meaning of category

1. 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.
2. Class; also, state, condition, or predicament; as, we are both in the same category.
3. category A category K is a collection of objects, objK, and a collection of morphisms or "arrows", morK 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 Russells 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.

Related Words

category | category 3 | category 5 |

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