Category Theory

θεωρία κατηγορία

katēgoréō (I accuse, J'accuse, I speak against)

θεώρημα

(mathematics) A collection of objects, together with a transitively closed collection of composable arrows between them, such that every object has an identity arrow, and such that arrow composition is associative.

1995, Michael Barr; Charles Wells, Category Theory for Computing Science, 2nd edition, University Press, Cambridge, Great Britain: Prentice Hall, §2.8.9, page 46: The use of the word 'factor' shows the explicit intention of categorists to work with functions in an algebraic manner: a category is an algebra of functions.

One well-known category has sets as objects and functions as arrows.

Just as a monoid consists of an underlying set with a binary operation "on top of it" which is closed, associative and with an identity, a category consists of an underlying digraph with an arrow composition operation "on top of it" which is transitively closed, associative, and with an identity at each object.

In fact, a category's composition operation, when restricted to a single one of its objects, turns that object's set of arrows (which would all be loops) into a monoid.