Definition
Group-like structures | |||||
Totality* | Associativity | Identity | Inverses | Commutativity | |
---|---|---|---|---|---|
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 homC(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 1x : x → x (some authors write idx) called the identity morphism for x, such that for every morphism f : a → b, we have 1b ∘ f = f = f ∘ 1a.
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.
Read more about this topic: Category (mathematics)
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