Basic Concepts of Enriched Category Theory

Although numerous contributions from divers authors, over the past fifteen years or so, have brought enriched category theory to a developed state, there is still no connected account of the theory, or even of a substantial part of it. As the applications of the theory continue to expand - some recent examples are given below - the lack of such an account is the more acutely felt. The present book is designed to supply the want in part, by giving a fairly complete treatment of the limited area to which the title refers. The basic concepts of category theory certainly include the notion of functor-category, of limit and colimit, of Kan extension, and of density; with their applications to completions, perhaps including those relative completions given by categories of algebras for limit-defined theories. If we read 'V-category' for 'category' here, this is essentially the list of our chapter-headings below, after the first chapter introducing V-categories. In fact our scope is wider than this might suggest; for what we give is also a selfcontained account of basic category theory as described above, assuming as prior knowledge only the most elementary categorical concepts, and treating the ordinary and enriched cases together from Chapter 3 on.