Quotient category

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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.

1 Definition
2 Properties
3 Examples
4 See also

[edit] 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

$f_1,f_2 : X \to Y\,$

are related in Hom(X, Y) and

$g_1,g_2 : Y \to Z\,$

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,

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

Composition of morphisms in C/R is well-defined since R is a congruence relation.

[edit] 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).

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 to C/~. This is may be regarded as the “first isomorphism theorem” for functors.

[edit] 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.