Category

A category consists of a collection of objects and, for each ordered pair , a set of morphisms , together with a composition and an identity for each object, subject to two laws:

A morphism records not what an object is but how it maps to another; the objects and their morphisms together are the datum. The opposite category has the same objects with every morphism reversed. Written in diagram order, composition is , matching the reading order of arrows.

References

• Eilenberg & Mac Lane, “General Theory of Natural Equivalences”, Trans. AMS (1945)
• Saunders Mac Lane, Categories for the Working Mathematician (1971; 2nd ed. 1998)