User:Steliosx/Sandbox

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Contents 1 Definition 2 Properties 3 Examples 4 See also

[edit] 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

are related in Hom(X, Y) and

are related in Hom(Y, Z) then g₁f₁ and g₂f₂ are related in Hom(X, Z).

That is:

.

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.

[edit] 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 → 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 “first isomorphism theorem” for functors.

[edit] 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.

[edit] See also

• Quotient object

[[Category:Category theory]]