This book is an introduction to the theory of categories, together with applications of this theory to some constructions with rings and modules. We start with a discussion of categories in general, and then concentrate on the types of category - additive, abelian - which enjoy to an increasing extent the properties of categories of modules. Our applications are the Morita theory, the localization and completion of rings and modules, and finally some ‘localglobal’ methods, in which the properties of a module are compared to those of its localizations and completions. We also develop the tools that we need for these applications, namely, the tensor product and limits, both direct and inverse.

The selection and presentation of our material is motivated by the needs of algebraic K-theory. Indeed, this book started out as some preliminary remarks within a text on that subject. Thus the categorical foundations are those needed to set up algebraic K-theory, and our applications are chosen since they underly some fundamental results in algebraic K-theory. However, the content of this text will be, we hope, of interest to a wider readership than potential K-theorists.

Here is a more detailed survey of the material that we cover. Our first chapter sets out the basics of category theory. There are three fundamental definitions, those of a category, a functor, and a natural transformation, and we show how to use these notions to define universal objects and universal constructions. An understanding of these is important for two reasons. On the one hand, many definitions in module theory (kernels, cokernels, …) can be reinterpreted in the language of universal objects and so extended to more general situations; on the other, many of the objects in K-theory itself arise as universal objects in one category or another.

A category of modules has a richer structure than an abstract category, since the additive structure on modules extends to their homomorphisms. The second chapter shows how this extra structure can be axiomatized and thus imposed on abstract categories. We first analyse the properties of the groups of homomorphisms in a module category to obtain a list of axioms that define additive and abelian categories; these are abstract categories that share most of the important properties of module categories.