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

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.

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

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.

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 *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,

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.

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* : *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 functors.

- Monoids and 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.
- 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 and consider the abelian category Mod(*k*) of all 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.]

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} + *f*_{2} ~ *g*_{1} + *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 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.

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.

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.

In category theory, a branch of mathematics, a **natural transformation** provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.

In mathematics, specifically category theory, **adjunction** is a relationship that two functors may have. Two functors that stand in this relationship are known as **adjoint functors**, one being the **left adjoint** and the other the **right adjoint**. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems, such as the construction of a free group on a set in algebra, or the construction of the Stone-Čech compactification of a topological space in topology.

In mathematics, specifically in category theory, a **preadditive category** is another name for an **Ab-category**, i.e., a category that is enriched over the category of abelian groups, **Ab**. That is, an **Ab-category****C** is a category such that every hom-set Hom(*A*,*B*) in **C** has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas:

In category theory, an **epimorphism** is a morphism *f* : *X* → *Y* that is right-cancellative in the sense that, for all objects *Z*, for all morphisms *g*_{1}, *g*_{2} : *Y* → *Z*,

In mathematics, specifically in category theory, an **additive category** is a preadditive category **C** admitting all finitary biproducts.

In mathematics, specifically in category theory, a **pre-abelian category** is an additive category that has all kernels and cokernels.

In category theory, the **coproduct**, or **categorical sum**, is a category-theoretic construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.

In mathematics, the idea of a **free object** is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure. It also has a formulation in terms of category theory, although this is in yet more abstract terms. Examples include free groups, tensor algebras, or free lattices. Informally, a free object over a set *A* can be thought of as being a "generic" algebraic structure over *A*: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure.

In category theory, an abstract branch of mathematics, an **equivalence of categories** is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

In category theory, a **coequalizer** is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer.

In mathematics, the category **Grp** has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.

In mathematics, the **Grothendieck group** construction in abstract algebra constructs an abelian group from a commutative monoid *M* in the most universal way in the sense that any abelian group containing a homomorphic image of *M* will also contain a homomorphic image of the Grothendieck group of *M*. The Grothendieck group construction takes its name from the more general construction in category theory, introduced by Alexander Grothendieck in his fundamental work of the mid-1950s that resulted in the development of K-theory, which led to his proof of the Grothendieck–Riemann–Roch theorem. This article treats both constructions.

In mathematics, a **triangulated category** is a category together with the additional structure of a "translation functor" and a class of "distinguished triangles". Prominent examples are the derived category of an abelian category and the stable homotopy category of spectra, both of which carry the structure of a triangulated category in a natural fashion. The distinguished triangles generate the long exact sequences of homology; they play a role akin to that of short exact sequences in abelian categories.

This is a glossary of properties and concepts in category theory in mathematics.

In mathematics, **algebraic spaces** form a generalization of the schemes of algebraic geometry, introduced by Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer étale topology. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology.

In mathematics, the **category of rings**, denoted by **Ring**, is the category whose objects are rings and whose morphisms are ring homomorphisms. Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper.

In category theory and homotopy theory the **Burnside category** of a finite group *G* is a category whose objects are finite *G*-sets and whose morphisms are spans of *G*-equivariant maps. It is a categorification of the Burnside ring of *G*.

In algebraic geometry, a **presheaf with transfers** is, roughly, a presheaf that, like cohomology theory, comes with pushforwards, “transfer” maps. Precisely, it is, by definition, a contravariant additive functor from the category of finite correspondences to the category of abelian groups.

- Mac Lane, Saunders (1998).
*Categories for the Working Mathematician*. Graduate Texts in Mathematics.**5**(Second ed.). Springer-Verlag.

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