Editing Category (mathematics) (section)

==Definition==
There are many equivalent definitions of a category.<ref>{{harvnb|Barr|Wells|2005|loc=Chapter 1}}</ref> One commonly used definition is as follows.  A '''category''' ''C'' consists of
* a [[Class (set theory)|class]] ob(''C'') of '''[[mathematical_object|object]]s''',
* a class mor(''C'') of '''[[morphism]]s''' or '''arrows''',
*a '''domain''' or '''source''' class function dom: mor(C) → ob(C),
*a '''codomain''' or '''target''' class function cod: mor(C) → ob(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'''. Here hom(''a'', ''b'') denotes the subclass of morphisms ''f'' in  mor(''C'') such that dom(f) = ''a'' and cod(f) = ''b''.  Morphisms in this subclass are written ''f'' : ''a'' → ''b'', and the composite of ''f'' : ''a'' → ''b'' and ''g'' : ''b'' → ''c'' is often written as ''g'' ∘ ''f'' or ''gf''. 
such that the following axioms hold:
* the ''[[associativity|associative law]]'': if ''f'' : ''a'' → ''b'', ''g'' : ''b'' → ''c'' and ''h'' : ''c'' → ''d'' then ''h'' ∘ (''g'' ∘ ''f'') = (''h'' ∘ ''g'') ∘ ''f'', and
* the '''([[identity (mathematics)|left and right unit laws]])''': 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 every morphism ''f'' : ''a'' → ''x'' satisfies 1<sub>''x''</sub> ∘ ''f'' = ''f'', and every morphism ''g'' : ''x'' → ''b'' satisfies ''g'' ∘ 1<sub>''x''</sub> = ''g''.

We write ''f'': ''a'' → ''b'', and we say "''f'' is a morphism from ''a'' to ''b''". We write hom(''a'', ''b'') (or hom<sub>''C''</sub>(''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''.<ref>Some authors write Mor(''a'', ''b'') or simply ''C''(''a'', ''b'') instead.</ref> 

Some authors write the composite of morphisms in "diagrammatic order", writing ''f;g'' or ''fg'' instead of ''g'' ∘ ''f''.

From these axioms, one can prove that there is exactly one identity morphism for every object.  Often the map assigning each object its identity morphism is treated as an extra part of the structure of a category, namely a class function i: ob(C) → mor(C).  
Some authors use a slight variant of the definition in which each object is identified with the corresponding identity morphism. This stems from the idea that the fundamental data of categories are morphisms and not objects. In fact, categories can be defined without reference to objects at all using a partial binary operation with additional properties.