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.

Read more about this topic: Category (mathematics)

Famous quotes containing the word definition:

“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)

“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)

“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)

Category (mathematics) - Definition

Famous quotes containing the word definition: