Comments:

yr2000, 136 pages, 1MB Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1-136

Abstract:

One of the very famous books on advanced category theory--enriched categories, originally published as: Cambridge University Press, Lecture Notes in Mathematics 64, 1982. Mathematics Subject Classification: 18-02, 18D10, 18D20. Key words and phrases: enriched categories, monoidal categories. From the Editors, a quote: "It is a pleasure to add this book to the Reprints in Theory and Applications of Categories. There have been requests for its inclusion since we began the series. As we did for Jon Beck's thesis, we asked for volunteers to retype the text and were again overwhelmed by the response - nearly 30 colleagues volunteered within a day of sending the request. We warmly thank everyone who volunteered and especially those we asked to do the work. Serendipitously, Richard Garner had begun to re-type the book for his own use and kindly offered his excellent work--much of the first three chapters." Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1--136 http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf Introduction 1 1 The elementary notions p.7 1.1 Monoidal categories p.7 1.2 The 2--category V--CAT for a monoidal V p. 8 1.3 The 2--functor (2F)0 : V--CAT --->CAT p.10 1.4 Symmetric monoidal categories: tensor product and duality 11 1.5 Closed and biclosedmonoidal categories 13 1.6 V as a V--category; representable V--functors p.15 1.7 Extraordinary V--naturality 17 1.8 The V--naturality of the canonical maps p. 19 1.9 The (weak) Yoneda lemma for V --CAT p. 21 1.10 Representability of V-functors; V--functor of the passive variables p. 22 1.11 Adjunctions and equivalences in V--CAT 23 2 Functor categories 27 2.1 Ends in V 27 2.2 The functor-category [A, B] for small A 29 2.3 The isomorphism [A Xtensor B, C] = [A, [B, C]] 31 2.4 The (strong) Yoneda lemma for V--CAT; the Yoneda embedding p.33 2.5 The free V--category on a Set--category 35 2.6 Universe--enlargement V ; [A, B] as a V'--category 35 3 Indexed limits and colimits 37 3.1 Indexing types; limits and colimits; Yoneda isomorphisms 37 3.2 Preservation of limits and colimits p.39 3.3 Limits in functor categories; double limits and iterated limits p. 40 3.4 The connexion with classical conical limits when V=Set p. 42 3.5 Full subcategories and limits; the closure of a full subcategory p.44 3.6 Strongly generating functors p. 46 3.7 Tensor and cotensor products p. 48 3.8 Conical limits in a V--category p. 49 3.9 The inadequacy of conical limits 50 3.10 Ends and coends in a general V--category; completeness 52 3.11 The existence of a limit--preserving universe--enlargement 54 4 Kan extensions 59 4.1 The definition of Kan extensions; their expressibility by limits and colimits 59 5 Density 85 5.1 Definition of density, and equivalent formulations, 5.5 Functor categories; small projectives; Morita equivalence 94 5.6 Left Kan extensions and adjoint functor theorems involving density 96 6 Essentially-algebraic theories defined by reguli and by sketches 113 6.1 Locally--bounded categories; the local boundedness of V p. 113 6.2 The reflectivity in [Aop, V] of the category of algebras for a regulus 115 6.3 The category of algebras for a sketch . 118 6.4 The F--theory generated by a small sketch 122 6.5 Symmetric monoidal closed structure on F-cocomplete categories 124 Bibliography 125 Index 131. "

Rights:

Open access; free PDF downloads at: http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. 2000 Originally published by Cambridge University Press, London Mathematical Society Lecture Notes Series 64, 1982 copyright by G.M. Kelly, 2005. Permission to copy for private use granted.

Links:

ISBN #:

No messages.