The set M is the disjoint union of sets of the form [A,B], where A and B are elements of A; if α is a member of [A,B], A is the domain of α, B is the codomain of α, and α is said to be a morphism from A to B. For each triple (A,B,C) of elements of A there is a dyadic operation ◦ from the Cartesian product [B,C] × [A,B]
to [A,C]. The image β◦α of the ordered pair (β,α) is the composition of β with α; the composition operation is associative. In addition, when the composition is defined there is an identity morphism for each A in A.
Examples of categories include the set of groups and homomorphisms on groups, and the set of rings and homomorphisms on rings. See functor.
cat·e·go·ry / ˈkatəˌgôrē/ • n. (pl. -ries) 1. a class or division of people or things regarded as having particular shared characteristics: five categories of intelligence.2. Philos. one of a possibly exhaustive set of classes among which all things might be distributed. ∎ one of the a priori conceptions applied by the mind to sense impressions. ∎ a relatively fundamental philosophical concept.DERIVATIVES: cat·e·go·ri·al / ˌkatəˈgôrēəl/ adj.