Quotient Category - Definition

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

are related in Hom(X, Y) and

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,

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 where the limit is over subobjects and such that . Then A/B is an Abelian category, and there is a canonical functor . This Abelian quotient satisfies the universal property that if C is any other Abelian category, and is an exact functor such that F(b) is a zero object of C for each, then there is a unique exact functor such that . (See .)

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