Quotient category

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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a quotient category is a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories'' (Aristotle) *Category (Kant) ...
obtained from another one by identifying sets of
morphism In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
s. Formally, it is a
quotient object In category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topolo ...
in the category of (locally small) categories, analogous to a
quotient group A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factor ...
or quotient space, but in the categorical setting.

# Definition

Let ''C'' be a category. A ''
congruence relation In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group (mathematics), group, ring (mathematics), ring, or vector space) that is compatible with the structure in the ...
'' ''R'' on ''C'' is given by: for each pair of objects ''X'', ''Y'' in ''C'', an
equivalence relation In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
''R''''X'',''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 ''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 class In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
es of morphisms in ''C''. That is, :$\mathrm_\left(X,Y\right) = \mathrm_\left(X,Y\right)/R_.$ Composition of morphisms in ''C''/''R'' is
well-defined In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
since ''R'' is a congruence relation.

# Properties

There is a natural quotient
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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 ''C'' → ''C''/~ in a unique manner. This may be regarded as the "first isomorphism theorem" for categories.

# Examples

* Monoids and group (mathematics), groups 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 A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factor ...
. * 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. *Let ''k'' be a Field (mathematics), field and consider the abelian category Mod(''k'') of all Vector space, vector spaces over ''k'' with ''k''-linear maps as morphisms. To "kill" all finite-dimensional spaces, we can call two linear maps ''f'',''g'' : ''X'' → ''Y'' congruent iff their difference has finite-dimensional image. In the resulting quotient category, all finite-dimensional vector spaces are isomorphic to 0. [This is actually an example of a quotient of additive categories, see below.]

# Related concepts

## 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 ''f''1, ''f''2, ''g''1 and ''g''2 are morphisms from ''X'' to ''Y'' with ''f''1 ~ ''f''2 and ''g''1 ~''g''2, then ''f''1 + ''g''1 ~ ''f''2 + ''g''2), then the quotient category ''C''/~ will also be additive, and the quotient functor ''C'' → ''C''/~ 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 Hom''C''(''X'', ''Y'') such that for all ''f'' ∈ ''I''(''X'',''Y''), ''g'' ∈ Hom''C''(''Y'', ''Z'') and ''h''∈ Hom''C''(''W'', ''X''), we have ''gf'' ∈ ''I''(''X'',''Z'') and ''fh'' ∈ ''I''(''W'',''Y''). Two morphisms in Hom''C''(''X'', ''Y'') are congruent iff their difference is in ''I''(''X'',''Y''). Every unital ring (mathematics), 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 Quotient of an abelian category, 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.

# References

* {{Category theory Category theory Quotient objects, Category