Quotient category

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:$f\_1,f\_2\; :\; X\; o\; Y,$are related in Hom("X", "Y") and:$g\_1,g\_2\; :\; Y\; o\; Z,$are related in Hom("Y", "Z") then "g"₁"f"₁ and "g"₂"f"₂ 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,:$mathrm\{Hom\}\_\{mathcal\; C/mathcal\; R\}(X,Y)\; =\; mathrm\{Hom\}\_\{mathcal\; C\}(X,Y)/R\_\{X,Y\}.$

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

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 $varinjlim\; mathrm\{Hom\}\_A(X\text{'},\; Y/Y\text{'})$ where the limit is over subobjects $X\text{'}\; subseteq\; X$ and $Y\text{'}\; subseteq\; Y$ such that $X/X\text{'},\; Y\text{'}\; in\; B$. Then "A/B" is an Abelian category, and there is a canonical functor $Q\; colon\; A\; o\; A/B$. This Abelian quotient satisfies the universal property that if "C" is any other Abelian category, and $F\; colon\; A\; o\; C$ 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 $overline\{F\}\; colon\; A/B\; o\; C$ such that $F\; =\; overline\{F\}\; circ\; Q$. (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).

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.

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.

ee also

*Quotient object

References

* Gabriel, Peter, "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.