# category

<history of philosophy, gnoseology, philosophy of science> predicate; hence, a fundamental class of things in our conceptual framework. In Aristotle's logic specifically, the categories are the ten general modes of being (substance, quantity, quality, relation, place, time, position, possession, doing, and undergoing) by reference to which any individual thing may be described. Following the lead of stoic thought, medieval logicians commonly employed only the first four of these ten, but allowed for additional, syncategorematic terms that belonged to none of them. Kant employed a schematized table of a dozen categories as the basis for our understanding of the phenomenal realm. Gilbert Ryle used the term much more broadly, warning of the category mistakes that occur when we fail to respect the unique features of kinds of things. Recommended Reading: F. E. Peters, Greek Philosophical Terms: A Historical Lexicon (NYU, 1967) and Aristotle, Categories, ed. by Hugh Tredennick (Harvard, 1938).

<mathematics, logic> 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.

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Nearby terms: categorical syllogism « categorical term « categoricity of systems « category » category mistake » catharsis » Cauchy sequence