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In mathematics, a category (sometimes called an abstract category to distinguish it from a
concrete category In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, ''see Relative concreteness below''). This functor makes it possible to think of the objects of t ...
) 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 In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...
, whose objects are sets and whose arrows are functions. '' Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the notion of category provides a fundamental and abstract way to describe mathematical entities and their relationships. In addition to formalizing mathematics, category theory is also used to formalize many other systems in computer science, such as the
semantics of programming languages In programming language theory, semantics is the rigorous mathematical study of the meaning of programming languages. Semantics assigns computational meaning to valid string (computer science), strings in a programming language syntax. Semantic ...
. Two categories are the same if they have the same collection of objects, the same collection of arrows, and the same associative method of composing any pair of arrows. Two ''different'' categories may also be considered "
equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry * Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equiva ...
" for purposes of category theory, even if they do not have precisely the same structure. Well-known categories are denoted by a short capitalized word or abbreviation in bold or italics: examples include Set, the category of sets and set functions; Ring, the category of
rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
and
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preser ...
s; and Top, the category of
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s and
continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
s. All of the preceding categories have the identity map as identity arrows and
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
as the associative operation on arrows. The classic and still much used text on category theory is '' Categories for the Working Mathematician'' by
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville ...
. Other references are given in the References below. The basic definitions in this article are contained within the first few chapters of any of these books. Any
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...
can be understood as a special sort of category (with a single object whose self-morphisms are represented by the elements of the monoid), and so can any preorder.


Definition

There are many equivalent definitions of a category. One commonly used definition is as follows. A category ''C'' consists of * a class ob(''C'') of objects, * a class hom(''C'') of morphisms, or arrows, or maps between the objects, *a domain, or source object class function \mathrm\colon \mathrm(C)\rightarrow \mathrm(C) , *a codomain, or target object class function \mathrm\colon \mathrm(C)\rightarrow \mathrm(C) , * for every three objects ''a'', ''b'' and ''c'', a binary operation hom(''a'', ''b'') × hom(''b'', ''c'') → hom(''a'', ''c'') called ''composition of morphisms''; the composition of ''f'' : ''a'' → ''b'' and ''g'' : ''b'' → ''c'' is written as ''g'' ∘ ''f'' or ''gf''. (Some authors use "diagrammatic order", writing ''f;g'' or ''fg''). Note: Here hom(''a'', ''b'') denotes the subclass of morphisms ''f'' in hom(''C'') such that \mathrm(f) = a and \mathrm(f) = b. Such morphisms are often written as ''f'' : ''a'' → ''b''. such that the following axioms hold: * (
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
) if ''f'' : ''a'' → ''b'', ''g'' : ''b'' → ''c'' and ''h'' : ''c'' → ''d'' then ''h'' ∘ (''g'' ∘ ''f'') = (''h'' ∘ ''g'') ∘ ''f'', and * ( identity) for every object ''x'', there exists a morphism 1''x'' : ''x'' → ''x'' (some authors write ''id''''x'') called the ''identity morphism for x'', such that every morphism ''f'' : ''a'' → ''x'' satisfies 1''x'' ∘ ''f'' = ''f'', and every morphism ''g'' : ''x'' → ''b'' satisfies ''g'' ∘ 1''x'' = ''g''. We write ''f'': ''a'' → ''b'', and we say "''f'' is a morphism from ''a'' to ''b''". We write hom(''a'', ''b'') (or hom''C''(''a'', ''b'') when there may be confusion about to which category hom(''a'', ''b'') refers) to denote the hom-class of all morphisms from ''a'' to ''b''.Some authors write Mor(''a'', ''b'') or simply ''C''(''a'', ''b'') instead. From these axioms, one can prove that there is exactly one identity morphism for every object. Some authors use a slight variation of the definition in which each object is identified with the corresponding identity morphism.


Small and large categories

A category ''C'' is called small if both ob(''C'') and hom(''C'') are actually sets and not proper classes, and large otherwise. A locally small category is a category such that for all objects ''a'' and ''b'', the hom-class hom(''a'', ''b'') is a set, called a homset. Many important categories in mathematics (such as the category of sets), although not small, are at least locally small. Since, in small categories, the objects form a set, a small category can be viewed as an algebraic structure similar to a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...
but without requiring closure properties. Large categories on the other hand can be used to create "structures" of algebraic structures.


Examples

The class of all sets (as objects) together with all
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
s between them (as morphisms), where the composition of morphisms is the usual function composition, forms a large category, Set. It is the most basic and the most commonly used category in mathematics. The category
Rel Rel or REL may mean: __NOTOC__ Science and technology * REL, a human gene * the rel descriptor of stereochemistry, see Relative configuration *REL (''Rassemblement Européen pour la Liberté''), European Rally for Liberty, a defunct French far-ri ...
consists of all sets (as objects) with binary relations between them (as morphisms). Abstracting from relations instead of functions yields
allegories As a literary device or artistic form, an allegory is a narrative or visual representation in which a character, place, or event can be interpreted to represent a hidden meaning with moral or political significance. Authors have used allegory th ...
, a special class of categories. Any class can be viewed as a category whose only morphisms are the identity morphisms. Such categories are called
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
. For any given set ''I'', the ''discrete category on I'' is the small category that has the elements of ''I'' as objects and only the identity morphisms as morphisms. Discrete categories are the simplest kind of category. Any preordered set (''P'', ≤) forms a small category, where the objects are the members of ''P'', the morphisms are arrows pointing from ''x'' to ''y'' when ''x'' ≤ ''y''. Furthermore, if ''≤'' is antisymmetric, there can be at most one morphism between any two objects. The existence of identity morphisms and the composability of the morphisms are guaranteed by the reflexivity and the transitivity of the preorder. By the same argument, any
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
and any equivalence relation can be seen as a small category. Any ordinal number can be seen as a category when viewed as an
ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
. Any
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...
(any algebraic structure with a single associative binary operation and an
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
) forms a small category with a single object ''x''. (Here, ''x'' is any fixed set.) The morphisms from ''x'' to ''x'' are precisely the elements of the monoid, the identity morphism of ''x'' is the identity of the monoid, and the categorical composition of morphisms is given by the monoid operation. Several definitions and theorems about monoids may be generalized for categories. Similarly any
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
can be seen as a category with a single object in which every morphism is ''invertible'', that is, for every morphism ''f'' there is a morphism ''g'' that is both left and right inverse to ''f'' under composition. A morphism that is invertible in this sense is called an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. A
groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *'' Group'' with a partial func ...
is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups,
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
s and equivalence relations. Actually, in the view of category the only difference between groupoid and group is that a groupoid may have more than one object but the group must have only one. Consider a topological space ''X'' and fix a base point x_0 of ''X'', then \pi_1(X,x_0) is the fundamental group of the topological space ''X'' and the base point x_0, and as a set it has the structure of group; if then let the base point x_0 runs over all points of ''X'', and take the union of all \pi_1(X,x_0), then the set we get has only the structure of groupoid (which is called as the fundamental groupoid of ''X''): two loops (under equivalence relation of homotopy) may not have the same base point so they cannot multiply with each other. In the language of category, this means here two morphisms may not have the same source object (or target object, because in this case for any morphism the source object and the target object are same: the base point) so they can not compose with each other. Any
directed graph In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pa ...
generates a small category: the objects are the vertices of the graph, and the morphisms are the paths in the graph (augmented with loops as needed) where composition of morphisms is concatenation of paths. Such a category is called the ''
free category In mathematics, the free category or path category generated by a directed graph or quiver is the category that results from freely concatenating arrows together, whenever the target of one arrow is the source of the next. More precisely, the objec ...
'' generated by the graph. The class of all preordered sets with
monotonic function In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
s as morphisms forms a category,
Ord Ord or ORD may refer to: Places * Ord of Caithness, landform in north-east Scotland * Ord, Nebraska, USA * Ord, Northumberland, England * Muir of Ord, village in Highland, Scotland * Ord, Skye, a place near Tarskavaig * Ord River, Western Austral ...
. It is a
concrete category In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, ''see Relative concreteness below''). This functor makes it possible to think of the objects of t ...
, i.e. a category obtained by adding some type of structure onto Set, and requiring that morphisms are functions that respect this added structure. The class of all groups with
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
s as morphisms and function composition as the composition operation forms a large category, Grp. Like Ord, Grp is a concrete category. The category Ab, consisting of all
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s and their group homomorphisms, is a full subcategory of Grp, and the prototype of an
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
. Other examples of concrete categories are given by the following table. Fiber bundles with
bundle map In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There ...
s between them form a concrete category. The category
Cat The cat (''Felis catus'') is a domestic species of small carnivorous mammal. It is the only domesticated species in the family Felidae and is commonly referred to as the domestic cat or house cat to distinguish it from the wild members of ...
consists of all small categories, with
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
s between them as morphisms.


Construction of new categories


Dual category

Any category ''C'' can itself be considered as a new category in a different way: the objects are the same as those in the original category but the arrows are those of the original category reversed. This is called the ''dual'' or ''opposite category'' and is denoted ''C''op.


Product categories

If ''C'' and ''D'' are categories, one can form the ''product category'' ''C'' × ''D'': the objects are pairs consisting of one object from ''C'' and one from ''D'', and the morphisms are also pairs, consisting of one morphism in ''C'' and one in ''D''. Such pairs can be composed componentwise.


Types of morphisms

A morphism ''f'' : ''a'' → ''b'' is called * a ''
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism ...
'' (or ''monic'') if it is left-cancellable, i.e. ''fg1'' = ''fg2'' implies ''g1'' = ''g2'' for all morphisms ''g''1, ''g2'' : ''x'' → ''a''. * an ''
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f ...
'' (or ''epic'') if it is right-cancellable, i.e. ''g1f'' = ''g2f'' implies ''g1'' = ''g2'' for all morphisms ''g1'', ''g2'' : ''b'' → ''x''. * a ''
bimorphism In mathematics, particularly in category theory, 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 ...
'' if it is both a monomorphism and an epimorphism. * a '' retraction'' if it has a right inverse, i.e. if there exists a morphism ''g'' : ''b'' → ''a'' with ''fg'' = 1''b''. * a ''
section Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...
'' if it has a left inverse, i.e. if there exists a morphism ''g'' : ''b'' → ''a'' with ''gf'' = 1''a''. * an ''
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
'' if it has an inverse, i.e. if there exists a morphism ''g'' : ''b'' → ''a'' with ''fg'' = 1''b'' and ''gf'' = 1''a''. * an ''
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
'' if ''a'' = ''b''. The class of endomorphisms of ''a'' is denoted end(''a''). * an '' automorphism'' if ''f'' is both an endomorphism and an isomorphism. The class of automorphisms of ''a'' is denoted aut(''a''). Every retraction is an epimorphism. Every section is a monomorphism. The following three statements are equivalent: * ''f'' is a monomorphism and a retraction; * ''f'' is an epimorphism and a section; * ''f'' is an isomorphism. Relations among morphisms (such as ''fg'' = ''h'') can most conveniently be represented with commutative diagrams, where the objects are represented as points and the morphisms as arrows.


Types of categories

* In many categories, e.g. Ab or Vect''K'', the hom-sets hom(''a'', ''b'') are not just sets but actually
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s, and the composition of morphisms is compatible with these group structures; i.e. is bilinear. Such a category is called preadditive. If, furthermore, the category has all finite
products Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
and
coproduct In category theory, the coproduct, or categorical sum, is a 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 coproduc ...
s, it is called an additive category. If all morphisms have a
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...
and a
cokernel The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the nam ...
, and all epimorphisms are cokernels and all monomorphisms are kernels, then we speak of an
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
. A typical example of an abelian category is the category of abelian groups. * A category is called complete if all small
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
exist in it. The categories of sets, abelian groups and topological spaces are complete. * A category is called cartesian closed if it has finite direct products and a morphism defined on a finite product can always be represented by a morphism defined on just one of the factors. Examples include Set and CPO, the category of
complete partial order In mathematics, the phrase complete partial order is variously used to refer to at least three similar, but distinct, classes of partially ordered sets, characterized by particular completeness properties. Complete partial orders play a central rol ...
s with Scott-continuous functions. * A
topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
is a certain type of cartesian closed category in which all of mathematics can be formulated (just like classically all of mathematics is formulated in the category of sets). A topos can also be used to represent a logical theory.


See also

*
Enriched category In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the ho ...
*
Higher category theory In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher cate ...
* Quantaloid *
Table of mathematical symbols A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula ...


Notes


References

* (now free on-line edition, GNU FDL). * . * . *. * . * * . * . * . * . * . * . * {{Authority control *