Quotient category

In mathematics, a quotient category is a obtained from another category by identifying sets of morphisms. Formally, it is a quotient object in the , analogous to 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 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). Every functor F : C → D determines a congruence on C by saying f ~ g iff F(f) = F(g). The functor F then factors through the quotient functor C → C/~ in a unique manner. This may be regarded as the "first isomorphism theorem" for categories. Monoids and groups 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 hTop is a quotient category of Top, the . The equivalence classes of morphisms are homotopy classes of continuous maps. Let k be a field and consider the Mod(k) of all vector spaces over k with k-linear maps as morphisms. To "kill" all finite-dimensional spaces, we can call two linear maps f,g : X → Y congruent iff their difference has finite-dimensional image. In the resulting quotient category, all finite-dimensional vector spaces are isomorphic to 0. [This is actually an example of a quotient of additive categories, see below.] If C is an and we require the congruence relation ~ on C to be additive (i.