category

In mathematics, a category is an abstract structure consisting of objects and morphisms—also called arrows—connecting these objects, such that the composition of morphisms is associative and each object has an identity morphism. Category theory, developed in the 1940s by Samuel Eilenberg and Saunders Mac Lane, provides a unifying framework for expressing and comparing mathematical structures of very different kinds—groups, topological spaces, sets and modules—using a common language. Morphisms between categories that preserve the composition structure are called functors: to each object of a source category, a functor assigns an object of a target category, and to each morphism of the source category, a morphism in the target category, compatibly with composition. Natural transformations, in turn, establish correspondences between functors. Category theory plays a fundamental role in modern mathematics, particularly in homological algebra, algebraic geometry and mathematical logic.