# Quotient category

In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. The notion is similar to that of a quotient group or quotient space, but in the categorical setting.

## Definition

Let C be a category. A congruence relation R on C is given by: for each pair of objects X, Y in C, an equivalence relation RX,Y on Hom(X,Y), such that the equivalence relations respect composition of morphisms. That is, if

$f_1,f_2 : X \to Y\,$

are related in Hom(X, Y) and

$g_1,g_2 : Y \to Z\,$

are related in Hom(Y, Z) then g1f1, g1f2, g2f1 and g2f2 are related in Hom(X, Z).

Given a congruence relation R on C we can define the quotient category C/R as the category whose objects are those of C and whose morphisms are equivalence classes of morphisms in C. That is,

$\mathrm{Hom}_{\mathcal{C}/\mathcal{R}}(X,Y) = \mathrm{Hom}_{\mathcal{C}}(X,Y)/R_{X,Y}.$

Composition of morphisms in C/R is well-defined since R is a congruence relation.

There is also a notion of taking the quotient of an Abelian category A by a Serre subcategory B. This is done as follows. The objects of A/B are the objects of A. Given two objects X and Y of A, we define the set of morphisms from X to Y in A/B to be $\varinjlim \mathrm{Hom}_A(X', Y/Y')$ where the limit is over subobjects $X' \subseteq X$ and $Y' \subseteq Y$ such that $X/X', Y' \in B$. Then A/B is an Abelian category, and there is a canonical functor $Q \colon A \to A/B$. This Abelian quotient satisfies the universal property that if C is any other Abelian category, and $F \colon A \to C$ is an exact functor such that F(b) is a zero object of C for each $b \in B$, then there is a unique exact functor $\overline{F} \colon A/B \to C$ such that $F = \overline{F} \circ Q$. (See [Gabriel].)

## Properties

There is a natural quotient functor from C to C/R which sends each morphism to its equivalence class. This functor is bijective on objects and surjective on Hom-sets (i.e. it is a full functor).