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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.

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,

${\mathrm{H}\mathrm{o}\mathrm{m}}_{𝒞/ℛ}(X,Y)={\mathrm{H}\mathrm{o}\mathrm{m}}_{𝒞}(X,Y)/R_{X,Y}.$

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

There is also a notion of taking the quotient of an Abelian category A by a Serre subcategory B. This is done as follows. The objects of A/B are the objects of A. Given two objects X and Y of A, we define the set of morphisms from X to Y in A/B to be $\underrightarrow{\lim }{\mathrm{H}\mathrm{o}\mathrm{m}}_A(X^{\prime },Y/Y^{\prime })$ where the limit is over subobjects $X^{\prime }\subseteq X$ and $Y^{\prime }\subseteq Y$ such that $X/X^{\prime },Y^{\prime }\in B$. Then A/B is an Abelian category, and there is a canonical functor $Q:A\to A/B$. This Abelian quotient satisfies the universal property that if C is any other Abelian category, and $F:A\to C$ is an exact functor such that F(b) is a zero object of C for each $b\in B$, then there is a unique exact functor $\bar{F}:A/B\to C$ such that $F=\bar{F}\circ Q$. (See [Gabriel].)

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 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

• Subobject

References

• Gabriel, Pierre, Des categories abeliennes, Bull. Soc. Math. France 90 (1962), 323-448.
• Mac Lane, Saunders (1998) Categories for the Working Mathematician. 2nd ed. (Graduate Texts in Mathematics 5). Springer-Verlag.