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What is Category Theory?

What is Category Theory?

Samuel Eilenberg and Saunders Mac Lane introduced categories, functors, and natural transformations from 1942–45 in their study of algebraic topology, with the goal of understanding the processes that preserve mathematical structure.

Category theory has practical applications in programming language theory. It may also be used as an axiomatic foundation for mathematics as an alternative to set theory and other proposed foundations.

Category theory models mathematical structure in terms of a labeled directed graph called a category. The nodes are called objects, and labelled directed edges are called arrows.

A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object.

Category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions.

Some terms used in category theory, including the term "morphism", are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself.

Content written and posted by Ken Abbott