category A collection of objects A, together with a related set of morphisms M. An object is a generalization of a set and a morphism is a generalization of a function that maps between sets.

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.

category A key aspect of the measurement process involves placing observations of measurements into groups or categories on the basis of unequivocally shared features. Hence a category is a homogeneous grouping of data. For example, the variable ‘sex’ would have two categories, male and female; the variable ‘social class’ might have three categories, upper, middle, and working. In the former case the rule for assigning cases or observations to the appropriate category is relatively simple. In the latter, the rules would be more complex, and dependent upon the class theory being used. See also CODING; RULES OF CORRESPONDENCE.

category XV. — F. catégorie or its source, late L. catēgoria — Gr. katēgoríā accusation, etc., f. katḗgoros accuser, etc., f. CATA- 4 + agorā́ assembly; see -Y 3.
So categoric XVII, categorical XVI. — F. catégorique or late L. catēgoricus.

category See rank.