More substantially, an adjunction between functors į:
{{defn1=A '''monad (category theory)monad''' in a category ''X'' is a '''monoid object''' in the monoidal category of endofunctors of ''X'' with the monoidal structure given by composition. For example, given a group ''G'', define an endofunctor ''T'' on '''Set''' by {$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$}, the identity map ''η'' on ''T'' to be a unit of the adjunction and also defines ''μ'' using the adjunction.}} Pakeistos 434435 eilutės iš
{{defn1=A '''monoidal category''', also called a tensor category, is a category ''C'' equipped with (1) a '''bifunctor''' į:
{{defn1=A '''monoidal category''', also called a tensor category, is a category ''C'' equipped with (1) a '''bifunctor''' {$\otimes: C \times C \to C$}, (2) an identity object and (3) natural isomorphisms that make ⊗ associative and the identity object an identity for ⊗, subject to certain coherence conditions.}} Pakeistos 440441 eilutės iš
{{defn1=A morphism ''f'' is a '''monomorphism''' (also called monic) if į:
{{defn1=A morphism ''f'' is a '''monomorphism''' (also called monic) if {$g=h$} whenever {$f\circ g=f\circ h$}; e.g., an '''Injective functioninjection''' in ''''''Category of setsSet''''''. In other words, ''f'' is the dual of an epimorphism.}} Pakeistos 462463 eilutės iš
: satisfying the condition: for each morphism ''f'': ''x'' → ''y'' in ''C'', į:
:{$\{ \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$} to <sup>*</sup>.}} Pakeistos 468469 eilutės iš
{{defn1=The '''nerve functor''' ''N'' is the functor from '''Cat''' to ''s'''''Set''' given by į:
{{defn1=The '''nerve functor''' ''N'' is the functor from '''Cat''' to ''s'''''Set''' given by {$N(C)_n = \operatorname{Hom}_{\mathbf{Cat}}([n], C)$}. For example, if {$\varphi$} is a functor in {$N(C)_2$} (called a 2simplex), let {$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 508510 eilutės iš
{{defnno=1The '''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 {{defnno=2The '''product of categoriesproduct of a family of categories''' ''C''<sub>''i''</sub>'s indexed by a set ''I'' is the category denoted by į:
{{defnno=1The '''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.}} {{defnno=2The '''product of categoriesproduct 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 homsets are {$\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)$}; the morphisms are composed componentwise. It is the dual of the disjoint union.}} Pakeistos 512513 eilutės iš
{{defn1=Given categories ''C'' and ''D'', a '''profunctor''' (or a distributor) from ''C'' to ''D'' is a functor of the form į:
{{defn1=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 515516 eilutės iš
{{defn1=An object ''A'' in an abelian category is '''projective objectprojective''' if the functor į:
{{defn1=An object ''A'' in an abelian category is '''projective objectprojective''' if the functor {$\operatorname{Hom}(A, )$} is exact. It is the dual of an injective object.}} Pakeistos 534535 eilutės iš
{{defn1=A setvalued contravariant functor ''F'' on a category ''C'' is said to be '''representable functorrepresentable''' if it belongs to the essential image of the '''Yoneda embedding''' į:
{{defn1=A setvalued contravariant functor ''F'' on a category ''C'' is said to be '''representable functorrepresentable''' 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:
{{defn1=Given a ''k''linear category ''C'' over a field ''k'', a '''Serre functor''' į:
{{defn1=Given a ''k''linear category ''C'' over a field ''k'', a '''Serre functor''' {$f: C \to C$} is an autoequivalence such that {$\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*$} for any objects ''A'', ''B''.}} Pakeistos 568569 eilutės iš
{{defn1=A '''simplicial object''' in a category ''C'' is roughly a sequence of objects į:
{{defn1=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 571572 eilutės iš
{{defn1=A '''simplicial set''' is a contravariant functor from Δ to '''Set''', where Δ is the '''simplex category''', a category whose objects are the sets [''n''] = { 0, 1, …, ''n'' } and whose morphisms are orderpreserving functions. One writes į:
{{defn1=A '''simplicial set''' is a contravariant functor from Δ to '''Set''', where Δ is the '''simplex category''', a category whose objects are the sets [''n''] = { 0, 1, …, ''n'' } and whose morphisms are orderpreserving 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 580581 eilutės iš
{{defn1=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 į:
{{defn1=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 593594 eilutės iš
{{defnA morphism ''f'' in a category admitting finite limits and finite colimits is '''strict morphismstrict''' if the natural morphism į:
{{defnA morphism ''f'' in a category admitting finite limits and finite colimits is '''strict morphismstrict''' if the natural morphism {$\operatorname{Coim}(f) \to \operatorname{Im}(f)$} is an isomorphism.}} Pakeistos 627628 eilutės iš
{{defnThe '''Tannakian duality''' states that, in an appropriate setup, to give a morphism į:
{{defnThe '''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 geometryderived 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 arxivlast=Bhattfirst=Bhargavdate=20140429title=Algebraization and Tannaka dualityeprint=1404.7483class=math.AG}}</ref>}} Pakeistos 636638 eilutės iš
{{defnGiven a monoidal category ''B'', the '''tensor product of functors''' : į:
{{defnGiven 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 641642 eilutės iš
{{defnno=2An object ''A'' in an ∞category ''C'' is terminal if į:
{{defnno=2An object ''A'' in an ∞category ''C'' is terminal if {$\operatorname{Map}_C(B, A)$} is '''contractible spacecontractible''' for every object ''B'' in ''C''.}} Pakeistos 657660 eilutės iš
{{defnno=1Given a functor {{defnno=2Stated more explicitly, given ''f'' as above, a morphism : is bijective. In particular, if į:
{{defnno=1Given 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''.}} {{defnno=2Stated 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:
: į:
:{$F(X) \simeq \operatorname{Nat}(\operatorname{Hom}_C(, X), F)$} Pakeista 682 eilutė iš:
: į:
:{$y: C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(, X)$} Pakeista 684 eilutė iš:
{{defnno=21=If į:
{{defnno=21=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 
Pakeistos 708717 eilutės iš
series=Lecture Notes in Mathematics volume=269  year = 1972  publisher = '''Springer Science+Business MediaSpringerVerlag'''  location = Berlin; New York  language = French  pages = xix+525  nopp = true doi= 10.1007/BFb0081551 isbn= 9783540058960 į:
title = Séminaire de Géométrie Algébrique du Bois Marie  196364  Théorie des topos et cohomologie étale des schémas  (SGA 4)  vol. 1 2019 vasario 10 d., 09:12
atliko 
Pakeistos 38 eilutės iš
This is a glossary of properties and concepts in * Especially for higher categories, the concepts from algebraic topology are also used in the category theory. For that see also į:
This is a glossary of properties and concepts in '''category theory''' in '''mathematics'''. *'''Notes on foundations''': In many expositions (e.g., Vistoli), the settheoretic 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 settheoretic 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 1112 eilutės iš
*'''Cat''', the *'''Fct'''(''C'', ''D''), the į:
*'''Cat''', the '''category of categoriescategory 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 į:
*''s'''''Set''', the category of '''simplicial set'''s. Pakeistos 1619 eilutės iš
*By an *The number {{Compact ToCshort1sym=yesx= į:
*By an '''∞category''', we mean a '''quasicategory''', the most popular model, unless other models are being discussed. *The number '''zero''' 0 is a natural number. {{Compact ToCshort1sym=yesx='''#XYZXYZ'''y=z=seealso=yesrefs=yes}} Pakeistos 2324 eilutės iš
{{defn1=A category is į:
{{defn1=A category is '''abelian categoryabelian''' if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.}} Pakeistos 2628 eilutės iš
{{defnno=1Given a {{defnno=2Given a į:
{{defnno=1Given a '''cardinal number''' κ, an object ''X'' in a category is '''accessible objectκaccessible''' (or κcompact or κpresentable) if <math>\operatorname{Hom}(X, )</math> commutes with κfiltered colimits.}} {{defnno=2Given a '''regular cardinal''' κ, a category is '''accessible categoryκaccessible''' if it has κfiltered colimits and there exists a small set ''S'' of κcompact objects that generates the category under colimits, meaning every object can be written as a colimit of diagrams of objects in ''S''.}} Pakeistos 3031 eilutės iš
{{defn1=A category is į:
{{defn1=A category is '''additive categoryadditive''' if it is preadditive (to be precise, has some preadditive 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:
{{defn1=An į:
{{defn1=An '''adjoint functoradjunction''' (also called an adjoint pair) is a pair of functors ''F'': ''C'' → ''D'', ''G'': ''D'' → ''C'' such that there is a "natural" bijection Pakeistos 3839 eilutės iš
{{defn1=Given a monad ''T'' in a category ''X'', an į:
{{defn1=Given a monad ''T'' in a category ''X'', an '''algebra for a monadalgebra 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 actionaction''' of ''G''.}} Pakeistos 5051 eilutės iš
{{defn1= į:
{{defn1='''Beck's monadicity theoremBeck's theorem''' characterizes the category of '''algebra for a monadalgebras for a given monad'''.}} Pakeistos 5354 eilutės iš
{{defn1=A į:
{{defn1=A '''bicategory''' is a model of a weak '''2category'''.}} Pakeistos 5657 eilutės iš
{{defn1=A į:
{{defn1=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 5960 eilutės iš
{{defn1=A į:
{{defn1=A '''bimorphism''' is a morphism that is both an epimorphism and a monomorphism.}} Pakeistos 6263 eilutės iš
{{defn1=See į:
{{defn1=See '''Bousfield localization'''.}} Pakeistos 6970 eilutės iš
{{defn1=The į:
{{defn1=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 7273 eilutės iš
{{defn1=A category is į:
{{defn1=A category is '''Cartesian closed categorycartesian closed''' if it has a terminal object and that any two objects have a product and exponential.}} Pakeistos 7880 eilutės iš
{{defnno=11=Given a functor π: ''C'' → ''D'' (e.g., a {{defnno=21=Given a functor π: ''C'' → ''D'' (e.g., a į:
{{defnno=11=Given a functor π: ''C'' → ''D'' (e.g., a '''prestack''' over schemes), a morphism ''f'': ''x'' → ''y'' in ''C'' is '''cartesian morphismπcartesian''' 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''.}} {{defnno=21=Given a functor π: ''C'' → ''D'' (e.g., a '''prestack''' over rings), a morphism ''f'': ''x'' → ''y'' in ''C'' is '''cartesian morphismπcoCartesian''' 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 πcartesian morphism.)}} Pakeistos 8586 eilutės iš
{{defn1= į:
{{defn1='''Categorical logic''' is an approach to '''mathematical logic''' that uses category theory.}} Pakeistos 8889 eilutės iš
{{defn1= į:
{{defn1='''Categorification''' is a process of replacing sets and settheoretic concepts with categories and categorytheoretic concepts in some nontrivial way to capture categoric flavors. Decategorification is the reverse of categorification.}} Pakeista 91 eilutė iš:
{{defn1=A į:
{{defn1=A '''category (mathematics)category''' consists of the following data Pakeistos 99100 eilutės iš
For example, a 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 102103 eilutės iš
{{defn1=The į:
{{defn1=The '''category of categoriescategory 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 105106 eilutės iš
{{defn1=The į:
{{defn1=The '''classifying space of a category''' ''C'' is the geometric realization of the nerve of ''C''.}} Pakeistos 108109 eilutės iš
{{defn1=Often used synonymous with op; for example, a į:
{{defn1=Often used synonymous with op; for example, a '''colimit''' refers to an oplimit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an opfibration is not the same thing as a '''cofibration'''.}} Pakeista 111 eilutė iš:
{{defn1=The coend of a functor <math>F: C^{\text{op}} \times C \to X</math> is the dual of the į:
{{defn1=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 į:
For example, if ''R'' is a ring, ''M'' a right ''R''module and ''N'' a left ''R''module, then the '''tensor product of modulestensor product''' of ''M'' and ''N'' is Pakeistos 118119 eilutės iš
{{defn1=The į:
{{defn1=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 121122 eilutės iš
{{defn1=The į:
{{defn1=The '''coimage''' of a morphism ''f'': ''X'' → ''Y'' is the coequalizer of <math>X \times_Y X \rightrightarrows X</math>.}} Pakeistos 124125 eilutės iš
{{defn1=Another term for į:
{{defn1=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:
{{defn1=Given functors <math>f: C \to B, g: D \to B</math>, the į:
{{defn1=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 131132 eilutės iš
{{defn1=A į:
{{defn1=A '''comonad''' in a category ''X'' is a '''comonid''' in the monoidal category of endofunctors of ''X''.}} Pakeistos 134135 eilutės iš
{{defn1=Probably synonymous with į:
{{defn1=Probably synonymous with '''#accessible'''.}} Pakeistos 137138 eilutės iš
{{defn1=A category is į:
{{defn1=A category is '''complete categorycomplete''' if all small limits exist.}} Pakeistos 145146 eilutės iš
{{defn1=A į:
{{defn1=A '''concrete category''' ''C'' is a category such that there is a faithful functor from ''C'' to ''''''Category of setsSet''''''; e.g., ''''''category of vector spacesVec'''''', ''''''category of groupsGrp'''''' and ''''''category of topological spacesTop''''''.}} Pakeista 148 eilutė iš:
{{defn1=A į:
{{defn1=A '''cone (category theory)cone''' is a way to express the '''universal property''' of a colimit (or dually a limit). One can show<ref>{{harvnbKashiwaraSchapira2006loc=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 153154 eilutės iš
{{defn1=A category is į:
{{defn1=A category is '''connected categoryconnected''' 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 156157 eilutės iš
{{defn1=A į:
{{defn1=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 159160 eilutės iš
{{defn1=A functor is į:
{{defn1=A functor is '''constant functorconstant''' 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:
{{defn1=A į:
{{defn1=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 167168 eilutės iš
{{defn1=The į:
{{defn1=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 groupsGrp'''''' is a '''free product'''.}} Pakeistos 170171 eilutės iš
{{defn1=The į:
{{defn1=The '''core (category theory)core''' of a category is the maximal groupoid contained in the category.}} Pakeistos 177178 eilutės iš
{{defnGiven a group or monoid ''M'', the į:
{{defnGiven 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 180181 eilutės iš
{{defn1=The į:
{{defn1=The '''density theorem (category theory)density theorem''' states that every presheaf (a setvalued 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:
{{defn1=Given categories ''I'', ''C'', the į:
{{defn1=Given categories ''I'', ''C'', the '''diagonal functor''' is the functor Pakeistos 188189 eilutės iš
{{defn1=Given a category ''C'', a į:
{{defn1=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 191192 eilutės iš
{{defn1=A į:
{{defn1=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 194195 eilutės iš
{{defn1=A category is į:
{{defn1=A category is '''discrete categorydiscrete''' 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:
{{defn1=A į:
{{defn1=A '''Dwyer–Kan equivalence''' is a generalization of an equivalence of categories to the simplicial context.<ref>{{cite arxivlast=Hinichfirst=V.date=20131117title=DwyerKan localization revisitedeprint=1311.4128class=math.QA}}</ref>}} Pakeistos 206207 eilutės iš
{{defn1=Another name for the category of į:
{{defn1=Another name for the category of '''algebra for a monadalgebras for a given monad'''.}} Pakeista 209 eilutė iš:
{{defn1=The į:
{{defn1=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 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:
{{defn1=Given a monoidal category (''C'', ⊗, 1), a į:
{{defn1=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 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 231232 eilutės iš
{{defnThe į:
{{defnThe '''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 234235 eilutės iš
{{defn1=A morphism ''f'' is an į:
{{defn1=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 237238 eilutės iš
{{defn1=The į:
{{defn1=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:
{{defnno=1A functor is an į:
{{defnno=1A functor is an '''equivalence of categoriesequivalence''' if it is faithful, full and essentially surjective.}} Pakeistos 244245 eilutės iš
{{defn1=A category is equivalent to another category if there is an į:
{{defn1=A category is equivalent to another category if there is an '''equivalence of categoriesequivalence''' between them.}} Pakeistos 247248 eilutės iš
{{defn1=A functor ''F'' is called į:
{{defn1=A functor ''F'' is called '''essentially surjective''' (or isomorphismdense) if for every object ''B'' there exists an object ''A'' such that ''F''(''A'') is isomorphic to ''B''.}} Pakeista 250 eilutė iš:
{{defn1=Given categories ''C'', ''D'' and an object ''A'' in ''C'', the į:
{{defn1=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 į:
For example, the '''Eilenberg–Steenrod axioms''' give an instance when the functor is an equivalence.}} Pakeistos 258259 eilutės iš
{{defn1=A functor is į:
{{defn1=A functor is '''faithful functorfaithful''' if it is injective when restricted to each '''homset'''.}} Pakeistos 261262 eilutės iš
{{defn1=The į:
{{defn1=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 264265 eilutės iš
{{defn1=The į:
{{defn1=The '''fundamental groupoid''' <math>\Pi_1 X</math> of a Kan complex ''X'' is the category where an object is a 0simplex (vertex) <math>\Delta^0 \to X</math>, a morphism is a homotopy class of a 1simplex (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 267268 eilutės iš
{{defn1=A functor π: ''C'' → ''D'' is said to exhibit ''C'' as a į:
{{defn1=A functor π: ''C'' → ''D'' is said to exhibit ''C'' as a '''fibered categorycategory fibered over''' ''D'' if, for each morphism ''g'': ''x'' → π(''y'') in ''D'', there exists a πcartesian 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 270271 eilutės iš
{{defn1=Given a category ''C'' and a set ''I'', the į:
{{defn1=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:
{{defnno=11=A į:
{{defnno=11=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 277278 eilutės iš
{{defn1=A į:
{{defn1=A '''finitary monad''' or an algebraic monad is a monad on '''Set''' whose underlying endofunctor commutes with filtered colimits.}} Pakeistos 283284 eilutės iš
{{defnThe į:
{{defnThe '''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 286287 eilutės iš
{{defn1=A į:
{{defn1=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 modulefree ''R''module''' generated by ''X'' is a free functor (whence the name).}} Pakeistos 289290 eilutės iš
{{defn1=A į:
{{defn1=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 292293 eilutės iš
{{defn1=See į:
{{defn1=See '''Fukaya category'''.}} Pakeistos 295297 eilutės iš
{{defnno=11=A functor is {{defnno=21=A category ''A'' is a į:
{{defnno=11=A functor is '''full functorfull''' if it is surjective when restricted to each '''homset'''.}} {{defnno=21=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:
{{defn1=Given categories ''C'', ''D'', a į:
{{defn1=Given categories ''C'', ''D'', a '''functor''' ''F'' from ''C'' to ''D'' is a structurepreserving 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 301302 eilutės iš
where <math>\mathfrak{P}(S)</math> is the 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 304305 eilutės iš
{{defn1=The į:
{{defn1=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 311312 eilutės iš
{{defn1=The į:
{{defn1=The '''Gabriel–Popescu theorem''' says an abelian category is a '''Serre quotient categoryquotient''' of the category of modules.}} Pakeistos 314315 eilutės iš
{{defn1=In a category ''C'', a family of objects <math>G_i, i \in I</math> is a į:
{{defn1=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 317318 eilutės iš
{{defn1=A categorytheoretic generalization of į:
{{defn1=A categorytheoretic generalization of '''Galois theory'''; see '''Grothendieck's Galois theory'''.}} Pakeistos 320321 eilutės iš
{{defn1=A į:
{{defn1=A '''Grothendieck category''' is a certain wellbehaved kind of an abelian category.}} Pakeistos 323324 eilutės iš
{{defn1=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 į:
{{defn1=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 326327 eilutės iš
{{defn1=A į:
{{defn1=A '''fibered category'''.}} Pakeistos 329331 eilutės iš
{{defnno=11=A category is called a {{defnno=21=An ∞category is called an į:
{{defnno=11=A category is called a '''groupoid''' if every morphism in it is an isomorphism.}} {{defnno=21=An ∞category is called an '''∞groupoid''' if every morphism in it is an equivalence (or equivalently if it is a '''Kan complex'''.)}} Pakeistos 337338 eilutės iš
{{defn1=See į:
{{defn1=See '''Ringel–Hall algebra'''.}} Pakeistos 340341 eilutės iš
{{defn1=The į:
{{defn1=The '''heart (category theory)heart''' of a '''tstructure''' (<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 343344 eilutės iš
{{defn1= į:
{{defn1='''Higher category theory''' is a subfield of category theory that concerns the study of '''ncategory''n''categories''' and '''∞categories'''.}} Pakeistos 346347 eilutės iš
{{defn1=The į:
{{defn1=The '''homological dimension''' of an abelian category with enough injectives is the least nonnegative 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 modulesMod<sub>''R''</sub>''' with a principal ideal domain ''R'' is at most one.}} Pakeistos 349350 eilutės iš
{{defn1=See<! for now > į:
{{defn1=See<! for now > '''homotopy category'''. It is closely related to a '''localization of a category'''.}} Pakeista 352 eilutė iš:
{{defn1=The į:
{{defn1=The '''homotopy hypothesis''' states an '''∞groupoid''' is a space (less equivocally, an ''n''groupoid can be used as a homotopy ''n''type.)}} Pakeistos 358361 eilutės iš
{{defnno=11=The {{defnno=2The [[identity functor]] on a category {{defnno=3Given a functor ''F'': ''C'' → ''D'', the į:
{{defnno=11=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>.}} {{defnno=2The '''identity functor''' on a category ''C'' is a functor from ''C'' to ''C'' that sends objects and morphisms to themselves.}} {{defnno=3Given 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 363364 eilutės iš
{{defn1=The į:
{{defn1=The '''image of a morphismimage''' of a morphism ''f'': ''X'' → ''Y'' is the equalizer of <math>Y \rightrightarrows Y \sqcup_X Y</math>.}} Pakeista 369 eilutė iš:
{{defn1=An į:
{{defn1=An '''∞category''' ''C'' is a '''simplicial set''' satisfying the following condition: for each 0 < ''i'' < ''n'', Pakeistos 371372 eilutės iš
where Δ<sup>''n''</sup> is the standard ''n''simplex and <math>\Lambda^n_i</math> is obtained from Δ<sup>''n''</sup> by removing the ''i''th face and the interior (see į:
where Δ<sup>''n''</sup> is the standard ''n''simplex and <math>\Lambda^n_i</math> is obtained from Δ<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 374376 eilutės iš
{{defnno=11=An object ''A'' is {{defn į:
{{defnno=11=An object ''A'' is '''initial objectinitial''' if there is exactly one morphism from ''A'' to each object; e.g., '''empty set''' in ''''''Category of setsSet''''''.}} {{defnno=21=An object ''A'' in an ∞category ''C'' is initial if <math>\operatorname{Map}_C(A, B)</math> is '''contractible spacecontractible''' for each object ''B'' in ''C''.}} Pakeistos 378379 eilutės iš
{{defn1=An object ''A'' in an abelian category is į:
{{defn1=An object ''A'' in an abelian category is '''injective objectinjective''' if the functor <math>\operatorname{Hom}(, A)</math> is exact. It is the dual of a projective object.}} Pakeistos 381382 eilutės iš
{{defn1=Given a į:
{{defn1=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 384385 eilutės iš
{{defn1=A morphism ''f'' is an į:
{{defn1=A morphism ''f'' is an '''inverse functioninverse''' 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:
{{defnno=11=An object is į:
{{defnno=11=An object is '''isomorphic''' to another object if there is an isomorphism between them.}} Pakeista 391 eilutė iš:
{{defn1=A morphism ''f'' is an į:
{{defn1=A morphism ''f'' is an '''isomorphism''' if there exists an ''inverse'' of ''f''.}} Pakeistos 397398 eilutės iš
{{defn1=A į:
{{defn1=A '''Kan complex''' is a '''fibrant object''' in the category of simplicial sets.}} Pakeista 400 eilutė iš:
{{defnno=1Given a category ''C'', the left į:
{{defnno=1Given 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/LectureXIHomological.pdf</ref> One can show: Pakeista 406 eilutė iš:
{{defn1=Given a monad ''T'', the į:
{{defn1=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 412413 eilutės iš
{{defn1=The term " į:
{{defn1=The term "'''lax functor'''" is essentially synonymous with "'''pseudofunctor'''".}} Pakeistos 415416 eilutės iš
{{defn1=An object in an abelian category is said to have <span id="finite length"></span>finite length if it has a į:
{{defn1=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>{{harvnbKashiwaraSchapira2006loc=exercise 8.20}}</ref>}} Pakeista 418 eilutė iš:
{{defnno=1The į:
{{defnno=1The '''limit (category theory)limit''' (or '''projective limit''') of a functor <math>f: I^{\text{op}} \to \mathbf{Set}</math> is Pakeista 421 eilutė iš:
{{defnno=3The į:
{{defnno=3The '''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:
{{defn1=See į:
{{defn1=See '''localization of a category'''.}} Pakeista 431 eilutė iš:
{{defn1=A į:
{{defn1=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 436438 eilutės iš
{{defnno=11=An adjunction is said to be {{defnno=21=A functor is said to be į:
{{defnno=11=An adjunction is said to be '''monadic adjunctionmonadic''' if it comes from the monad that it determines by means of the '''Eilenberg–Moore category''' (the category of algebras for the monad).}} {{defnno=21=A functor is said to be '''monadic functormonadic''' if it is a constituent of a monadic adjunction.}} Pakeistos 440441 eilutės iš
{{defn1=A į:
{{defn1=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 443444 eilutės iš
{{defn1=A į:
{{defn1=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 446447 eilutės iš
{{defn1=A morphism ''f'' is a į:
{{defn1=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 functioninjection''' in ''''''Category of setsSet''''''. In other words, ''f'' is the dual of an epimorphism.}} Pakeista 449 eilutė iš:
{{defn1=A į:
{{defn1=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 462465 eilutės iš
}}{{defnno=11=A {{defnno=21=The notion of a {{defnno=31=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 į:
}}{{defnno=11=A '''strict ncategorystrict ''n''category''' is defined inductively: a strict 0category 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 1category is an ordinary category.}} {{defnno=21=The notion of a '''weak ncategoryweak ''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.}} {{defnno=31=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 '''quasicategory''') in the beginning, then a weak ''n''category can be defined as a type of a truncated ∞category.}} Pakeista 467 eilutė iš:
{{defnno=1A natural transformation is, roughly, a map between functors. Precisely, given a pair of functors ''F'', ''G'' from a category ''C'' to category ''D'', a į:
{{defnno=1A 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 469472 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 {{defnno=2A 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>.}} {{defnno=2A '''natural isomorphism''' is a natural transformation that is an isomorphism (i.e., admits the inverse).}} '''Image:Nerve2simplex.pngthumbrightThe composition is encoded as a 2simplex.''' Pakeistos 474475 eilutės iš
{{defn1=The į:
{{defn1=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 2simplex), 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:
{{defn1=A category is į:
{{defn1=A category is '''normal categorynormal''' if every monic is normal.{{citation neededdate=October 2015}}}} Pakeistos 484485 eilutės iš
{{defnno=21=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 į:
{{defnno=21=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 '''Γobject''' 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 487488 eilutės iš
{{defn1=A functor π:''C'' → ''D'' is an į:
{{defn1=A functor π:''C'' → ''D'' is an '''opfibration''' if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one πcoCartesian 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:
{{defn1=The į:
{{defn1=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 496497 eilutės iš
{{defnSometimes synonymous with "compact". See į:
{{defnSometimes synonymous with "compact". See '''perfect complex'''.}} Pakeistos 502503 eilutės iš
{{defn1=A functor from the category of finitedimensional vector spaces to itself is called a į:
{{defn1=A functor from the category of finitedimensional 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 505506 eilutės iš
{{defn1=A category is į:
{{defn1=A category is '''preadditive categorypreadditive''' if it is '''enriched categoryenriched''' over the '''monoidal category''' of '''abelian group'''s. More generally, it is '''preadditive category#Rlinear categories''R''linear''' if it is enriched over the monoidal category of '''module (mathematics)''R''modules''', for ''R'' a '''commutative ring'''.}} Pakeistos 508509 eilutės iš
{{defnGiven a į:
{{defnGiven a '''regular cardinal''' κ, a category is '''presentabl categoryκpresentable''' if it admits all small colimits and is '''#accessibleκaccessible'''. A category is presentable if it is κpresentable for some regular cardinal κ (hence presentable for any larger cardinal). '''Note''': Some authors call a presentable category a '''locally presentable category'''.}} Pakeistos 511512 eilutės iš
{{defn1=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 į:
{{defn1=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 topologytopology''' on ''C'', if any, tells which presheaf is a sheaf (with respect to that topology).}} Pakeistos 514516 eilutės iš
{{defnno=1The {{defnno=2The į:
{{defnno=1The '''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.}} {{defnno=2The '''product of categoriesproduct 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 homsets are <math>\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)</math>; the morphisms are composed componentwise. It is the dual of the disjoint union.}} Pakeistos 518519 eilutės iš
{{defn1=Given categories ''C'' and ''D'', a į:
{{defn1=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 521522 eilutės iš
{{defn1=An object ''A'' in an abelian category is į:
{{defn1=An object ''A'' in an abelian category is '''projective objectprojective''' if the functor <math>\operatorname{Hom}(A, )</math> is exact. It is the dual of an injective object.}} Pakeista 524 eilutė iš:
{{defn1=A į:
{{defn1=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:
{{defn1= į:
{{defn1='''Quillen’s theorem A''' provides a criterion for a functor to be a weak equivalence.}} Pakeistos 540541 eilutės iš
{{defn1=A setvalued contravariant functor ''F'' on a category ''C'' is said to be į:
{{defn1=A setvalued contravariant functor ''F'' on a category ''C'' is said to be '''representable functorrepresentable''' 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:
{{defn1= į:
{{defn1='''File:Section retract.svg150pxthumb''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 549550 eilutės iš
{{defn1=A morphism is a į:
{{defn1=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 552553 eilutės iš
{{defn1= į:
{{defn1='''Segal space'''s were certain simplicial spaces, introduced as models for '''(infinity,1)category(∞, 1)categories'''.}} Pakeistos 555556 eilutės iš
{{defn1=An abelian category is į:
{{defn1=An abelian category is '''semisimple categorysemisimple''' if every short exact sequence splits. For example, a ring is '''semisimple ringsemisimple''' if and only if the category of modules over it is semisimple.}} Pakeista 558 eilutė iš:
{{defn1=Given a ''k''linear category ''C'' over a field ''k'', a į:
{{defn1=Given a ''k''linear category ''C'' over a field ''k'', a '''Serre functor''' <math>f: C \to C</math> is an autoequivalence such that <math>\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*</math> for any objects ''A'', ''B''.}} Pakeista 560 eilutė iš:
{{defn1=In a category, a į:
{{defn1=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 562563 eilutės iš
{{defn1=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 į:
{{defn1=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 565566 eilutės iš
{{defn1=The į:
{{defn1=The '''simplex category''' Δ 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 orderpreserving function.}} Pakeistos 571572 eilutės iš
{{defn1= į:
{{defn1='''Simplicial localization''' is a method of localizing a category.}} Pakeistos 574575 eilutės iš
{{defn1=A į:
{{defn1=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 577578 eilutės iš
{{defn1=A į:
{{defn1=A '''simplicial set''' is a contravariant functor from Δ to '''Set''', where Δ is the '''simplex category''', a category whose objects are the sets [''n''] = { 0, 1, …, ''n'' } and whose morphisms are orderpreserving 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 580581 eilutės iš
{{defn1=A category equipped with a į:
{{defn1=A category equipped with a '''Grothendieck topology'''.}} Pakeistos 583584 eilutės iš
{{defn1=A category is į:
{{defn1=A category is '''Skeleton (category theory)skeletal''' if isomorphic objects are necessarily identical.}} Pakeistos 586587 eilutės iš
{{defn1=Given a category ''C'' and an object ''A'' in it, the į:
{{defn1=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 589591 eilutės iš
{{defnno=11=A {{defnno=2An object in a category is said to be į:
{{defnno=11=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.unibremen.de/acc/ format=PDF year=2004 publisher= Wiley & Sons location=New York isbn=0471609226 page=40}}</ref> (NB: some authors use the term "quasicategory" with a different meaning.<ref>{{cite journaldoi=10.1016/S00224049(02)001354last=Joyalfirst=A.title=Quasicategories and Kan complexesjournal=Journal of Pure and Applied Algebravolume=175year=2002issue=1–3pages=207–222ref=harv}}</ref>)}} {{defnno=2An object in a category is said to be '''small objectsmall''' if it is κcompact 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 593594 eilutės iš
{{defn1=A į:
{{defn1=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 596597 eilutės iš
{{defn1=An ∞category is į:
{{defn1=An ∞category is '''stable ∞categorystable''' 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 599600 eilutės iš
{{defnA morphism ''f'' in a category admitting finite limits and finite colimits is į:
{{defnA morphism ''f'' in a category admitting finite limits and finite colimits is '''strict morphismstrict''' if the natural morphism <math>\operatorname{Coim}(f) \to \operatorname{Im}(f)</math> is an isomorphism.}} Pakeistos 602603 eilutės iš
{{defnA strict 0category is a set and for any integer ''n'' > 0, a į:
{{defnA strict 0category is a set and for any integer ''n'' > 0, a '''strict ncategorystrict ''n''category''' is a category enriched over strict (''n''1)categories. For example, a strict 1category is an ordinary category. '''Note''': the term "''n''category" typically refers to "'''weak ncategoryweak ''n''category'''"; not strict one.}} Pakeistos 605606 eilutės iš
{{defn1=A topology on a category is į:
{{defn1=A topology on a category is '''subcanonical''' if every representable contravariant functor on ''C'' is a sheaf with respect to that topology.<ref>{{harvnbVistoli2004loc=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 608609 eilutės iš
{{defn1=A category ''A'' is a į:
{{defn1=A category ''A'' is a '''subcategory''' of a category ''B'' if there is an inclusion functor from ''A'' to ''B''.}} Pakeistos 611612 eilutės iš
{{defn1=Given an object ''A'' in a category, a į:
{{defn1=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 614615 eilutės iš
{{defn1=A į:
{{defn1=A '''subquotient''' is a quotient of a subobject.}} Pakeistos 617618 eilutės iš
{{defn1=A į:
{{defn1=A '''subterminal object''' is an object ''X'' such that every object has at most one morphism into ''X''.}} Pakeistos 620621 eilutės iš
{{defn1=A į:
{{defn1=A '''symmetric monoidal category''' is a '''monoidal category''' (i.e., a category with ⊗) that has maximally symmetric braiding.}} Pakeistos 623624 eilutės iš
{{defn1=A į:
{{defn1=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 630631 eilutės iš
{{defn1=A į:
{{defn1=A '''tstructure''' is an additional structure on a '''triangulated category''' (more generally '''stable ∞category''') that axiomatizes the notions of complexes whose cohomology concentrated in nonnegative degrees or nonpositive degrees.}} Pakeistos 633634 eilutės iš
{{defnThe į:
{{defnThe '''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 geometryderived 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 arxivlast=Bhattfirst=Bhargavdate=20140429title=Algebraization and Tannaka dualityeprint=1404.7483class=math.AG}}</ref>}} Pakeistos 636637 eilutės iš
{{defn1=Usually synonymous with į:
{{defn1=Usually synonymous with '''monoidal category''' (though some authors distinguish between the two concepts.)}} Pakeistos 639640 eilutės iš
{{defn1=A į:
{{defn1=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 646648 eilutės iš
{{defnno=1An object ''A'' is {{defnno=2An object ''A'' in an ∞category ''C'' is terminal if <math>\operatorname{Map}_C(B, A)</math> is į:
{{defnno=1An object ''A'' is '''terminal objectterminal''' (also called final) if there is exactly one morphism from each object to ''A''; e.g., '''singleton (mathematics)singleton'''s in ''''''Category of setsSet''''''. It is the dual of an '''initial object'''.}} {{defnno=2An object ''A'' in an ∞category ''C'' is terminal if <math>\operatorname{Map}_C(B, A)</math> is '''contractible spacecontractible''' for every object ''B'' in ''C''.}} Pakeistos 650651 eilutės iš
{{defn1=A full subcategory of an abelian category is į:
{{defn1=A full subcategory of an abelian category is '''thick subcategorythick''' if it is closed under extensions.}} Pakeistos 653654 eilutės iš
{{defn1=A į:
{{defn1=A '''thin categorythin''' is a category where there is at most one morphism between any pair of objects.}} Pakeistos 656657 eilutės iš
{{defn1=A į:
{{defn1=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:
į:
[+U+] Pakeista 663 eilutė iš:
{{defnno=1Given a functor <math>f: C \to D</math> and an object ''X'' in ''D'', a į:
{{defnno=1Given 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 672673 eilutės iš
{{defn1=A į:
{{defn1=A '''Waldhausen category''' is, roughly, a category with families of cofibrations and weak equivalences.}} Pakeista 675 eilutė iš:
{{defn1=A category is wellpowered if for each object there is only a set of pairwise nonisomorphic į:
{{defn1=A category is wellpowered if for each object there is only a set of pairwise nonisomorphic '''subobject'''s.}} Pakeista 683 eilutė iš:
author= į:
author='''Barry Mazur''' Pakeista 685 eilutė iš:
width=33%}}The į:
width=33%}}The '''Yoneda lemma''' says: for each setvalued contravariant functor ''F'' on ''C'' and an object ''X'' in ''C'', there is a natural bijection Pakeista 690 eilutė iš:
{{defnno=21=If <math>F: C \to D</math> is a functor and ''y'' is the Yoneda embedding of ''C'', then the į:
{{defnno=21=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:
į:
[+Z+] Pakeista 696 eilutė iš:
{{defn1=A į:
{{defn1=A '''zero object''' is an object that is both initial and terminal, such as a '''trivial group''' in ''''''Category of groupsGrp''''''.}} Pakeista 707 eilutė iš:
editor= į:
editor='''Alexandre Grothendieck''' editor2='''JeanLouis Verdier''' Pakeista 711 eilutė iš:
 publisher = į:
 publisher = '''Springer Science+Business MediaSpringerVerlag''' Pakeistos 728732 eilutės iš
* * *Lurie, J., '' * {{cite book  editor1last=Pedicchio  editor1first=Maria Cristina  editor2last=Tholen  editor2first=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= į:
*'''André JoyalA. Joyal''', [http://mat.uab.cat/~kock/crm/hocat/advancedcourse/Quadern452.pdf The theory of quasicategories II] (Volume I is missing??) *'''Jacob LurieLurie, 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='''SpringerVerlag'''  year=1998  isbn=0387984038  zbl=0906.18001 ref=harv}} * {{cite book  editor1last=Pedicchio  editor1first=Maria Cristina  editor2last=Tholen  editor2first=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=0521834147  zbl=1034.18001 }} Pakeista 735 eilutė iš:
į:
[+Further reading+] Pakeista 738 eilutė iš:
* į:
*'''History of topos theory''' Pakeista 742 eilutė iš:
* [http://www.andrew.cmu.edu/user/awodey/catlog/ Categorical Logic] lecture notes by į:
* [http://www.andrew.cmu.edu/user/awodey/catlog/ Categorical Logic] lecture notes by '''Steve Awodey''' 2019 vasario 10 d., 09:04
atliko 
Pridėtos 1742 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 settheoretic 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 settheoretic 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 categoriescategory 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 [[quasicategory]], the most popular model, unless other models are being discussed. *The number [[zero]] 0 is a natural number. {{Compact ToCshort1sym=yesx=[[#XYZXYZ]]y=z=seealso=yesrefs=yes}} ==A== {{glossary}} {{term1=abelian}} {{defn1=A category is [[abelian categoryabelian]] if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.}} {{term1=accessible}} {{defnno=1Given a [[cardinal number]] κ, an object ''X'' in a category is [[accessible objectκaccessible]] (or κcompact or κpresentable) if <math>\operatorname{Hom}(X, )</math> commutes with κfiltered colimits.}} {{defnno=2Given a [[regular cardinal]] κ, a category is [[accessible categoryκaccessible]] if it has κfiltered colimits and there exists a small set ''S'' of κcompact objects that generates the category under colimits, meaning every object can be written as a colimit of diagrams of objects in ''S''.}} {{term1=additive}} {{defn1=A category is [[additive categoryadditive]] if it is preadditive (to be precise, has some preadditive 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>}} {{term1=adjunction}} {{defn1=An [[adjoint functoradjunction]] (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.)}} {{term1=algebra for a monad}} {{defn1=Given a monad ''T'' in a category ''X'', an [[algebra for a monadalgebra 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 actionaction]] of ''G''.}} {{term1=amnestic}} {{defn1=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}} {{term1=balanced}} {{defn1=A category is balanced if every bimorphism is an isomorphism.}} {{term1=Beck's theorem}} {{defn1=[[Beck's monadicity theoremBeck's theorem]] characterizes the category of [[algebra for a monadalgebras for a given monad]].}} {{term1=bicategory}} {{defn1=A [[bicategory]] is a model of a weak [[2category]].}} {{term1=bifunctor}} {{defn1=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'''.}} {{term1=bimorphism}} {{defn1=A [[bimorphism]] is a morphism that is both an epimorphism and a monomorphism.}} {{term1=Bousfield localization}} {{defn1=See [[Bousfield localization]].}} {{glossary end}} ==C== {{glossary}} {{term1=calculus of functors}} {{defn1=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".}} {{term1=cartesian closed}} {{defn1=A category is [[Cartesian closed categorycartesian closed]] if it has a terminal object and that any two objects have a product and exponential.}} {{term1=cartesian functor}} {{defn1=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.}} {{term1=cartesian morphism}} {{defnno=11=Given a functor π: ''C'' → ''D'' (e.g., a [[prestack]] over schemes), a morphism ''f'': ''x'' → ''y'' in ''C'' is [[cartesian morphismπcartesian]] 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''.}} {{defnno=21=Given a functor π: ''C'' → ''D'' (e.g., a [[prestack]] over rings), a morphism ''f'': ''x'' → ''y'' in ''C'' is [[cartesian morphismπcoCartesian]] 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 πcartesian morphism.)}} {{term1=Cartesian square}} {{defn1=A commutative diagram that is isomorphic to the diagram given as a fiber product.<! really need a diagram here >}} {{term1=categorical logic}} {{defn1=[[Categorical logic]] is an approach to [[mathematical logic]] that uses category theory.}} {{term1=categorification}} {{defn1=[[Categorification]] is a process of replacing sets and settheoretic concepts with categories and categorytheoretic concepts in some nontrivial way to capture categoric flavors. Decategorification is the reverse of categorification.}} {{term1=category}} {{defn1=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.}} {{term1=category of categories}} {{defn1=The [[category of categoriescategory 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.}} {{term1=classifying space}} {{defn1=The [[classifying space of a category]] ''C'' is the geometric realization of the nerve of ''C''.}} {{term1=co}} {{defn1=Often used synonymous with op; for example, a [[colimit]] refers to an oplimit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an opfibration is not the same thing as a [[cofibration]].}} {{term1=coend}} {{defn1=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 modulestensor 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''.}} {{term1=coequalizer}} {{defn1=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.}} {{term1=coimage}} {{defn1=The [[coimage]] of a morphism ''f'': ''X'' → ''Y'' is the coequalizer of <math>X \times_Y X \rightrightarrows X</math>.}} {{term1=colored operad}} {{defn1=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.}} {{term1=comma}} {{defn1=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''. }} {{term1=comonad}} {{defn1=A [[comonad]] in a category ''X'' is a [[comonid]] in the monoidal category of endofunctors of ''X''.}} {{term1=compact}} {{defn1=Probably synonymous with [[#accessible]].}} {{term1=complete}} {{defn1=A category is [[complete categorycomplete]] if all small limits exist.}} {{term1=composition}} {{defnno=11=A composition of morphisms in a category is part of the datum defining the category.}} {{defnno=21=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>.}} {{defnno=31=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>.}} {{term1=concrete}} {{defn1=A [[concrete category]] ''C'' is a category such that there is a faithful functor from ''C'' to '''[[Category of setsSet]]'''; e.g., '''[[category of vector spacesVec]]''', '''[[category of groupsGrp]]''' and '''[[category of topological spacesTop]]'''.}} {{term1=cone}} {{defn1=A [[cone (category theory)cone]] is a way to express the [[universal property]] of a colimit (or dually a limit). One can show<ref>{{harvnbKashiwaraSchapira2006loc=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 righthand side is then the set of cones with vertex ''X''.<ref>{{harvnbMac Lane1998loc=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 >}} {{term1=connected}} {{defn1=A category is [[connected categoryconnected]] 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''.}} {{term1=conservative functor}} {{defn1=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.}} {{term1=constant}} {{defn1=A functor is [[constant functorconstant]] 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'' }.}} {{term1=contravariant functor}} {{defn1=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 preimage <math>f^{1}(A)</math>. With this, <math>\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}</math> is a contravariant functor.}} {{term1=coproduct}} {{defn1=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 groupsGrp]]''' is a [[free product]].}} {{term1=core}} {{defn1=The [[core (category theory)core]] of a category is the maximal groupoid contained in the category.}} {{glossary end}} ==D== {{glossary}} {{term1=Day convolution}} {{defnGiven 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>}} {{term1=density theorem}} {{defn1=The [[density theorem (category theory)density theorem]] states that every presheaf (a setvalued 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.}} {{term1=diagonal functor}} {{defn1=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''.}} {{term1=diagram}} {{defn1=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''.}} {{term1=differential graded category}} {{defn1=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.}} {{term1=discrete}} {{defn1=A category is [[discrete categorydiscrete]] if each morphism is an identity morphism (of some object). For example, a set can be viewed as a discrete category.}} {{term1=distributor}} {{defn1=Another term for "profunctor".}} {{term1=Dwyer–Kan equivalence}} {{defn1=A [[Dwyer–Kan equivalence]] is a generalization of an equivalence of categories to the simplicial context.<ref>{{cite arxivlast=Hinichfirst=V.date=20131117title=DwyerKan localization revisitedeprint=1311.4128class=math.QA}}</ref>}} {{glossary end}} ==E== {{glossary}} {{term1=Eilenberg–Moore category}} {{defn1=Another name for the category of [[algebra for a monadalgebras for a given monad]].}} {{term1=end}} {{defn1=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/intuitionforcoends this mathoverflow thread]. The dual of an end is a coend.}} {{term1=endofunctor}} {{defn1=A functor between the same category.}} {{term1=enriched category}} {{defn1=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.}} {{term1=empty}} {{defnThe [[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.}} {{term1=epimorphism}} {{defn1=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.}} {{term1=equalizer}} {{defn1=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.}} {{term1=equivalence}} {{defnno=1A functor is an [[equivalence of categoriesequivalence]] if it is faithful, full and essentially surjective.}} {{defnno=2A morphism in an ∞category ''C'' is an equivalence if it gives an isomorphism in the homotopy category of ''C''.}} {{term1=equivalent}} {{defn1=A category is equivalent to another category if there is an [[equivalence of categoriesequivalence]] between them.}} {{term1=essentially surjective}} {{defn1=A functor ''F'' is called [[essentially surjective]] (or isomorphismdense) if for every object ''B'' there exists an object ''A'' such that ''F''(''A'') is isomorphic to ''B''.}} {{term1=evaluation}} {{defn1=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}} {{term1=faithful}} {{defn1=A functor is [[faithful functorfaithful]] if it is injective when restricted to each [[homset]].}} {{term1=fundamental category}} {{defn1=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>.}} {{term1=fundamental groupoid}} {{defn1=The [[fundamental groupoid]] <math>\Pi_1 X</math> of a Kan complex ''X'' is the category where an object is a 0simplex (vertex) <math>\Delta^0 \to X</math>, a morphism is a homotopy class of a 1simplex (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''.>}} {{term1=fibered category}} {{defn1=A functor π: ''C'' → ''D'' is said to exhibit ''C'' as a [[fibered categorycategory fibered over]] ''D'' if, for each morphism ''g'': ''x'' → π(''y'') in ''D'', there exists a πcartesian 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).}} {{term1=fiber product}} {{defn1=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]].}} {{term1=filtered}} {{defnno=11=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''.}} {{defnno=21=Given a cardinal number π, a category is said to be πfiltrant 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''.}} {{term1=finitary monad}} {{defn1=A [[finitary monad]] or an algebraic monad is a monad on '''Set''' whose underlying endofunctor commutes with filtered colimits.}} {{term1=finite}} {{defn1=A category is finite if it has only finitely many morphisms.}} {{term1=forgetful functor}} {{defnThe [[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.}} {{term1=free functor}} {{defn1=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 modulefree ''R''module]] generated by ''X'' is a free functor (whence the name).}} {{term1=Frobenius category}} {{defn1=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.}} {{term1=Fukaya category}} {{defn1=See [[Fukaya category]].}} {{term1=full}} {{defnno=11=A functor is [[full functorfull]] if it is surjective when restricted to each [[homset]].}} {{defnno=21=A category ''A'' is a [[full subcategory]] of a category ''B'' if the inclusion functor from ''A'' to ''B'' is full.}} {{term1=functor}} {{defn1=Given categories ''C'', ''D'', a [[functor]] ''F'' from ''C'' to ''D'' is a structurepreserving 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>.}} {{term1=functor category}} {{defn1=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}} {{term1=Gabriel–Popescu theorem}} {{defn1=The [[Gabriel–Popescu theorem]] says an abelian category is a [[Serre quotient categoryquotient]] of the category of modules.}} {{term1=generator}} {{defn1=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.}} {{term1=Grothendieck's Galois theory}} {{defn1=A categorytheoretic generalization of [[Galois theory]]; see [[Grothendieck's Galois theory]].}} {{term1=Grothendieck category}} {{defn1=A [[Grothendieck category]] is a certain wellbehaved kind of an abelian category.}} {{term1=Grothendieck construction}} {{defn1=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]].}} {{term1=Grothendieck fibration}} {{defn1=A [[fibered category]].}} {{term1=groupoid}} {{defnno=11=A category is called a [[groupoid]] if every morphism in it is an isomorphism.}} {{defnno=21=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}} {{term1=Hall algebra of a category}} {{defn1=See [[Ringel–Hall algebra]].}} {{term1=heart}} {{defn1=The [[heart (category theory)heart]] of a [[tstructure]] (<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.}} {{term1=Higher category theory}} {{defn1=[[Higher category theory]] is a subfield of category theory that concerns the study of [[ncategory''n''categories]] and [[∞categories]].}} {{term1=homological dimension}} {{defn1=The [[homological dimension]] of an abelian category with enough injectives is the least nonnegative 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 modulesMod<sub>''R''</sub>]] with a principal ideal domain ''R'' is at most one.}} {{term1=homotopy category}} {{defn1=See<! for now > [[homotopy category]]. It is closely related to a [[localization of a category]].}} {{term1=homotopy hypothesis}} {{defn1=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}} {{term1=identity}} {{defnno=11=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>.}} {{defnno=2The [[identity functor]] on a category ''C'' is a functor from ''C'' to ''C'' that sends objects and morphisms to themselves.}} {{defnno=3Given 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''.}} {{term1=image}} {{defn1=The [[image of a morphismimage]] of a morphism ''f'': ''X'' → ''Y'' is the equalizer of <math>Y \rightrightarrows Y \sqcup_X Y</math>.}} {{term1=indlimit}} {{defn1=A colimit (or inductive limit) in <math>\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})</math>.}} {{term1=∞category}} {{defn1=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 Δ<sup>''n''</sup> is the standard ''n''simplex and <math>\Lambda^n_i</math> is obtained from Δ<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.}} {{term1=initial}} {{defnno=11=An object ''A'' is [[initial objectinitial]] if there is exactly one morphism from ''A'' to each object; e.g., [[empty set]] in '''[[Category of setsSet]]'''.}} {{defnno=21=An object ''A'' in an ∞category ''C'' is initial if <math>\operatorname{Map}_C(A, B)</math> is [[contractible spacecontractible]] for each object ''B'' in ''C''.}} {{term1=injective}} {{defn1=An object ''A'' in an abelian category is [[injective objectinjective]] if the functor <math>\operatorname{Hom}(, A)</math> is exact. It is the dual of a projective object.}} {{term1=internal Hom}} {{defn1=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.}} {{term1=inverse}} {{defn1=A morphism ''f'' is an [[inverse functioninverse]] 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.}} {{term1=isomorphic}} {{defnno=11=An object is [[isomorphic]] to another object if there is an isomorphism between them.}} {{defnno=21=A category is isomorphic to another category if there is an isomorphism between them.}} {{term1=isomorphism}} {{defn1=A morphism ''f'' is an [[isomorphism]] if there exists an ''inverse'' of ''f''.}} {{glossary end}} ==K== {{glossary}} {{term1=Kan complex}} {{defn1=A [[Kan complex]] is a [[fibrant object]] in the category of simplicial sets.}} {{term1=Kan extension}} {{defnno=1Given 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/LectureXIHomological.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.}} {{defnno=2The right Kan extension functor is the right adjoint (if it exists) to <math>f^*</math>.}} {{term1=Kleisli category}} {{defn1=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}} {{term1=lax}} {{defn1=The term "[[lax functor]]" is essentially synonymous with "[[pseudofunctor]]".}} {{term1=length}} {{defn1=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>{{harvnbKashiwaraSchapira2006loc=exercise 8.20}}</ref>}} {{term1=limit}} {{defnno=1The [[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_ii) \in \prod_{i} f(i)  f(s)(x_j) = x_i \text{ for any } s: i \to j \}.</math>}} {{defnno=2The 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>}} {{defnno=3The [[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''. }} {{term1=localization of a category}} {{defn1=See [[localization of a category]].}} {{glossary end}} ==M== {{glossary}} {{term1=monad}} {{defn1=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.}} {{term1=monadic}} {{defnno=11=An adjunction is said to be [[monadic adjunctionmonadic]] if it comes from the monad that it determines by means of the [[Eilenberg–Moore category]] (the category of algebras for the monad).}} {{defnno=21=A functor is said to be [[monadic functormonadic]] if it is a constituent of a monadic adjunction.}} {{term1=monoidal category}} {{defn1=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.}} {{term1=monoid object}} {{defn1=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''.}} {{term1=monomorphism}} {{defn1=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 functioninjection]] in '''[[Category of setsSet]]'''. In other words, ''f'' is the dual of an epimorphism.}} {{term1=multicategory}} {{defn1=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}} {{term1=''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/10/1/1001abs.html A survey of definitions of ''n''category] author=Tom Leinster align=right width=33% }}{{defnno=11=A [[strict ncategorystrict ''n''category]] is defined inductively: a strict 0category 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 1category is an ordinary category.}} {{defnno=21=The notion of a [[weak ncategoryweak ''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.}} {{defnno=31=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 [[quasicategory]]) in the beginning, then a weak ''n''category can be defined as a type of a truncated ∞category.}} {{term1=natural}} {{defnno=1A 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>.}} {{defnno=2A [[natural isomorphism]] is a natural transformation that is an isomorphism (i.e., admits the inverse).}} [[Image:Nerve2simplex.pngthumbrightThe composition is encoded as a 2simplex.]] {{term1=nerve}} {{defn1=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 2simplex), 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.}} {{term1=normal}} {{defn1=A category is [[normal categorynormal]] if every monic is normal.{{citation neededdate=October 2015}}}} {{glossary end}} ==O== {{glossary}} {{term1=object}} {{defnno=11=An object is part of a data defining a category.}} {{defnno=21=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 [[Γobject]] is a pointed contravariant functor from [[Γ (category theory)Γ]] (roughly the pointed category of pointed finite sets) to ''C'' provided ''C'' is pointed.}} {{term1=opfibration}} {{defn1=A functor π:''C'' → ''D'' is an [[opfibration]] if, for each object ''x'' in ''C'' and each morphism ''g'' : π(''x'') → ''y'' in ''D'', there is at least one πcoCartesian morphism ''f'': ''x'' → ''y<nowiki>'</nowiki>'' in ''C'' such that π(''f'') = ''g''. In other words, π is the dual of a [[Grothendieck fibration]].}} {{term1=opposite}} {{defn1=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}} {{term1=perfect}} {{defnSometimes synonymous with "compact". See [[perfect complex]].}} {{term1=pointed}} {{defn1=A category (or ∞category) is called pointed if it has a zero object.}} {{term1=polynomial}} {{defn1=A functor from the category of finitedimensional 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.}} {{term1=preadditive}} {{defn1=A category is [[preadditive categorypreadditive]] if it is [[enriched categoryenriched]] over the [[monoidal category]] of [[abelian group]]s. More generally, it is [[preadditive category#Rlinear categories''R''linear]] if it is enriched over the monoidal category of [[module (mathematics)''R''modules]], for ''R'' a [[commutative ring]].}} {{term1=presentable}} {{defnGiven a [[regular cardinal]] κ, a category is [[presentabl categoryκpresentable]] if it admits all small colimits and is [[#accessibleκaccessible]]. A category is presentable if it is κpresentable for some regular cardinal κ (hence presentable for any larger cardinal). '''Note''': Some authors call a presentable category a [[locally presentable category]].}} {{term1=presheaf}} {{defn1=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 topologytopology]] on ''C'', if any, tells which presheaf is a sheaf (with respect to that topology).}} {{term1=product}} {{defnno=1The [[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.}} {{defnno=2The [[product of categoriesproduct 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 homsets are <math>\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)</math>; the morphisms are composed componentwise. It is the dual of the disjoint union.}} {{term1=profunctor}} {{defn1=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>.}} {{term1=projective}} {{defn1=An object ''A'' in an abelian category is [[projective objectprojective]] if the functor <math>\operatorname{Hom}(A, )</math> is exact. It is the dual of an injective object.}} {{term1=PROP}} {{defn1=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}} {{term1=Quillen}} {{defn1=[[Quillen’s theorem A]] provides a criterion for a functor to be a weak equivalence.}} {{glossary end}} ==R== {{glossary}} {{term1=reflect}} {{defnno=11=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.}} {{defnno=21=A functor is said to reflect isomorphisms if it has the property: ''F''(''k'') is an isomorphism then ''k'' is an isomorphism as well.}} {{term1=representable}} {{defn1=A setvalued contravariant functor ''F'' on a category ''C'' is said to be [[representable functorrepresentable]] 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''.}} {{term1=retraction}} {{defn1=[[File:Section retract.svg150pxthumb''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}} {{term1=section}} {{defn1=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.}} {{term1=Segal space}} {{defn1=[[Segal space]]s were certain simplicial spaces, introduced as models for [[(infinity,1)category(∞, 1)categories]].}} {{term1=semisimple}} {{defn1=An abelian category is [[semisimple categorysemisimple]] if every short exact sequence splits. For example, a ring is [[semisimple ringsemisimple]] if and only if the category of modules over it is semisimple.}} {{term1=Serre functor}} {{defn1=Given a ''k''linear category ''C'' over a field ''k'', a [[Serre functor]] <math>f: C \to C</math> is an autoequivalence such that <math>\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*</math> for any objects ''A'', ''B''.}} <!{{term1=sieve}} {{defn1=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? > {{term1=simple object}} {{defn1=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.}} {{term1=simplex category}} {{defn1=The [[simplex category]] Δ 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 orderpreserving function.}} {{term1=simplicial category}} {{defn1=A category enriched over simplicial sets.}} {{term1=Simplicial localization}} {{defn1=[[Simplicial localization]] is a method of localizing a category.}} {{term1=simplicial object}} {{defn1=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.}} {{term1=simplicial set}} {{defn1=A [[simplicial set]] is a contravariant functor from Δ to '''Set''', where Δ is the [[simplex category]], a category whose objects are the sets [''n''] = { 0, 1, …, ''n'' } and whose morphisms are orderpreserving 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>.}} {{term1=site}} {{defn1=A category equipped with a [[Grothendieck topology]].}} {{term1=skeletal}} {{defn1=A category is [[Skeleton (category theory)skeletal]] if isomorphic objects are necessarily identical.}} {{term1=slice}} {{defn1=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''.}} {{term1=small}} {{defnno=11=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.unibremen.de/acc/ format=PDF year=2004 publisher= Wiley & Sons location=New York isbn=0471609226 page=40}}</ref> (NB: some authors use the term "quasicategory" with a different meaning.<ref>{{cite journaldoi=10.1016/S00224049(02)001354last=Joyalfirst=A.title=Quasicategories and Kan complexesjournal=Journal of Pure and Applied Algebravolume=175year=2002issue=1–3pages=207–222ref=harv}}</ref>)}} {{defnno=2An object in a category is said to be [[small objectsmall]] if it is κcompact for some regular cardinal κ. The notion prominently appears in Quiilen's [[small object argument]] (cf. https://ncatlab.org/nlab/show/small+object+argument)}} {{termspecies}} {{defn1=A [[combinatorial species(combinatorial) species]] is an endofunctor on the groupoid of finite sets with bijections. It is categorically equivalent to a [[symmetric sequence]].}} {{term1=stable}} {{defn1=An ∞category is [[stable ∞categorystable]] 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.}} {{termstrict}} {{defnA morphism ''f'' in a category admitting finite limits and finite colimits is [[strict morphismstrict]] if the natural morphism <math>\operatorname{Coim}(f) \to \operatorname{Im}(f)</math> is an isomorphism.}} {{termstrict ''n''category}} {{defnA strict 0category is a set and for any integer ''n'' > 0, a [[strict ncategorystrict ''n''category]] is a category enriched over strict (''n''1)categories. For example, a strict 1category is an ordinary category. '''Note''': the term "''n''category" typically refers to "[[weak ncategoryweak ''n''category]]"; not strict one.}} {{term1=subcanonical}} {{defn1=A topology on a category is [[subcanonical]] if every representable contravariant functor on ''C'' is a sheaf with respect to that topology.<ref>{{harvnbVistoli2004loc=Definition 2.57.}}</ref> Generally speaking, some [[flat topology]] may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical.}} {{term1=subcategory}} {{defn1=A category ''A'' is a [[subcategory]] of a category ''B'' if there is an inclusion functor from ''A'' to ''B''.}} {{term1=subobject}} {{defn1=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''.}} {{term1=subquotient}} {{defn1=A [[subquotient]] is a quotient of a subobject.}} {{term1=subterminal object}} {{defn1=A [[subterminal object]] is an object ''X'' such that every object has at most one morphism into ''X''.}} {{term1=symmetric monoidal category}} {{defn1=A [[symmetric monoidal category]] is a [[monoidal category]] (i.e., a category with ⊗) that has maximally symmetric braiding.}} {{term1=symmetric sequence}} {{defn1=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}} {{term1=tstructure}} {{defn1=A [[tstructure]] is an additional structure on a [[triangulated category]] (more generally [[stable ∞category]]) that axiomatizes the notions of complexes whose cohomology concentrated in nonnegative degrees or nonpositive degrees.}} {{term1=Tannakian duality}} {{defnThe [[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 geometryderived 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 arxivlast=Bhattfirst=Bhargavdate=20140429title=Algebraization and Tannaka dualityeprint=1404.7483class=math.AG}}</ref>}} {{term1=tensor category}} {{defn1=Usually synonymous with [[monoidal category]] (though some authors distinguish between the two concepts.)}} {{term1=tensor triangulated category}} {{defn1=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.}} {{termtensor product}} {{defnGiven 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>}} {{term1=terminal}} {{defnno=1An object ''A'' is [[terminal objectterminal]] (also called final) if there is exactly one morphism from each object to ''A''; e.g., [[singleton (mathematics)singleton]]s in '''[[Category of setsSet]]'''. It is the dual of an [[initial object]].}} {{defnno=2An object ''A'' in an ∞category ''C'' is terminal if <math>\operatorname{Map}_C(B, A)</math> is [[contractible spacecontractible]] for every object ''B'' in ''C''.}} {{term1=thick subcategory}} {{defn1=A full subcategory of an abelian category is [[thick subcategorythick]] if it is closed under extensions.}} {{term1=thin}} {{defn1=A [[thin categorythin]] is a category where there is at most one morphism between any pair of objects.}} {{term1=triangulated category}} {{defn1=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}} {{term1=universal}} {{defnno=1Given 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''.}} {{defnno=2Stated 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}} {{term1=Waldhausen category}} {{defn1=A [[Waldhausen category]] is, roughly, a category with families of cofibrations and weak equivalences.}} {{term1=wellpowered}} {{defn1=A category is wellpowered if for each object there is only a set of pairwise nonisomorphic [[subobject]]s.}} {{glossary end}} ==Y== {{glossary}} {{term1=Yoneda}} {{defnno=11={{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 setvalued 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> }} {{defnno=21=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}} {{term1=zero}} {{defn1=A [[zero object]] is an object that is both initial and terminal, such as a [[trivial group]] in '''[[Category of groupsGrp]]'''.}} {{glossary end}} ==Notes== {{reflist}} ==References== *{{cite book  first = Michael  last = Artin  authorlink = Michael Artin editor=[[Alexandre Grothendieck]] editor2=[[JeanLouis Verdier]]  title = Séminaire de Géométrie Algébrique du Bois Marie  196364  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 MediaSpringerVerlag]]  location = Berlin; New York  language = French  pages = xix+525  nopp = true doi= 10.1007/BFb0081551 isbn= 9783540058960 }} *{{Cite book  last=Kashiwara  first=Masaki  last2=Schapira  first2=Pierre  title=Categories and sheaves  year=2006  ref=harv authorlink = Masaki Kashiwaraauthorlink2 = Pierre Schapira (mathematician)}} *[[André JoyalA. Joyal]], [http://mat.uab.cat/~kock/crm/hocat/advancedcourse/Quadern452.pdf The theory of quasicategories II] (Volume I is missing??) *[[Jacob LurieLurie, 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=[[SpringerVerlag]]  year=1998  isbn=0387984038  zbl=0906.18001 ref=harv}} * {{cite book  editor1last=Pedicchio  editor1first=Maria Cristina  editor2last=Tholen  editor2first=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=0521834147  zbl=1034.18001 }} * {{Cite arxiv title = Notes on Grothendieck topologies, fibered categories and descent theory eprint = math/0412512date = 20041228first = Angelolast = Vistoli ref=harv}} == Further reading == * Groth, M., [http://www.math.unibonn.de/~mgroth/InfinityCategories.pdf A Short Course on ∞categories] * [http://www.math.univtoulouse.fr/~dcisinsk/1097.pdf Cisinski's notes] *[[History of topos theory]] *http://plato.stanford.edu/entries/categorytheory/ *{{Cite book last=Leinsterfirst=Tomdate=2014title=Basic Category Theoryseries=Cambridge Studies in Advanced Mathematics publisher=Cambridge University Pressvolume=143 arxiv=1612.09375bibcode=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
