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Definition

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The statement of the theorem

Group-like structures TotalAssociativeIdentityDivisiblePartial magmaUnneededUnneededUnneededUnneeded SemigroupoidUnneededRequiredUnneededUnneeded Small categoryUnneededRequiredRequiredUnneeded GroupoidUnneededRequiredRequiredRequired MagmaRequiredUnneededUnneededUnneeded QuasigroupRequiredUnneededUnneededRequired Unital magmaRequiredUnneededRequiredUnneeded LoopRequiredUnneededRequiredRequired SemigroupRequiredRequiredUnneededUnneeded Associative quasigroupRequiredRequiredUnneededRequired MonoidRequiredRequiredRequiredUnneeded GroupRequiredRequiredRequiredRequired There are many equivalent definitions of a category. One commonly used definition is as follows. A category C{\mathcal {C}} consists of - a class ob⁡(C)\operatorname {ob} ({\mathcal {C}}) of objects, - a class mor⁡(C)\operatorname {mor} ({\mathcal {C}}) of morphisms or arrows, - a domain or source class function dom⁡:mor⁡(C)→ob⁡(C)\operatorname {dom} :\operatorname {mor} ({\mathcal {C}})\to \operatorname {ob} ({\mathcal {C}}) , - a codomain or target class function cod⁡:mor⁡(C)→ob⁡(C)\operatorname {cod} :\operatorname {mor} ({\mathcal {C}})\to \operatorname {ob} ({\mathcal {C}}) , - for every three objects a,b,ca,b,c , a binary operation hom⁡(a,b)×hom⁡(b,c)→hom⁡(a,c)\operatorname {hom} (a,b)\times \operatorname {hom} (b,c)\to \operatorname {hom} (a,c) called composition of morphisms. Here hom⁡(a,b)\operatorname {hom} (a,b) denotes the subclass of morphisms ff in mor⁡(C)\operatorname {mor} ({\mathcal {C}}) such that dom⁡(f)=a\operatorname {dom} (f)=a and cod⁡(f)=b\operatorname {cod} (f)=b . Morphisms in this subclass are written f:a→bf:a\to b , and the composite of f:a→bf:a\to b and g:b→cg:b\to c is often written as g∘fg\circ f or gfgf . such that the following axioms hold: - the associative law: if f:a→bf:a\to b , g:b→cg:b\to c and h:c→dh:c\to d then h∘(g∘f)=(h∘g)∘fh\circ (g\circ f)=(h\circ g)\circ f , and - the left and right unit laws: for every object xx , there exists a morphism 1x:x→x1_{x}:x\to x (some authors write id⁡x\operatorname {id} _{x} ) called the identity morphism for xx , such that every morphism f:a→xf:a\to x satisfies 1x∘f=f1_{x}\circ f=f , and every morphism g:x→bg:x\to b satisfies g∘1x=gg\circ 1_{x}=g . We write f:a→bf:a\to b , and we say " ff is a morphism from aa to bb ". We write hom⁡(a,b)\operatorname {hom} (a,b) (or hom⁡C(a,b)\operatorname {hom} _{\mathcal {C}}(a,b) when there may be confusion about to which category hom⁡(a,b)\operatorname {hom} (a,b) refers) to denote the hom-class of all morphisms from aa to bb . Some authors write the composite of morphisms in "diagrammatic order", writing f ;gf\,;g (sometimes with ⨟ ) or fgfg instead of g∘fg\circ 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)i:\operatorname {ob} ({\mathcal {C}})\to \operatorname {mor} ({\mathcal {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. - ^Barr & Wells 2005, Chapter 1 - ^Some authors write Mor⁡(a,b)\operatorname {Mor} (a,b) or simply C(a,b){\mathcal {C}}(a,b) instead. - ^Fong, Brendan; Spivak, David I. (2018). "Seven Sketches in Compositionality: An Invitation to Applied Category Theory". p. 12. arXiv:1803.05316 [math.CT].