Category theory in Algebraic Topology

Abstract

The concepts category, functor, and natural transformation were introduced (in reverse order) during the early 1940s by Samuel Eilenberg and Saunders Mac Lane, aiming at resolving certain conceptual problems in algebraic topology. Before explaining in detail the points concerned, it might be useful to develop some hypotheses. In view of the intention of the category concept, the idea comes to mind that category theory should have emerged from some study of mappings. In algebraic topology, there was indeed a strong tendency beginning in the 1920s to study mappings, as exemplified in the Lefschetz fixed point theorem and the study of homotopy classes of mappings initiated by Brouwer, Hopf and others.