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any general or comprehensive division; a class.
a classificatory division in any field of knowledge, as a phylum or any of its subdivisions in biology.
Metaphysics.

(in Aristotelian philosophy) any of the fundamental modes of existence, such as substance, quality, and quantity, as determined by analysis of the different possible kinds of predication.
(in Kantian philosophy) any of the fundamental principles of the understanding, as the principle of causation.
any classification of terms that is ultimate and not susceptible to further analysis.

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.
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.
Grammar. part of speech.
Contemporary Examples

That declaration pushed the firm into a category of elite contractors that get their work renewed without bidding.
Supreme Group Probed Over No-Bid Contracts to Feed Troops in Afghanistan Aram Roston November 26, 2011

By the same token, the “national Obama white vote” category, sometimes discussed these days, is skewed by the South.
The Gallup Mystery and Other Matters Michael Tomasky October 26, 2012

The best idea for Depp, then, would be to embrace this idea that the category of “box-office star” is kaput, for him at least.
Johnny Depp and the ‘Lone Ranger’ Flop: Is His Career Doomed? Kevin Fallon July 8, 2013

Anthos was also nominated for a James Beard Award in the category of Best New Restaurant.
Fresh Picks Michael Psilakis November 9, 2009

Here are the facts—with important exceptions in every category.
Memo to the South: Go Ahead, Secede Already! Lee Siegel April 29, 2013

Historical Examples

Betteredge’s present effort at corresponding with me came within this category.
The Moonstone Wilkie Collins

The category of quantity, therefore, does not admit of variation of degree.
The Categories Aristotle

This might take it out of the category of crosses as a symbol of any religion of which we have knowledge.
The Swastika Thomas Wilson

So it is with all other contraries falling under the category of quality.
The Categories Aristotle

For those who commonly go by the name of the Seven Sages are not admitted into the category of the wise by fastidious critics.
Treatises on Friendship and Old Age Marcus Tullius Cicero

noun (pl) -ries
a class or group of things, people, etc, possessing some quality or qualities in common; a division in a system of classification
(metaphysics) any one of the most basic classes into which objects and concepts can be analysed

(in the philosophy of Aristotle) any one of ten most fundamental modes of being, such as quantity, quality, and substance
(in the philosophy of Kant) one of twelve concepts required by human beings to interpret the empirical world
any set of objects, concepts, or expressions distinguished from others within some logical or linguistic theory by the intelligibility of a specific set of statements concerning them See also category mistake

n.

1580s, from Middle French catégorie, from Late Latin categoria, from Greek kategoria “accusation, prediction, category,” verbal noun from kategorein “to speak against; to accuse, assert, predicate,” from kata “down to” (or perhaps “against;” see cata-) + agoreuein “to harangue, to declaim (in the assembly),” from agora “public assembly” (see agora). Original sense of “accuse” weakened to “assert, name” by the time Aristotle applied kategoria to his 10 classes of things that can be named.

category should be used by no-one who is not prepared to state (1) that he does not mean class, & (2) that he knows the difference between the two …. [Fowler]

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)