# 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

${\displaystyle f_{1},f_{2}:X\to Y\,}$

are related in Hom(X, Y) and

${\displaystyle g_{1},g_{2}:Y\to Z\,}$

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,

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

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 ${\displaystyle \varinjlim \mathrm {Hom} _{A}(X',Y/Y')}$ where the limit is over subobjects ${\displaystyle X'\subseteq X}$ and ${\displaystyle Y'\subseteq Y}$ such that ${\displaystyle X/X',Y'\in B}$. Then A/B is an Abelian category, and there is a canonical functor ${\displaystyle Q\colon A\to A/B}$. This Abelian quotient satisfies the universal property that if C is any other Abelian category, and ${\displaystyle F\colon A\to C}$ is an exact functor such that F(b) is a zero object of C for each ${\displaystyle b\in B}$, then there is a unique exact functor ${\displaystyle {\overline {F}}\colon A/B\to C}$ such that ${\displaystyle F={\overline {F}}\circ Q}$. (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).