|n.||1.||(Logic.) One of the highest classes to which the objects of knowledge or thought can be reduced, and by which they can be arranged in a system; an ultimate or undecomposable conception; a predicament.|
The categories or predicaments - the former a Greek word, the latter its literal translation in the Latin language - were intended by Aristotle and his followers as an enumeration of all things capable of being named; an enumeration by the summa genera i.e., the most extensive classes into which things could be distributed.
|2.||Class; also, state, condition, or predicament; |
|Noun||1.||category - a collection of things sharing a common attribute; "there are two classes of detergents"|
|2.||category - a general concept that marks divisions or coordinations in a conceptual scheme|
|(theory)||category - A category K is a collection of objects, obj(K), and
a collection of morphisms (or "arrows"), mor(K) such that|
1. Each morphism f has a "typing" on a pair of objects A, B written f:A->B. This is read 'f is a morphism from A to B'. A is the "source" or "domain" of f and B is its "target" or "co-domain".
2. There is a partial function on morphisms called composition and denoted by an infix ring symbol, o. We may form the "composite" g o f : A -> C if we have g:B->C and f:A->B.
3. This composition is associative: h o (g o f) = (h o g) o f.
4. Each object A has an identity morphism id_A:A->A associated with it. This is the identity under composition, shown by the equations id_B o f = f = f o id_A.
In general, the morphisms between two objects need not form a set (to avoid problems with Russell's paradox). An example of a category is the collection of sets where the objects are sets and the morphisms are functions.
Sometimes the composition ring is omitted. The use of capitals for objects and lower case letters for morphisms is widespread but not universal. Variables which refer to categories themselves are usually written in a script font.