Category (mathematics)

Definition

	Totality*	Associativity	Identity	Inverses	Commutativity
Group-like structures
Magma	Yes	No	No	No	No
Semigroup	Yes	Yes	No	No	No
Monoid	Yes	Yes	Yes	No	No
Group	Yes	Yes	Yes	Yes	No
Abelian Group	Yes	Yes	Yes	Yes	Yes
Loop	Yes	No	Yes	Yes	No
Quasigroup	Yes	No	No	Yes	No
Groupoid	No	Yes	Yes	Yes	No
Category	No	Yes	Yes	No	No
Semicategory	No	Yes	No	No	No

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. Each morphism f has a unique source object a and target object b where a and b are in ob(C). 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.)
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.)

such that the following axioms hold:

(associativity) 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 for every morphism f : a → b, we have 1_b ∘ f = f = f ∘ 1_a.

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.

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Category (mathematics) - Definition

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