See: Category theory

**Investigation: Organize the concepts in category theory to reveal underlying themes**

- 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.
- Think through in what sense Product means "and" and Coproduct means "or".

This is a glossary of properties and concepts in **category theory** in **mathematics**. It is based especially on the following sources:

The notations and the conventions used throughout the article are:

- [
*n*] = { 0, 1, 2, …,*n*}, which is viewed as a category (by writing {$i \to j \Leftrightarrow i \le j$}.) **Cat**, the**category of (small) categories**, where the objects are categories (which are small with respect to some universe) and the morphisms**functor**s.**Fct**(*C*,*D*), the**functor category**: the category of**functor**s from a category*C*to a category*D*.**Set**, the category of (small) sets.*s***Set**, the category of**simplicial set**s.- "weak" instead of "strict" is given the default status; e.g., "
*n*-category" means "weak*n*-category", not the strict one, by default. - By an
**∞-category**, we mean a**quasi-category**, the most popular model, unless other models are being discussed. - The number
**zero**0 is a natural number.

Types of object

- An
**object**is part of a data defining a category. - An object is
**isomorphic**to another object if there is an isomorphism between them. - An object
*A*is**initial**if there is exactly one morphism from*A*to each object; e.g.,**empty set**in**Set**. - 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**. - 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
**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. - 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 inis an*R*-mod**associative algebra**over a commutative ring*R*. - Given a monad
*T*in a category*X*, an**algebra for a algebra for**or a*T**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*. - A
**Kan complex**is a**fibrant object**in the category of simplicial sets. - Given a
**cardinal number**κ, an object*X*in a category is**κ-accessible**(or κ-compact or κ-presentable) if {$\operatorname{Hom}(X, -)$} commutes with κ-filtered colimits. **compact**Probably synonymous with**#accessible**.**perfect**Sometimes synonymous with "compact". See**perfect complex**.- An object in a category is said to be
**small**if it is κ-compact for some regular cardinal κ. The notion prominently appears in Quiilen's**small object argument**n-lab: small object argument.

Variations of objects

- 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*. - 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 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)

Types of morphism

- 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. - A
**bimorphism**is a morphism that is both an epimorphism and a monomorphism. - A monomorphism is normal if it is the kernel of some morphism.
- An epimorphism is conormal if it is the cokernel of some morphism.
- 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. - Given a functor π:
*C*→*D*(e.g., a**prestack**over schemes), a morphism*f*:*x*→*y*in*C*is**π-cartesian**if, for each object*z*in*C*, each morphism*g*:*z*→*y*in*C*and each morphism*v*: π(*z*) → π(*x*) in*D*such that π(*g*) = π(*f*) ∘*v*, there exists a unique morphism*u*:*z*→*x*such that π(*u*) =*v*and*g*=*f*∘*u*. - Given a functor π:
*C*→*D*(e.g., a**prestack**over rings), a morphism*f*:*x*→*y*in*C*is**π-coCartesian**if, for each object*z*in*C*, each morphism*g*:*x*→*z*in*C*and each morphism*v*: π(*y*) → π(*z*) in*D*such that π(*g*) =*v*∘ π(*f*), there exists a unique morphism*u*:*y*→*z*such that π(*u*) =*v*and*g*=*u*∘*f*. (In short,*f*is the dual of a π-cartesian morphism.) - 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. - 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 {$u_X$} 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 {$\operatorname{Hom}_D(X, f(-))$}.

Types of category

- A
**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. **composition**A composition of morphisms in a category is part of the datum defining the 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. - A
**small category**is a category in which the class of all morphisms is a**set**(i.e., not a**proper class**); otherwise**large**. A category is**locally small**if the morphisms between every pair of objects*A*and*B*form a set. Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a**quasicategory**is a category whose objects and morphisms merely form a**conglomerate**. - A category is
**wellpowered**if for each object there is only a set of pairwise non-isomorphic**subobject**s. - 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. - 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 order-preserving function. - 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**is defined inductively: a strict 0-category is a set and a strict*n*-category*n*-category is a category whose Hom sets are strict (*n*-1)-categories. Precisely, a strict*n*-category is a category enriched over strict (*n*-1)-categories. For example, a strict 1-category is an ordinary category. - A category is
**isomorphic**to another category if there is an isomorphism between them. - 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 is
**equivalent**to another category if there is an**equivalence**between them. - A category
*A*is a**subcategory**of a category*B*if there is an inclusion functor from*A*to*B*. - 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. - A category is
**skeletal**if isomorphic objects are necessarily identical. - A category is
**finite**if it has only finitely many morphisms. - 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. - A
**thin**is a category where there is at most one morphism between any pair of objects. - A category is
**balanced**if every bimorphism is an isomorphism. - A category is
**normal**if every monomorphism is normal. - A category is called a
**groupoid**if every morphism in it is an isomorphism. - The
**core**of a category is the maximal groupoid contained in the category. - The
**fundamental groupoid**{$\Pi_1 X$} of a Kan complex*X*is the category where an object is a 0-simplex (vertex) {$\Delta^0 \to X$}, a morphism is a homotopy class of a 1-simplex (path) {$\Delta^1 \to X$} and a composition is determined by the Kan property. - A
**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
**closed**if it has an internal Hom functor. - A category is
**cartesian closed**if it has a terminal object and that any two objects have a product and exponential. - 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. - 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. - The
**product of a family of categories**{$C_i$}'s indexed by a set*I*is the category denoted by {$\prod_i C_i$} whose class of objects is the product of the classes of objects of {$C_i$}'s and whose hom-sets are {$\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)$}; the morphisms are composed component-wise. It is the dual of the disjoint union. - 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*. - A category is
**complete**if all small limits exist. - A
*'finitely complete category*is a category C which admits all finite limits. - A
**regular category**is a finitely complete category which admits a good notion of image factorization. A primary raison d’être behind regular categories C is to have a decently behaved calculus of relations in C. - A
**coherent category**(also called a**pre-logos**) is a regular category in which the subobject posets Sub(X) all have finite unions which are preserved by the base change functors {$f^*:Sub(Y)\to Sub(X)$}. - A
**geometric category**is a regular category in which the subobject posets Sub(X) have all small unions which are stable under pullback. - Well-powered.
- 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. **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. - 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. - A full subcategory of an abelian category is
**thick**if it is closed under extensions. - A
**Grothendieck category**is a certain well-behaved kind of an abelian category. - 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. - The
**heart**of a**t-structure**({$D^{\ge 0}$}, {$D^{\le 0}$}) on a triangulated category is the intersection {$D^{\ge 0} \cap D^{\le 0}$}. It is an abelian category. - A
**derived category**is a triangulated category that is not necessary an abelian category. - 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**. - A category is
**additive**if it is preadditive (to be precise, has some pre-additive structure) and admits all finite**coproduct**s. Although "preadditive" is an additional structure, one can show "additive" is a*property*of a category; i.e., one can ask whether a given category is additive or not. - An
**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. - The
**homological dimension**of an abelian category with enough injectives is the least non-negative integer*n*such that every object in the category admits an injective resolution of length at most*n*. The dimension is ∞ if no such integer exists. For example, the homological dimension of {$Mod_R$} with a principal ideal domain*R*is at most one. - 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.
**simplicial category**A category enriched over simplicial sets.- 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. - A
**Fukaya category**is a certain kind of category of Lagrangian submanifolds of a symplectic manifold. **Grothendieck fibration**A**fibered category**, which is used for a general framework of descent theory, to discuss vector bundles, principal bundles and sheaves over topological spaces.**site**A category equipped with a**Grothendieck topology**, which makes its objects act like open sets of a topological space.- A
**Waldhausen category**is, roughly, a category C with families of cofibrations and weak equivalences, which makes it possible to calculate the K-spectrum of C. **homotopy category**. It is closely related to a**localization of a category**.- Given a
**regular cardinal**κ, a category is**κ-accessible**if it has κ-filtered colimits and there exists a small set*S*of κ-compact objects that generates the category under colimits, meaning every object can be written as a colimit of diagrams of objects in*S*. - Given a
**regular cardinal**κ, a category is**κ-presentable**if it admits all small colimits and is**κ-accessible**. A category is presentable if it is κ-presentable for some regular cardinal κ (hence presentable for any larger cardinal).**Note**: Some authors call a presentable category a**locally presentable category**. - Given a cardinal number π, a category is said to be π-filtrant if, for each category
*J*whose set of morphisms has cardinal number strictly less than π, the set {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object*i*in*I*.

Variation of categories

- A topology on a category is
**subcanonical**if every representable contravariant functor on*C*is a sheaf with respect to that topology. Generally speaking, some**flat topology**may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical. - A
**t-structure**is an additional structure on a**triangulated category**(more generally**stable ∞-category**) that axiomatizes the notions of complexes whose cohomology concentrated in non-negative degrees or non-positive degrees.

Generalizations of a category

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

Types of functor

- Given categories
*C*,*D*, a**functor***F*from*C*to*D*is a structure-preserving map from*C*to*D*; i.e., it consists of an object*F*(*x*) in*D*for each object*x*in*C*and a morphism*F*(*f*) in*D*for each morphism*f*in*C*satisfying the conditions: (1) {$F(f \circ g) = F(f) \circ F(g)$} whenever {$f \circ g$} is defined and (2) {$F(\operatorname{id}_x) = \operatorname{id}_{F(x)}$}. - For example, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}, \, S \mapsto \mathfrak{P}(S)$}, where {$\mathfrak{P}(S)$} is the
**power set**of*S*is a functor if we define: for each function {$f: S \to T$}, {$\mathfrak{P}(f): \mathfrak{P}(S) \to \mathfrak{P}(T)$} by {$\mathfrak{P}(f)(A) = f(A)$}. - The
**identity functor**on a category*C*is a functor from*C*to*C*that sends objects and morphisms to themselves. - 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*}. - 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*. - A functor is said to
**reflect identities**if it has the property: if*F*(*k*) is an identity then*k*is an identity as well. - A functor is said to
**reflect isomorphisms**if it has the property:*F*(*k*) is an isomorphism then*k*is an isomorphism as well. - A
**conservative functor**is a functor that reflects isomorphisms. Many forgetful functors are conservative, but the forgetful functor from**Top**to**Set**is not conservative. - A functor is amnestic if it has the property: if
*k*is an isomorphism and*F*(*k*) is an identity, then*k*is an identity. - A functor is
**faithful**if it is injective when restricted to each**hom-set**. - A functor is
**full**if it is surjective when restricted to each**hom-set**. - A functor
*F*is called**essentially surjective**(or isomorphism-dense) if for every object*B*there exists an object*A*such that*F*(*A*) is isomorphic to*B*. - A functor is an
**equivalence**if it is faithful, full and essentially surjective. - 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 functor π:
*C*→*D*is an**op-fibration**if, for each object*x*in*C*and each morphism*g*: π(*x*) →*y*in*D*, there is at least one π-coCartesian morphism*f*:*x*→*y*in*C*such that π(*f*) =*g*. In other words, π is the dual of a**Grothendieck fibration**. **endofunctor.**A functor between the same category.- A
**contravariant functor***F*from a category*C*to a category*D*is a (covariant) functor from*C*<sup>op</sup> to*D*. It is sometimes also called a**presheaf**especially when*D*is**Set**or the variants. For example, for each set*S*, let {$\mathfrak{P}(S)$} be the power set of*S*and for each function {$f: S \to T$}, define:{$\mathfrak{P}(f): \mathfrak{P}(T) \to \mathfrak{P}(S)$} by sending a subset*A*of*T*to the pre-image {$f^{-1}(A)$}. With this, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}$} is a contravariant functor. - 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
**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**. - 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 order-preserving functions. One writes {$X_n = X([n])$} and an element of the set {$X_n$} is called an*n*-simplex. For example, {$\Delta^n = \operatorname{Hom}_{\Delta}(-, [n])$} is a simplicial set called the standard*n*-simplex. By Yoneda's lemma, {$X_n \simeq \operatorname{Nat}(\Delta^n, X)$}. - The functor {$y: C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(-, X)$} is fully faithful and is called the Yoneda embedding. Technical note: the lemma implicitly involves a choice of
**Set**; i.e., a choice of universe. - A set-valued contravariant functor
*F*on a category*C*is said to be**representable**if it belongs to the essential image of the**Yoneda embedding**{$C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}; i.e., {$F \simeq \operatorname{Hom}_C(-, Z)$} for some object*Z*. The object*Z*is said to be the representing object of*F*. - An [adjective] object in a category
*C*is a contravariant functor (or presheaf) from some fixed category corresponding to the "adjective" to*C*. For example, a**simplicial object**in*C*is a contravariant functor from the simplicial category to*C*and a**Γ-object**is a pointed contravariant functor from**Γ**(roughly the pointed category of pointed finite sets) to*C*provided*C*is pointed. - Given categories
*C*and*D*, a**profunctor**(or a distributor) from*C*to*D*is a functor of the form {$D^{\text{op}} \times C \to \mathbf{Set}$}. **distributor**. Another term for "profunctor".- If {$f: C \to D, \, g: D \to E$} are functors, then the
**composition**{$g \circ f$} or {$gf$} is the functor defined by: for an object*x*and a morphism*u*in*C*, {$(g \circ f)(x) = g(f(x)), \, (g \circ f)(u) = g(f(u))$}. - Given categories
*C*,*D*and an object*A*in*C*, the**evaluation**at*A*is the functor:{$\mathbf{Fct}(C, D) \to D, \,\, F \mapsto F(A).$} For example, the**Eilenberg–Steenrod axioms**give an instance when the functor is an equivalence. - A
**(combinatorial) species**is an endofunctor on the groupoid of finite sets with bijections. It is categorically equivalent to a**symmetric sequence**. - A functor from the category of finite-dimensional vector spaces to itself is called a
**polynomial functor**if, for each pair of vector spaces*V*,*W*, {{nowrap|*F*: Hom(*V*,*W*) → Hom(*F*(*V*),*F*(*W*))}} is a polynomial map between the vector spaces. A**Schur functor**is a basic example. - 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). - A functor is said to be
**monadic**if it is a constituent of a monadic adjunction. - 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**generated by*R*-module*X*is a free functor (whence the name). - A functor π:
*C*→*D*is said to exhibit*C*as a**category fibered over***D*if, for each morphism*g*:*x*→ π(*y*) in*D*, there exists a π-cartesian morphism*f*:*x<nowiki>'</nowiki>*→*y*in*C*such that π(*f*) =*g*. If*D*is the category of affine schemes (say of finite type over some field), then π is more commonly called a**prestack**.**Note**: π is often a forgetful functor and in fact the**Grothendieck construction**implies that every fibered category can be taken to be that form (up to equivalences in a suitable sense). - 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. - 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*.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 {$f^*$}. - 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. - 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*. - The
**nerve functor***N*is the functor from**Cat**to*s***Set**given by {$N(C)_n = \operatorname{Hom}_{\mathbf{Cat}}([n], C)$}. For example, if {$\varphi$} is a functor in {$N(C)_2$} (called a 2-simplex), let {$x_i = \varphi(i), \, 0 \le i \le 2$}. Then {$\varphi(0 \to 1)$} is a morphism {$f: x_0 \to x_1$} in*C*and also {$\varphi(1 \to 2) = g: x_1 \to x_2$} for some*g*in*C*. Since {$0 \to 2$} is {$0 \to 1$} followed by {$1 \to 2$} and since {$\varphi$} is a functor, {$\varphi(0 \to 2) = g \circ f$}. In other words, {$\varphi$} encodes*f*,*g*and their compositions. - The
**fundamental category functor**{$\tau_1: s\mathbf{Set} \to \mathbf{Cat}$} is the left adjoint to the nerve functor*N*. For every category*C*, {$\tau_1 NC = C$}. - 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
**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*. - Given a
*k*-linear category*C*over a field*k*, a**Serre functor**{$f: C \to C$} is an auto-equivalence such that {$\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*$} for any objects*A*,*B*.

Generalizations of a functor

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

Natural transformations

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

Types of diagram

- Given a category
*C*, a**diagram**in*C*is a functor {$f: I \to C$} from a small category*I*. **Cartesian square**A commutative diagram that is isomorphic to the diagram given as a fiber product.- A
**cone**is a way to express the**universal property**of a colimit (or dually a limit). One can show that the colimit {$\varinjlim$} is the left adjoint to the diagonal functor {$\Delta: C \to \operatorname{Fct}(I, C)$}, which sends an object*X*to the constant functor with value*X*; that is, for any*X*and any functor {$f: I \to C$}, {$\operatorname{Hom}(\varinjlim f, X) \simeq \operatorname{Hom}(f, \Delta_X),$} provided the colimit in question exists. The right-hand side is then the set of cones with vertex*X*. - The
**limit**(or**projective limit**) of a functor {$f: I^{\text{op}} \to \mathbf{Set}$} is {$\varprojlim_{i \in I} f(i) = \{ (x_i|i) \in \prod_{i} f(i) | f(s)(x_j) = x_i \text{ for any } s: i \to j \}.$} - The limit {$\varprojlim_{i \in I} f(i)$} of a functor {$f: I^{\text{op}} \to C$} is an object, if any, in
*C*that satisfies: for any object*X*in*C*, {$\operatorname{Hom}(X, \varprojlim_{i \in I} f(i)) = \varprojlim_{i \in I} \operatorname{Hom}(X, f(i))$}; i.e., it is an object representing the functor {$X \mapsto \varprojlim_i \operatorname{Hom}(X, f(i)).$} - The
**colimit**(or**inductive limit**) {$\varinjlim_{i \in I} f(i)$} is the dual of a limit; i.e., given a functor {$f: I \to C$}, it satisfies: for any*X*, {$\operatorname{Hom}(\varinjlim f(i), X) = \varprojlim \operatorname{Hom}(f(i), X)$}. Explicitly, to give {$\varinjlim f(i) \to X$} is to give a family of morphisms {$f(i) \to X$} such that for any {$i \to j$}, {$f(i) \to X$} is {$f(i) \to f(j) \to X$}. Perhaps the simplest example of a colimit is a**coequalizer**. For another example, take*f*to be the identity functor on*C*and suppose {$L = \varinjlim_{X \in C} f(X)$} exists; then the identity morphism on*L*corresponds to a compatible family of morphisms {$\alpha_X: X \to L$} such that {$\alpha_L$} is the identity. If {$f: X \to L$} is any morphism, then {$f = \alpha_L \circ f = \alpha_X$}; i.e.,*L*is a final object of*C*. **ind-limit**A colimit (or inductive limit) in {$\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}.- The
**equalizer**of a pair of morphisms {$f, g: A \to B$} is the limit of the pair. It is the dual of a coequalizer. - 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
**image**of a morphism*f*:*X*→*Y*is the equalizer of {$Y \rightrightarrows Y \sqcup_X Y$}. - The
**coimage**of a morphism*f*:*X*→*Y*is the coequalizer of {$X \times_Y X \rightrightarrows X$}. - 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. - 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**Grp**is a**free product**. - Given a monoidal category
*B*, the**tensor product of functors**{$F: C^{\text{op}} \to B$} and {$G: C \to B$} is the coend: {$F \otimes_C G = \int^{c \in C} F(c) \otimes G(c).$} - 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**. - The
**direct limit of algebraic objects**is a colimit. - The
**direct limit in an arbitrary category**is a colimit. - The
**inverse limit of algebraic objects**is a limit. - The
**inverse limit in an arbitrary category**is a limit. - 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 this mathoverflow thread. The dual of an end is a coend. - 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*.

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**.

Theorems

**Beck's theorem**characterizes the category of**algebras for a given monad**.- The
**density theorem**states that every presheaf (a set-valued contravariant functor) is a colimit of representable presheaves. Yoneda's lemma embeds a category*C*into the category of presheaves on*C*. The density theorem then says the image is "dense", so to say. The name "density" is because of the analogy with the**Jacobson density theorem**(or other variants) in abstract algebra. - The
**homotopy hypothesis**states an**∞-groupoid**is a space (less equivocally, an*n*-groupoid can be used as a homotopy*n*-type.) **Quillen’s theorem A**provides a criterion for a functor to be a weak equivalence.- The
**Gabriel–Popescu theorem**says an abelian category is a**quotient**of the category of modules. - Yoneda’s Lemma asserts ... in more evocative terms, a mathematical object X is best thought of in the context of a category surrounding it, and is determined by the network of relations it enjoys with all the objects of that category. Moreover, to understand X it might be more germane to deal directly with the functor representing it. This is reminiscent of Wittgenstein’s ’language game’; i.e., that the meaning of a word is—in essence—determined by, in fact is nothing more than, its relations to all the utterances in a language.
- The
**Yoneda lemma**says: for each set-valued contravariant functor*F*on*C*and an object*X*in*C*, there is a natural bijection {$F(X) \simeq \operatorname{Nat}(\operatorname{Hom}_C(-, X), F)$} where Nat means the set of natural transformations.

Techniques

- 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". **Categorification**is a process of replacing sets and set-theoretic concepts with categories and category-theoretic concepts in some nontrivial way to capture categoric flavors. Decategorification is the reverse of categorification.**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**.**localization of a category****Bousfield localization****Simplicial localization**is a method of localizing a category.

Dualities

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

Areas of math

**Categorical logic**is an approach to**mathematical logic**that uses category theory.**Grothendieck's Galois theory**A category-theoretic generalization of**Galois theory**.- Higher category theory is a subfield of category theory that concerns the study of n-categories and ∞-categories.

Parsiųstas iš http://www.ms.lt/sodas/Book/CategoryTheoryGlossary

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