Let C be a category. A congruence relationR on C is given by: for each pair of objectsX, Y in C, an equivalence relationRX,Y on Hom, such that the equivalence relations respect composition of morphisms. That is, if are related in Hom and are related in Hom, then g1f1 and g2f2 are related in Hom. Given a congruence relation R on C we can define the quotient categoryC/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.
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. Every functor F : C → D determines a congruence on C by saying f ~ giffF = F. 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 functors.
Examples
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.
Let k be a field and consider the abelian categoryMod of all vector spaces over k with k-linear maps as morphisms. To "kill" all finite-dimensional spaces, we can call two linear mapsf,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.
If C is an additive category and we require the congruence relation ~ on C to be additive, then the quotient category C/~ will also be additive, and the quotient functor C → C/~ will be an additive functor. The concept of an additive congruence relation is equivalent to the concept of a two-sided ideal of morphisms: for any two objects X and Y we are given an additive subgroup I of HomC such that for all f ∈ I, g ∈ HomC and h∈ HomC, we have gf ∈ I and fh ∈ I. Two morphisms in HomC are congruent iff their difference is in I. Every unital ring may be viewed as an additive category with a single object, and the quotient of additive categories defined above coincides in this case with the notion of a quotient ring modulo a two-sided ideal.
Localization of a category
The localization of a category introduces new morphisms to turn several of the original category's morphisms into isomorphisms. This tends to increase the number of morphisms between objects, rather than decrease it as in the case of quotient categories. But in both constructions it often happens that two objects become isomorphic that weren't isomorphic in the original category.
The Serre quotient of an abelian category by a Serre subcategory is a new abelian category which is similar to a quotient category but also in many cases has the character of a localization of the category.
