Introduction Category Theory and Homological Algebra

Abstract

The purpose of these notes is to provide as rapid an introduction to category theory and homological algebra as possible without overwhelming the reader entirely unfamiliar with these subjects. We begin with the definition of a category, and end with the basic properties of derived functors, in particular, Tor and Ext. This was the spirit of the four lectures on which thenotes are based, although there is, needless to say, much more material contained herein an was touched on in the lectures. For example, we have included a fairly complete treatment of the basic facts pertaining to adjoint functors, including Freyd's adjoint functor theorems. Application of category theory in the direction of topos theory and logic were treated in the accompanying lectures of Tierney, and Buchsbaum in his lectures indicated some outlets for homological algebra in commutative algebra and local ring theory. We have therefore not felt compelled to emphasize any specific topic. We have, nevertheless, presented module theory as something associated with ringoids (small, additive categories) rather than with the more conventional and restrictive notion of a ring. This point of view has enabled us recently to incorporate several new examples into the traditional setting of homological algebra as found in the book of Cartan-Eilenberg [2]. One can consult [15] in this regard.