Quotient category

In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of (small) categories, analogous to a quotient group or quotient space, but in the categorical setting.

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.

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).

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.
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.

References

Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. 5 (Second ed.). Springer-Verlag.

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Quotient category

Definition

Properties

Examples

See also

References