category theory

mathematics

Written by

The Information Architects of Encyclopaedia Britannica

Last updated

July 23, 2026

Quick Summary

Category theory is a branch of mathematics that studies abstract structures and the relationships between them. Introduced by Samuel Eilenberg and Saunders Mac Lane after World War II, it provides a framework for understanding mathematical concepts by focusing on the interrelations among their elements rather than individual properties.12

A category consists of objects and arrows (also called morphisms) that connect these objects. These arrows represent transformations or mappings that preserve certain structures. Examples include the category of sets with mappings, groups with homomorphisms, and vector spaces with linear transformations. Category theory allows mathematicians to generalize concepts and techniques across different mathematical domains, proving particularly useful in areas like algebraic topology and theoretical computer science.12

Summary created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

2 Britannica Sources

Ask Britannica AI about this topic

Britannica doesn't have a full article on this topic, so we've curated a list of our relevant coverage.

foundations of mathematics
Category theory
One recent tendency in the development of mathematics has been the gradual process of abstraction. The Norwegian mathematician Niels Henrik Abel (1802–29) proved that equations of the fifth degree cannot, in general, be solved by radicals. The French mathematician
Read More
One recent tendency in the development of mathematics has been the gradual process of abstraction. The Norwegian mathematician Niels Henrik Abel (1802–29) proved that equations of the fifth degree cannot, in general, be solved by radicals. The French mathematician
Read More

algebra
Category theory
…notion of structure developed within category theory. The first paper on the subject was published in the United States in 1942 by Mac Lane and Samuel Eilenberg. The idea behind their approach was that the essential features of any particular mathematical domain (a category) could be identified by focusing on…
Read More
…notion of structure developed within category theory. The first paper on the subject was published in the United States in 1942 by Mac Lane and Samuel Eilenberg. The idea behind their approach was that the essential features of any particular mathematical domain (a category) could be identified by focusing on…
Read More

Saunders Mac Lane
…topology, the two men created category theory, the general cohomology of groups, and the basis for the Eilenberg-Steenrod axioms for homology of topological spaces. Mac Lane worked with categorical duality and defined categorical universal properties. He defined and named Abelian categories, further developed by Alexandre Grothendieck to become central to…
Read More
…topology, the two men created category theory, the general cohomology of groups, and the basis for the Eilenberg-Steenrod axioms for homology of topological spaces. Mac Lane worked with categorical duality and defined categorical universal properties. He defined and named Abelian categories, further developed by Alexandre Grothendieck to become central to…
Read More