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Category - Dictionary Definition and Overview |
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Category : (noun)
1: a collection of things sharing a common attribute; "there
are two classes of detergents" [syn: class, family]
2: a general concept that marks divisions or coordinations in a
conceptual scheme
Based on WordNet 2.0
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Category : \Cat"e*go*ry\, n.; pl. Categories. [L. categoria, Gr.
?, fr. ? to accuse, affirm, predicate; ? down, against _ ? to
harrangue, assert, fr. ? assembly.]
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. --J. S. Mill.
2. Class; also, state, condition, or predicament; as, we are
both in the same category.
There is in modern literature a whole class of
writers standing within the same category. --De
Quincey.
Based on Webster's Revised Unabridged Dictionary
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Category :
A category K is a collection of objects, obj(K), and
morphisms_(or_"{arrows">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)
Based on Webster's Revised Unabridged Dictionary (1913)
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