Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. Several typographical errors corrected 2012-05-13. 2000 Mathematics Subject Classification: 18-02, 18D10, 18D20.

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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.

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Preliminaries. Categories. Functors. Diagrams. Naturality and Sketches. Products and Sums. Catesian Closed Categories. Finite Discrete Sketches. Limits and Colimits. More About Sketches. Fibrations. Adjoints. Algebras for Endofunctors. Toposes.

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Category Theory For Computing Science

The present paper will be referred to as Part I. A subsequent paper entitled, “Lie algebra cohomology and generalized Schubert cells,” will be referred to as Part II.

Lie Algebra Cohomology and the Generalized Borel-Weil Theorem

1. Categories.- 2. Toposes.- 3. Triples.- 4. Theories.- 5. Properties of Toposes.- 6. Permanence Properties of Toposes.- 7. Representation Theorems.- 8. Cocone Theories.- 9. More on Triples.- Index to Exercises.

Charles Ehresmann

Papers

Sur la Topologie de Certains Espaces Homogenes

Categories et Structures.

Gattungen von lokalen Strukturen.

Categories of sketched structures

Cahiers de topologie et géometrie différentielle