Suvarna Garge (Editor)

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.

Contents

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 RX,Y on Hom(X,Y), such that the equivalence relations respect composition of morphisms. That is, if

f 1 , f 2 : X → Y

are related in Hom(X, Y) and

g 1 , g 2 : Y → 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,

H o m C / R ( X , Y ) = H o m 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

    Quotient category Wikipedia
    (Text) CC BY-SA


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