Introduction to the Theory of Categories and Functors (Pure & Applied Mathematics Monograph)

The theory of categories has arisen in the last twenty-five years and now constitutes an autonomous branch of mathematics. It owes its origin and early inspiration to developments in algebraic topology. When the basic concepts of category, functor, natural transformation and natural equivalence were first formulated by Eilenberg and MacLane they served immediately to provide the appropriate framework for describing the way in which algebraic tools were used, and could be used, in the study of topology. It was surely evident from the outset, to the inventors of these fundamental notions and to others, that their domain of application certainly extended far beyond that of algebraic topology. Indeed there were clearly many applications within algebra itself, and homological algebra began to emerge as a mathematical discipline in its own right concerned with abelian categories and their specializations to categories of modules. However, it was not clear in the early stages that there was a "pure" theory latent within the domain of categories and functors which was capable of assuming substantial proportions within the body of mathematics. Of course, the original basic concepts came to be reinforced by auxiliary notions suggested by applications of those concepts; and many arguments traditionally carried out in a more specialized setting were seen to fit naturally into the more abstract framework of category theory. Nevertheless, it is only in the last ten years, or less, that the source of inspiration for advances in category theory has come to any considerable extent from within the theory itself. Once this process had begun it accelerated rapidly, so that now the corpus of knowledge has increased enormously and with this advance has come a great increase in the scope of application of the theory, to include such widely scattered parts of mathematics as functional analysis and mathematical logic. It seems therefore that the time is ripe for a book devoted to category theory, and suitable both for those wishing to work within the theory itself and those wishing to use the theory—or at least its basic aspects— in other mathematical disciplines such as algebra, topology, algebraic geometry, logic etc. The present volume is intended to serve such a double purpose and, hopefully, to be suitable not only as a reference source but also as a text for a graduate course. The mathematical background required is very slight, being certainly no more than that available to a student on completion of his first degree; but it should be said from the outset that some sophistication is called for from the reader if he is to appreciate the rather abstract viewpoint and arguments of category theory. The purpose of this book is not the same as that of the valuable texts by Freyd and Mitchell (to which reference is made in the bibliography). Freyd's book is devoted more or less exclusively to abelian categories and Mitchell's book is also rather different in scope, emphasis and direction from the present one. In order to render their text as useful as possible to specialists in various fields and to indicate as clearly as possible the flavour of the theory, the authors have included, as well as basic topics such as the representabihty and prorepresentabiUty of functors and embedding theorems for categories, specialized applications such as the study of Grothendieck topologies and the theory of sheaves. The authors have taken the theory right up to (but excluding) the modern theory of triples, algebras and equationally-defined categories