User:Steliosx/Sandbox

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
 * $$f_1,f_2 : X \to Y\,$$

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).

That is:
 * $$\text{If } f_1 R_{X,Y} f_2 \text{ and } g_1 R_{Y,Z}\, g_2 \text{ then } g_1f_1 R_{X,Z}\, g_2f_2$$.

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).

Every functor F : C &rarr; D determines a congruence on C by saying f ~ g iff F(f) = F(g). The functor F then factors through the quotient functor to C/~. This is may be regarded as the &ldquo;first isomorphism theorem&rdquo; for functors.

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.