Quotient category

For the quotient of an abelian category by a Serre subcategory, see Quotient of an abelian category.

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

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, g₁f₂, g₂f₁ 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.

See also

• Subobject
• Localization of a category

References

• Mac Lane, Saunders (1998) Categories for the Working Mathematician. 2nd ed. (Graduate Texts in Mathematics 5). Springer-Verlag.