Quotient category

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In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but in the categorical setting.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

Category (mathematics) algebraic structure

In mathematics, a category is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions.

In mathematics, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.

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

In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes for the relation.

Equivalence relation reflexive, symmetric and transitive relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c:

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 categoryC/R as the category whose objects are those of C and whose morphisms are equivalence classes of morphisms in C. That is,

Equivalence class mathematical concept

In mathematics, when the elements of some set S have a notion of equivalence defined on them, then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if and only if a and b are equivalent.

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

In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well-defined or ambiguous. A function is well-defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance if f takes real numbers as input, and if f(0.5) does not equal f(1/2) then f is not well-defined. The term well-defined is also used to indicate whether a logical statement is unambiguous.

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).

In mathematics, a functor is a map between categories. Functors were first considered in algebraic topology, where algebraic objects are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.

Every functor F : CD determines a congruence on C by saying f ~ g iff F(f) = F(g). The functor F then factors through the quotient functor CC/~ in a unique manner. This may be regarded as the first isomorphism theorem for functors.

Examples

Quotients of additive categories modulo ideals

If C is an additive category and we require the congruence relation ~ on C to be additive (i.e. if f1, f2, g1 and g2 are morphisms from X to Y with f1 ~ f2 and g1 ~g2, then f1 + f2 ~ g1 + g2), then the quotient category C/~ will also be additive, and the quotient functor CC/~ will be an additive functor.

The concept of an additive congruence relation is equivalent to the concept of a two-sided ideal of morphisms: for any two objects X and Y we are given an additive subgroup I(X,Y) of HomC(X, Y) such that for all fI(X,Y), g HomC(Y, Z) and h HomC(W, X), we have gfI(X,Z) and fhI(W,Y). Two morphisms in HomC(X, Y) are congruent iff their difference is in I(X,Y).

Every unital ring may be viewed as an additive category with a single object, and the quotient of additive categories defined above coincides in this case with the notion of a quotient ring modulo a two-sided ideal.

Localization of a category

The localization of a category introduces new morphisms to turn several of the original category's morphisms into isomorphisms. This tends to increase the number of morphisms between objects, rather than decrease it as in the case of quotient categories. But in both constructions it often happens that two objects become isomorphic that weren't isomorphic in the original category.

Serre quotients of abelian categories

The Serre quotient of an abelian category by a Serre subcategory is a new abelian category which is similar to a quotient category but also in many cases has the character of a localization of the category.

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