# Categories and Functors

# Categories

# Definitions

Definitions

A category C\mcC consists of the following data:

  • A set of ObC\Ob \mcC wholse elements are called objects of C\mcC;
  • A collection of sets Hom(X,Y)\Hom(X,Y), one for each ordered pair of objects X,YObCX, Y \in \Ob \mcC, whose elements are called morphisms (from XX to YY); they are denoted φ:XY\vp: X \rightarrow Y;
  • A collection of mappings

Hom(X,Y)×Hom(Y,Z)Hom(X,Z), \Hom(X,Y) \times \Hom(Y,Z) \longrightarrow \Hom(X,Z),

one for each ordered triple of objects X,Y,ZObCX,Y,Z \in \Ob \mcC. Any mapping in this collection associates to a pair φ:XY\vp: X \rightarrow Y, ψ:YZ\psi: Y \rightarrow Z, a morphism from XX to ZZ, denoted ψφ:XZ\psi \circ \vp : X \rightarrow Z, and called the composition or product of φ\vp and ψ.\psi.

This data should satisfy the following conditions:

  • Any morphism φ\vp uniquely determines X,YObCX,Y \in \Ob \mcC such that φHom(X,Y)\vp \in \Hom(X,Y). In other words, the sets Hom(X,Y)\Hom(X,Y) are pairwise disjoint.
  • For any XObCX \in \Ob \mcC there exists the identity morphism idX:XX\id_{X} : X \rightarrow X of XX; it is determined uniquely by the conditions idXφ=φ\id_{X} \circ \vp = \vp, ψidX=ψ\psi \circ \id_{X} = \psi wherever these compositions are defined.
  • The composition of morphisms is associateive

(ξψ)φ=ξ(ψφ) (\xi \circ \psi) \circ \vp = \xi \circ (\psi \circ \vp)

for any φ:XY\vp : X \rightarrow Y, ψ:YZ\psi: Y \rightarrow Z, ξ:ZU\xi: Z \rightarrow U.

Notation

Sometimes it is abbreviated XCX \in \mcC rather thatn XObCX \in \Ob \mcC, and HomC(X,Y)\Hom_{C}(X,Y) or C(X,Y)\mcC(X,Y) instead of Hom(X,Y)\Hom(X,Y). A morphism φHom(X,Y)\vp \in \Hom(X,Y) may sometimes be called an arrow starting at XX and ending at YY. The set X,YCHom(X,Y)\cup_{X,Y \in \mcC} \Hom(X,Y) is denoted Mor(C)\Mor(\mcC).

# Category of Sets and Categories of Sets with a Structure

An important class of categories is formed by categories whose objects are sets with some additional structure, and whose morphisms are maps respecting this structure.

Examples

  • TopTop, the category of topological spaces with continuous maps between them;
  • DiffDiff, the category of smooth manifolds and smooth maps;
  • AbAb, the category of abelian groups and homomorphisms;
  • AA-mod, the category of left-modules over some fixed ring AA;
  • GrGr, the category of groups and homomorphisms.

# More Examples

In this group of examples, the objects of the categories are the same as in the list above, i.e. sets with additional structure, but here we define different morphims.

  • The category TophToph:

Ob Toph=Ob Top={topological spaces},HomToph(X,Y)= the set of homotopy classes of continuous mappings from X to Y.\begin{aligned} \Ob\ Toph = \Ob\ Top &= \{\text{topological spaces}\}, \\ \Hom_{Toph}(X,Y) &= \text{ the set of homotopy classes of continuous mappings from } X \text{ to } Y. \end{aligned}

  • The category of relations RelRel:

ObRel=ObSet=sets in the given Universe,HomRel(X,Y)={subsets of the direct product X×Y}.\begin{aligned} \Ob Rel = \Ob Set &= \text{sets in the given Universe},\\ \Hom_{Rel}(X,Y) &= \{\text{subsets of the direct product } X \times Y \}. \end{aligned}

The composition of φ:XY\vp : X \rightarrow Y and ψ:YZ\psi : Y \rightarrow Z is defined as follows:

ψφ={(x,z)X×Z:there exists yY such that (x,y)φ,(y,z)ψ}X×Z. \psi \circ \vp = \{ (x,z) \in X \times Z : \text{there exists } y \in Y \text{ such that } (x,y) \in \vp, (y,z) \in \psi \} \subset X \times Z.

The identity morphism is the diagonal

idX={(x,x):xX}X×X. \id_{X} = \{ (x,x) : x \in X \} \subset X \times X.

  • The category RelAbRel\ Ab of addititve relations:

ObRelAb=ObAb={abelian groups},HomRelAb(X,Y)=the set of subgroups in X×Y.\begin{aligned} \Ob Rel\ Ab = \Ob Ab &= \{\text{abelian groups}\}, \\ \Hom_{Rel\ Ab}(X,Y) &= \text{the set of subgroups in } X \times Y. \end{aligned}

The composition of morphisms and the identity morphism are defined the same way as in the previous example.

  • The category C(I)\mcC(I) of a partially ordered set II:

ObC(I)=I,HomC(I)(i,j)consists of one element if ij and is empty otherwise.\begin{aligned} \Ob \mcC(I) \quad=\quad &I,\\ \Hom_{\mcC(I)}(i,j) \qquad &\text{consists of one element if } i \leq j \text{ and is empty otherwise.} \end{aligned}

The composition of morphisms and the identity morphism are defined in the only possible way. An important special case of C(I)\mcC(I) is:

  • The category TopXTop_{X}. Let XX be a topological space. Define

ObTopX={open subsets of X},Hom(U,V)=the inclusion UV if UV,Hom(U,V)is empty if U⊄V. \begin{aligned} \Ob Top_{X} \quad &= \quad \{ \text{open subsets of } X \}, \\ \Hom(U,V) \quad &= \quad \text{the inclusion } U \rightarrow V \text{ if } U \subset V, \\ \Hom(U,V) \quad & \qquad \text{is empty if } U \not\subset V. \end{aligned}

# Functors

Definition

A covariant functor FF from a category C\mcC to a category D\mcD (notation: F:CDF: \mcC \rightarrow \mcD) consists of the following data:

  • A mapping ObCD:XF(X)\Ob \mcC \rightarrow \mcD : X \rightarrow F(X);
  • A mapping MorCD:φF(φ)\Mor \mcC \rightarrow \mcD: \vp \rightarrow F(\vp) such that for φHomC(X,Y)\vp \in \Hom_{\mcC}(X,Y) we have that F(φ)HomD(F(X),F(Y))F(\vp) \in \Hom_{\mcD}(F(X), F(Y)).
    This data should satisfy the following conditions: F(φψ)=F(ψ)F(\vp \circ \psi) = F(\psi) for any φ,ψMorC\vp, \psi \in \Mor \mcC for which φψ\vp \circ \psi is defined. In particular, F(idX)=idF(X)F(\id_{X}) = \id_{F(X)}.

Example

Consider the mapping hX:CSet,XObCh_{X} : \mcC \rightarrow Set, X \in \Ob \mcC,

hX(Y)=HomC(X,Y),hX(f)(φ)=fφ, where f:YY,φHomC(X,Y). \begin{aligned} h_{X}(Y) &= \Hom_{\mcC}(X,Y), \\ h_{X}(f)(\vp) &= f \circ \vp, \text{ where } f : Y \rightarrow Y^{\prime}, \vp \in \Hom_{\mcC}(X,Y). \end{aligned}

Example

Given any category C\mcC, we can construct the opposite category of C\mcC, denoted Cop\mcC^{op}. It has the same objects as C\mcC, but for any two X,YObCX, Y \in \Ob\mcC, MorCop(X,Y)=MorC(Y,X)\Mor_{\mcC^{op}}(X,Y) = \Mor_{\mcC}(Y,X).

Definition

A contravariant functor from a category C\mcC to a category D\mcD is a covariant functor from C\mcC to Dop\mcD^{op}.

Examples

  • Any presheaf of abelian groups on a topological space XX is a functor

F:TopXAb. F: Top_{X} \longrightarrow Ab.

  • Let II be a partially ordered set, C(I)\mcC(I) be the corresponding category. A functor G:C(I)AbG : \mcC(I) \rightarrow Ab is a family of abelian groups {G(i):iI}\{ G(i) : i \in I \} and of maps gij:GiGjg_{ij} : G_{i} \rightarrow G_{j}, one for each ordered pair iji \leq j. These maps should satisfy the conditions gijgij=gikg_{ij} \circ g_{ij} = g_{ik} for ijki \leq j \leq k, gii=idGig_{ii} = \id_{G_{i}}. Such families usually appear as raw material for inductive and/or projective limits.
  • By simply forgetting one or more structures on an object, we obtain many functors for the initial category. For example,

Top,Diff,Ab,GrSet, Top,\ Diff, \ Ab, \ Gr \longrightarrow Set,

are functors that simply forget the topological, differentiable, abelian group, and group structures of the objects in the initial category, respectively. Since these functors "forget" some or all of the structure(s) of the object in the initial category, these are examples of forgetful functors.

Definition

Let F,GF,G be two functors from C\mcC to D\mcD. A morphism of functors from FF to GG (notation: f:FGf:F \rightarrow G) is a family of morphisms in D\mcD:

f(X):F(X)G(X), f(X): F(X) \longrightarrow G(X),

one for each XObCX \in \Ob \mcC, satisfying the following condition: for any morphism φ:XY\vp : X \rightarrow Y in C\mcC, the diagram (insert diagram) is commutative.

The composition of morphisms of functors, as well as the identity morphism of functors, are defined in the obvious way. Thus, functors from C\mcC to D\mcD form a category in their own right, which is usually denoted Funct(C,D)Funct(\mcC,\mcD).

# Subcategories

Definition

A category C\mcC is said to be a subcategory of a category D\mcD if

  • ObCObD\Ob \mcC \subset \Ob \mcD;
  • HomC(X,Y)HomD(X,Y)\Hom_{\mcC}(X,Y) \subset \Hom_{\mcD}(X,Y) for any X,YObCX,Y \in \Ob \mcC;
  • The composition of morphisms in C\mcC coincide with their composition in D\mcD, for XObCX \in \Ob \mcC the identity morphism idX\id_{X} in C\mcC coincides with the identity morphism idX\id_{X} in D\mcD.

Definitions

A subcategory C\mcC is said to be full if HomC(X,Y)\Hom_{\mcC}(X,Y) = HomD(X,Y)\Hom_{\mcD}(X,Y) for any X,YObCX,Y \in \Ob \mcC.

A functor f:CDf : \mcC \rightarrow \mcD is said to be faithful if for any X,YObCX,Y \in \Ob \mcC the map

F:HomC(X,Y)HomD(FX,FY) F : \Hom_{\mcC}(X,Y) \longrightarrow \Hom_{\mcD}(FX,FY)

is injective, and full if this map is surjective.

An object α\alpha of a category C\mcC is said to be an initial object if HomC(α,X)\Hom_{\mcC}(\alpha, X) is a one-element set of any XObCX \in \Ob \mcC. Similarly, an object ω\w is said to be a final object in C\mcC if HomC(X,ω)\Hom_{\mcC}(X,\w) is a one-element set for any XObCX \in \Ob \mcC.

Both initial and final objects are determined uniquely up to isomorphism, if they exist.

# To Do

  • Isomorphisms of categories
  • Equivalence of categories
  • Quasi-inverses of functors