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.
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
are related in Hom(X, Y) and
are related in Hom(Y, Z) then g1f1 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,
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].)
- 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.