[kat-i-gawr-ee, -gohr-ee] 
| 1. | any general or comprehensive division; a class. |
| 2. | a classificatory division in any field of knowledge, as a phylum or any of its subdivisions in biology. |
| 3. | Metaphysics.
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| 4. | categories. Also called Guggenheim. (used with a singular verb ) a game in which a key word and a list of categories, as dogs, automobiles, or rivers, are selected, and in which each player writes down a word in each category that begins with each of the letters of the key word, the player writing down the most words within a time limit being declared the winner. |
| 5. | Mathematics. a type of mathematical object, as a set, group, or metric space, together with a set of mappings from such an object to other objects of the same type. |
| 6. | Grammar. part of speech. |
gor(os) accuser, affirmer (katēgor(eîn) to accuse, affirm, lit., speak publicly against, equiv. to kata- cata- + -agoreîn to speak before the agora + -os n. suffix) + -ia -y 3 | cat·e·go·ry (kāt'ĭ-gôr'ē, -gōr'ē) n. pl. cat·e·go·ries
[French catégorie, from Old French, from Late Latin catēgoria, class of predicables, from Greek katēgoriā, accusation, charge, from katēgorein, to accuse, predicate : kat-, kata-, down, against; see cata- + agoreuein, ēgor-, to speak in public (from agorā, marketplace, assembly; see ger- in Indo-European roots).] |
category theory
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.
(1997-10-06)