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

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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

$f\_1,f\_2\; :\; X\; \backslash to\; Y\backslash ,$

are related in Hom(X, Y) and

$g\_1,g\_2\; :\; Y\; \backslash to\; Z\backslash ,$

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,

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

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

Properties

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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.

Examples

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• 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

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• Quotient object

Category theory