In

_{''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.

^{op}.

_{1}'' = ''fg_{2}'' implies ''g_{1}'' = ''g_{2}'' for all morphisms ''g''_{1}, ''g_{2}'' : ''x'' → ''a''.
* an ''_{1}f'' = ''g_{2}f'' implies ''g_{1}'' = ''g_{2}'' for all morphisms ''g_{1}'', ''g_{2}'' : ''b'' → ''x''.
* a ''_{''b''}.
* a ''_{''a''}.
* an ''_{''b''} and ''gf'' = 1_{''a''}.
* an ''

_{''K''}, the hom-sets hom(''a'', ''b'') are not just sets but actually

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a category (sometimes called an abstract category to distinguish it from a concrete category
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

) 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 setsIn the mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is ...

, whose objects are sets and whose arrows are functions
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

.
''Category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

'' 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 that studies , 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 , is mostly concerned with those that are relevant to ...

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
Programming language theory (PLT) is a branch of computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as pract ...

.
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
*''Equivalent ...

" 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 most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to:
*Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck
Ring may also refer to:
Sounds
* Ri ...

and ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function s ...

s; and Top
A spinning top, or simply a top, is a toy
A toy is an item that is used primarily by children though may also be marketed to adults under certain circumstances. Playing with toys can be an enjoyable means of training young children for li ...

, the category of topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

s and continuous map
In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value o ...

s. All of the preceding categories have the as identity arrows and composition
Composition or Compositions may refer to:
Arts
* Composition (dance), practice and teaching of choreography
* Composition (music), an original piece of music and its creation
*Composition (visual arts)
The term composition means "putting togethe ...

as the associative operation on arrows.
The classic and still much used text on category theory is ''Categories for the Working Mathematician
''Categories for the Working Mathematician'' (''CWM'') is a textbook in category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), ...

'' by Saunders Mac Lane
Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ...

. Other references are given in the References
Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to ''refer to'' the second object. It is called a ''name
...

below. The basic definitions in this article are contained within the first few chapters of any of these books.
Any monoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

.
Definition

There are many equivalent definitions of a category. One commonly used definition is as follows. A category ''C'' consists of * aclass
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differently f ...

ob(''C'') of objects,
* a class hom(''C'') of morphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

s, or arrows, or maps between the objects,
*a domain, or source object class function $\backslash mathrm\backslash colon\; \backslash mathrm(C)\backslash rightarrow\; \backslash mathrm(C)$,
*a codomain, or target object class function $\backslash mathrm\backslash colon\; \backslash mathrm(C)\backslash rightarrow\; \backslash mathrm(C)$,
* 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 $\backslash mathrm(f)\; =\; a$ and $\backslash mathrm(f)\; =\; b$. 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 Validity (logic), valid rule ...

) if ''f'' : ''a'' → ''b'', ''g'' : ''b'' → ''c'' and ''h'' : ''c'' → ''d'' then ''h'' ∘ (''g'' ∘ ''f'') = (''h'' ∘ ''g'') ∘ ''f'', and
* (identity
Identity may refer to:
Social sciences
* Identity (social science), personhood or group affiliation in psychology and sociology
Group expression and affiliation
* Cultural identity, a person's self-affiliation (or categorization by others ...

) for every object ''x'', there exists a morphism 1Small and large categories

A category ''C'' is called small if both ob(''C'') and hom(''C'') are actually sets and notproper class
Proper may refer to:
Mathematics
* Proper map
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and qu ...

es, 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

similar to a monoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

but without requiring closure properties. Large categories on the other hand can be used to create "structures" of algebraic structures.
Examples

Theclass
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differently f ...

of all sets (as objects) together with all function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

s between them (as morphisms), where the composition of morphisms is the usual function composition
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

, 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 Rel or REL may mean:
__NOTOC__ Science and technology
* , a human gene
* the rel descriptor of , see
*REL (''Rassemblement Européen pour la Liberté''), , a defunct French far-right party ...

consists of all sets (as objects) with binary relation
Binary may refer to:
Science and technology
Mathematics
* Binary number
In mathematics and digital electronics
Digital electronics is a field of electronics
The field of electronics is a branch of physics and electrical engineeri ...

s between them (as morphisms). Abstracting from relations
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...

instead of functions yields allegories
As a literary device, an allegory is a narrative in which a character, place, or event is used to deliver a broader message about real-world issues and occurrences. Authors have used allegory throughout history in all forms of art to illustrate ...

, 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 in science is the opposite of :wikt:continuous, continuous: something that is separate; distinct; individual.
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic c ...

. 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

(''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
upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not.
In mathem ...

and any equivalence relation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

can be seen as a small category. Any ordinal number
In set theory
Set theory is the branch of that studies , 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 , is mostly concerned with those that ...

can be seen as a category when viewed as an ordered set
Image:Hasse diagram of powerset of 3.svg, upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparab ...

.
Any monoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

(any algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

with a single associative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

binary operation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

and an identity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

) 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
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

.
A groupoid
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups, group action
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s and equivalence relation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s. 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 $\backslash pi\_1(X,x\_0)$ is the fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence class
In mathematics, when the elements of some set (mathematics), set have a notion of equivalence (formalized ...

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 $\backslash pi\_1(X,x\_0)$, then the set we get has only the structure of groupoid (which is called as the fundamental groupoidIn algebraic topology
250px, A torus, one of the most frequently studied objects in algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebr ...

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 (discrete mathematics), graph that is made up of a set of Vertex (graph theory), vertices connected by directed Edge (graph theory), edges often called ...

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 categoryIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

'' generated by the graph.
The class of all preordered sets with monotonic function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

, 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
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

s as morphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

s and function composition
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s and their group homomorphisms, is a full subcategory
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of Grp, and the prototype of an abelian category
In mathematics, an abelian category is a Category (mathematics), category in which morphisms and Object (category theory), objects can be added and in which Kernel (category theory), kernels and cokernels exist and have desirable properties. The mo ...

. Other examples of concrete categories are given by the following table.
Fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in English in the Commonwealth of Nations, Commonwealth English: fibre bundle) is a Space (mathematics), space that is ''locally'' a product space, but ''globally'' may have a dif ...

s with bundle map
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s between them form a concrete category.
The category Cat
The cat (''Felis catus'') is a domestic
Domestic may refer to:
In the home
* Anything relating to the human home
A home, or domicile, is a space used as a permanent or semi-permanent residence for an individual, group or famil ...

consists of all small categories, with functor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

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''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

Amorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

''f'' : ''a'' → ''b'' is called
* a ''monomorphism
In the context of abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, rin ...

'' (or ''monic'') if it is left-cancellable, i.e. ''fgepimorphism
220px
In category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labe ...

'' (or ''epic'') if it is right-cancellable, i.e. ''gbimorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

'' if it is both a monomorphism and an epimorphism.
* a ''retraction
In academic publishing
Academic publishing is the subfield of publishing which distributes academic research and scholarship. Most academic work is published in academic journal articles, books or thesis' form. The part of academic written o ...

'' if it has a right inverse, i.e. if there exists a morphism ''g'' : ''b'' → ''a'' with ''fg'' = 1section
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 sign ...

'' if it has a left inverse, i.e. if there exists a morphism ''g'' : ''b'' → ''a'' with ''gf'' = 1isomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

'' if it has an inverse, i.e. if there exists a morphism ''g'' : ''b'' → ''a'' with ''fg'' = 1endomorphism
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 group ...

'' if ''a'' = ''b''. The class of endomorphisms of ''a'' is denoted end(''a'').
* an ''automorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

'' 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 diagram
350px, The commutative diagram used in the proof of the five lemma.
In mathematics, and especially in category theory, a commutative diagram is a Diagram (category theory), diagram such that all directed paths in the diagram with the same start an ...

s, where the objects are represented as points and the morphisms as arrows.
Types of categories

* In many categories, e.g. Ab or Vectabelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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
* Produc ...

and coproduct
In category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...

s, it is called an additive category
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

. 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 learnin ...

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 (mathematics), category in which morphisms and Object (category theory), objects can be added and in which Kernel (category theory), kernels and cokernels exist and have desirable properties. The mo ...

. 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 (music)
In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre (music), genre of music, or the harmonies that can be made using a particular ...

exist in it. The categories of sets, abelian groups and topological spaces are complete.
* A category is called cartesian closed
In category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dir ...

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 orderIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s with Scott-continuous functions.
* A topos
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

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 categoryIn category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...

* Higher category theory In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...

* Quantaloid
* Table of mathematical symbols
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that ...

Notes

References

* (now free on-line edition,GNU FDL
The GNU Free Documentation License (GNU FDL or simply GFDL) is a copyleft license for free documentation, designed by the Free Software Foundation (FSF) for the GNU Project. It is similar to the GNU General Public License, giving readers the ri ...

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