Categories

Abstract

This chapter completes the introduction to the theory of categories, which was begun in Section 1.7. Categories and functors first appeared in the work of Eilenberg-Mac-Lane in algebraic topology in the 1940s. It was soon apparent that these concepts had far wider applications. Many different mathematical topics may be interpreted in terms of categories so that the techniques and theorems of the theory of categories may be applied to these topics. For example, two proofs in disparate areas frequently use “similar” methods. Categorical algebra provides a means of precisely expressing these similarities. Consequently it is frequently possible to provide a proof in a categorical setting, which has as special cases the previously known results from two different areas. This unification process provides a means of comprehending wider areas of mathematics as well as new topics whose fundamentals are expressible in categorical terms.