# 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 *R*_{X,Y} on Hom(*X*,*Y*), such that the equivalence relations respect composition of morphisms. That is, if

are related in Hom(*X*, *Y*) and

are related in Hom(*Y*, *Z*) then *g*_{1}*f*_{1}, *g*_{1}*f*_{2}, *g*_{2}*f*_{1} and *g*_{2}*f*_{2} 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,

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 where the limit is over subobjects and such that . Then *A/B* is an Abelian category, and there is a canonical functor . This Abelian quotient satisfies the universal property that if *C* is any other Abelian category, and is an exact functor such that *F(b)* is a zero object of *C* for each , then there is a unique exact functor such that . (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).

## Examples

- Monoids and group may be regarded as categories with one object. In this case the quotient category coincides with the notion of a quotient monoid or a quotient group.
- The homotopy category of topological spaces
**hTop**is a quotient category of**Top**, the category of topological spaces. The equivalence classes of morphisms are homotopy classes of continuous maps.

## See also

## References

- Gabriel, Pierre,
*Des categories abeliennes*, Bull. Soc. Math. France**90**(1962), 323-448. - Mac Lane, Saunders (1998)
*Categories for the Working Mathematician*. 2nd ed. (Graduate Texts in Mathematics 5). Springer-Verlag.