Quotient categories are a generalization of quotient structures found in many mathematical systems, such as sets and groups. For example, if we have a set $S= \lbrace a,b,c \rbrace$ , by identifying $a$ and $c$ as being the ``same'', we obtain a two-element set $\lbrace \lbrace a,c\rbrace ,b\rbrace$ , called the quotient set of $A$ by identifying $a$ and $b$ . Here is a more useful example: take the set $S$ of integers. Identify integers $m$ and $n$ if their difference is divisible by $2$ . Then we obtain a quotient of $S$ consisting of two elements: the set of odd numbers and the set of even numbers.

In both of the examples above, we start with a set $S$ and a binary relation $R$ (in the first example, $R$ is the singleton $\lbrace (a,b)\rbrace$ , and in the second, $R$ consists of pairs of integers $(m,m\!+\!2n)$ , where $m,n$ are integers) on $S$ . The quotient $Q$ of $S$ by $R$ then satisfies the following functional properties:

1. there is an onto map $p:S\to Q$ such that $aRb$ implies $p(a)=p(b)$ , and
2. if we have another onto map $q:S\to T$ satisfying 1 above, then there is a unique map $f:Q\to T$ such that $f\circ p=q$ .

The two properties above are exactly what we need to define the quotient of a category $\mathcal{C}$ by a binary relation $R$ on $\mathcal{C}$ . But what is $R$ precisely? We define a binary relation on a category $\mathcal{C}$ to be a binary relation $R$ on $\operatorname{Mor}(\mathcal{C})$ , the class of morphisms of $\mathcal{C}$ . In other words, $R$ is a class consisting of pairs $(f,g)$ where $f$ and $g$ are morphisms in $\mathcal{C}$ . If $(f,g)$ is in $R$ , we write $fRg$ .

Definition. Let $\mathcal{C}$ be a category and $R$ a binary relation on $\mathcal{C}$ . Then a quotient of $\mathcal{C}$ by $R$ is a pair $(\mathcal{Q},P)$ where $\mathcal{Q}$ is a category and $P:\mathcal{C} \to \mathcal{Q}$ is a functor, satisfying the following two conditions:

1. $P$ is a surjection on objects; that is, for every object $Q$ in $\mathcal{Q}$ , there is a object $C$ in $\mathcal{C}$ , such that $P(C)=Q$ .
2. if $fRg$ , then $P(f)=P(g)$ ,
3. if $(\mathcal{D},F)$ is another pair satisfying conditions 1 and 2 above, then there is a unique functor $G:\mathcal{Q}\to \mathcal{D}$ such that $G\circ P = F$ .

To see how $\mathcal{C}/R$ may possibly be constructed, we make the following few observations:

• First, if we take the reflexive, symmetric, and transitive closure $R^*$ of $R$ (so that $R^*$ is the smallest equivalence relation containing $R$ ), then $f R^* g$ implies $P(f)=P(g)$ .
• In addition, if $f_1 R^* g_1$ and $f_2 R^* g_2$ such that $f_1 \circ f_2$ and $g_1\circ g_2$ are both defined, then $P(f_1\circ f_2) = P(g_1\circ g_2)$ . This means that we may reduce $R^*$ to a congruence relation $R'$ with respect to morphism composition. Once again, $f R' g$ implies $P(f)=P(g)$ .
• Furthermore, if $f:A\to B$ and $g:C\to D$ and $f R' g$ , then $P(f)=P(g)$ forces $P(A)=P(C)$ and $P(B)=P(D)$ . In other words, the domains of $f$ and $g$ are identified under $P$ . The same is true for the corresponding codomains. This implies that, if $[f]$ denotes the equivalence class of $f$ under $R'$ , then the domains (codomains) of all $g\in [f]$ are identified as one object in $\mathcal{C}/R$ , if it exists.
• As a result, all the corresponding identity morphisms are identified as well, for if $P(A)=P(B)$ , then $P(1_A)=1_{P(A)} = 1_{P(B)}= P(1_B)$ . This means we can further reduce $R'$ to $R''$ by such identifications. Note that $R''$ is still a congruence relation.

1. $R$ is a set, or
2. if $f R g$ implies that $f$ and $g$ are parallel (they have the same domain and codomain),

In the following discussion, we denote objects and morphisms in $\mathcal{C}/R$ by $[A]_R$ and $[f]_R$ (images of $A$ and $f$ under $P$ ), or simply $[A]$ and $[f]$ when there is no confusion.

Suppose $\mathcal{C}$ is a category with a binary relation $R$ on its morphisms. Below are some examples, as well as a non-example:

1. Suppose $\mathcal{C}$ is generated by four objects $A,B,C,D$ and three morphisms $f:A\to B$ , $g:C\to D$ , and $h: B\to C$ . Furthermore, suppose $R=\lbrace (f,g)\rbrace$ . Then $\mathcal{C}/R$ exists and is generated by objects $X=[A]=[C]$ and $Y=[B]=[D]$ , and morphisms $\alpha=[f]:X\to Y$ and $\beta=[h]:Y\to X$ . Note that whereas $\mathcal{C}$ has only a finite number of morphisms ($f,g,h$ and four identities), the quotient $\mathcal{C}/R$ has infinitely many. For example, on $X$ , the morphisms are $(\beta\alpha)^n$ , for any non-negative integer $n$ .
2. If $R$ is defined so that $f R g$ iff $f$ is parallel to $g$ , then $R$ is easily seen to be a congruence relation. Then $\mathcal{C}/R$ is the category whose objects are objects of $\mathcal{C}$ , such that each hom set has at most one element. As a result, if both $\hom(A,B)$ and $\hom(B,A)$ are non-empty, then $A$ is isomorphic to $B$ .
3. The two examples above are artificial, here's a concrete one: let $\mathcal{C}$ be the category of topological spaces and continuous maps, define $R$ so that $f R g$ whenever $f$ is homotopic to $g$ . Then $\mathcal{C}/R$ is the category of topological spaces with homotopic classes of continuous maps.
4. Here's an example where $\mathcal{C}/R$ does not exist. Let $\mathcal{C}$ be a category whose collection of objects is a proper class, and there is a non-identity morphism $h_A$ on each object $A$ . Define $R$ so that $f R g$ iff there are objects $A,B$ with $f=1_A$ and $g=1_B$ . Then if $\mathcal{C}/R$ were to exist, it would have only one object, say $X$ . Furthermore, $\hom(X,X)$ contains a morphism $[h_A]$ for every $A$ in $\mathcal{C}$ . Since $[h_A]=[h_B]$ iff $A=B$ , $\hom(X,X)$ is not a set. Therefore, $\mathcal{C}/R$ is not well-defined.

Bibliography

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S. Mac Lane. Categories for the Working Mathematician, 2nd ed. Springer-Verlag, 1997