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*([[identidad]]) para todo objeto A en ob('''C''') existe una flecha en '''C'''(A,A) comunmente denotada 1<sub>A</sub> tal que para toda flecha f en '''C'''(A;B) f=1<sub>B</sub>of y f=fo1<sub>A</sub>
 
De estos axiomas se puede deducir facilmente que existe una única flecha identidad para cada objeto.
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==Historia==
such that the following axioms hold:
La noción de categoría, y en general, las primeras nociones de teoría de categorías, aparecieron por primera vez en 1945 en un artículo de [[Samuel Eilenberg]] y [[Saunders Mac Lane]] llamado "General Theory of Natural Equivalences" (''Teoría general de las equivalencias naturales'').<ref>Sica (2006), p. 223; Awodey (2006), p. 1.</ref>
* ([[associativity]]) if ''f'' : ''a'' → ''b'', ''g'' : ''b'' → ''c'' and ''h'' : ''c'' → ''d'' then ''h'' o (''g'' o ''f'') = (''h'' o ''g'') o ''f'', and
* ([[identity (mathematics)|identity]]) for every object ''x'', there exists a morphism 1<sub>''x''</sub> : ''x'' → ''x'' (some authors write ''id''<sub>''x''</sub>) called the ''identity morphism for x'', such that for every morphism ''f'' : ''a'' → ''b'', we have 1<sub>''b''</sub> o ''f'' = ''f'' = ''f'' o 1<sub>''a''</sub>.
 
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.
 
<!--==Categorías pequeñas==
==History==
Category theory first appeared in a paper entitled "General Theory of Natural Equivalences", written by [[Samuel Eilenberg]] and [[Saunders Mac Lane]] in 1945.<ref>Sica (2006), p. 223; Awodey (2006), p. 1.</ref>
 
==Small and large categories==
A category ''C'' is called '''small''' if both ob(''C'') and hom(''C'') are actually [[Set (mathematics)|sets]] and not [[proper class]]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.
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==Ejemplos==
*La categoría '''Con''' es aquella cuyos objetos son todos los conjuntos y si A y B son conjuntos, entonces '''Con'''(A,B) es el conjunto de funciones con dominio A y codominio B. Ésta es la categoría más comúnmente usada en matemáticas.
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==Examples==
The [[class (set theory)|class]] of all sets together with all [[function (mathematics)|function]]s between sets, where composition is the usual [[function composition]], forms a large category, [[category of sets|'''Set''']].<ref>Jacobson (2009), p. 11, ex. 1.</ref> It is the most basic and the most commonly used category in mathematics. The category [[category of relations|'''Rel''']] consists of all [[Set (mathematics)|sets]], with [[binary relation]]s as morphisms. Abstracting from [[Relation (mathematics)|relations]] instead of functions yields [[Allegory (category theory)|allegories]] instead of categories.
 
* A [[topos]] 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]]
* [[Higher category theory]]
* [[Table of mathematical symbols]]
 
 
==Notes==
 
 
 
 
{{Portal|Category theory}}
 
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==Véase también==
* [[Categoría enriquecida]]
* [[Teoría de categorías de orden superior]]
 
==NotesNotas==
<references/>
 
==ReferencesReferencias==
* {{Citation| last1=Adámek |first1=Jiří |last2=Herrlich |first2=Horst |last3=Strecker |first3=George E. |date=1990 |title=Abstract and Concrete Categories |publisher=John Wiley & Sons|isbn=0-471-60922-6|url=http://katmat.math.uni-bremen.de/acc/acc.pdf}} (now free on-line edition, [[GNU Free Documentation License|GNU FDL]]).
* {{Citation| last1=Asperti| first1=Andrea| last2=Longo| first2=Giuseppe| date=1991| title=Categories, Types and Structures| publisher=MIT Press| url=ftp://ftp.di.ens.fr/pub/users/longo/CategTypesStructures/book.pdf| isbn=0262011255}}.
 
 
[[Categoría:Teoría de categorías]]
{{Portal|Category theory}}
 
[[Category:Category theory| ]]
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[[bg:Категория (математика)]]
[[cs:Kategorie (matematika)]]
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