# # Categories and Functors

## # Categories

### # Definitions

Definitions

A category $\mcC$ consists of the following data:

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

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

one for each ordered triple of objects $X,Y,Z \in \Ob \mcC$. Any mapping in this collection associates to a pair $\vp: X \rightarrow Y$, $\psi: Y \rightarrow Z$, a morphism from $X$ to $Z$, denoted $\psi \circ \vp : X \rightarrow Z$, and called the composition or product of $\vp$ and $\psi.$

This data should satisfy the following conditions:

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

$(\xi \circ \psi) \circ \vp = \xi \circ (\psi \circ \vp)$

for any $\vp : X \rightarrow Y$, $\psi: Y \rightarrow Z$, $\xi: Z \rightarrow U$.

Notation

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

### # 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

• $Top$, the category of topological spaces with continuous maps between them;
• $Diff$, the category of smooth manifolds and smooth maps;
• $Ab$, the category of abelian groups and homomorphisms;
• $A$-mod, the category of left-modules over some fixed ring $A$;
• $Gr$, 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 $Toph$:

\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 $Rel$:

\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 $\vp : X \rightarrow Y$ and $\psi : Y \rightarrow Z$ is defined as follows:

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

The identity morphism is the diagonal

$\id_{X} = \{ (x,x) : x \in X \} \subset X \times X.$

• The category $Rel\ Ab$ of addititve relations:

\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 $\mcC(I)$ of a partially ordered set $I$:

\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 $\mcC(I)$ is:

• The category $Top_{X}$. Let $X$ be a topological space. Define

\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 $F$ from a category $\mcC$ to a category $\mcD$ (notation: $F: \mcC \rightarrow \mcD$) consists of the following data:

• A mapping $\Ob \mcC \rightarrow \mcD : X \rightarrow F(X)$;
• A mapping $\Mor \mcC \rightarrow \mcD: \vp \rightarrow F(\vp)$ such that for $\vp \in \Hom_{\mcC}(X,Y)$ we have that $F(\vp) \in \Hom_{\mcD}(F(X), F(Y))$.
This data should satisfy the following conditions: $F(\vp \circ \psi) = F(\psi)$ for any $\vp, \psi \in \Mor \mcC$ for which $\vp \circ \psi$ is defined. In particular, $F(\id_{X}) = \id_{F(X)}$.

Example

Consider the mapping $h_{X} : \mcC \rightarrow Set, X \in \Ob \mcC$,

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

Definition

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

Examples

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

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

• Let $I$ be a partially ordered set, $\mcC(I)$ be the corresponding category. A functor $G : \mcC(I) \rightarrow Ab$ is a family of abelian groups $\{ G(i) : i \in I \}$ and of maps $g_{ij} : G_{i} \rightarrow G_{j}$, one for each ordered pair $i \leq j$. These maps should satisfy the conditions $g_{ij} \circ g_{ij} = g_{ik}$ for $i \leq j \leq k$, $g_{ii} = \id_{G_{i}}$. 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,

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

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 $F,G$ be two functors from $\mcC$ to $\mcD$. A morphism of functors from $F$ to $G$ (notation: $f:F \rightarrow G$) is a family of morphisms in $\mcD$:

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

one for each $X \in \Ob \mcC$, satisfying the following condition: for any morphism $\vp : X \rightarrow Y$ in $\mcC$, 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 $\mcC$ to $\mcD$ form a category in their own right, which is usually denoted $Funct(\mcC,\mcD)$.

## # Subcategories

Definition

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

• $\Ob \mcC \subset \Ob \mcD$;
• $\Hom_{\mcC}(X,Y) \subset \Hom_{\mcD}(X,Y)$ for any $X,Y \in \Ob \mcC$;
• The composition of morphisms in $\mcC$ coincide with their composition in $\mcD$, for $X \in \Ob \mcC$ the identity morphism $\id_{X}$ in $\mcC$ coincides with the identity morphism $\id_{X}$ in $\mcD$.

Definitions

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

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

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

is injective, and full if this map is surjective.

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

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
Last Updated: 12/18/2019, 9:12:29 AM