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Republished in: Reprints in Theory and Applications of Categories, No. 10 (2005), pp. 1-136

Abstract:

Basic Concepts of Enriched Category Theory by G.M. Kelly Originally published as: Cambridge University Press, Lecture Notes in Mathematics 64, 1982. Keywords: enriched categories, monoidal categories 2000 MSC: 18-02, 18D10, 18D20 Author's Contents cited: "Contents Introduction 1 1 The elementary notions 7 1.1 Monoidal categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 The 2-category V-CAT for a monoidal V . . . . . . . . . . . . . . . . . . . 8 1.3 The 2-functor 0 : V-CAT--->CAT . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . 15 1.7 Extraordinary V-naturality . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.8 The V-naturality of the canonical maps . . . . . . . . . . . . . . . . . . . . 19 1.9 The (weak) Yoneda lemma for V -CAT . . . . . . . . . . . . . . . . . . . . 21 1.10 Representability of V-functors; V-functor of the passive variables . . . . . . 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 [AxB, C] [A, [B, C]] . . . . . . . . . . . . . . . . . . . 31 2.4 The (strong) Yoneda lemma for V -CAT; the Yoneda embedding . . . . . 33 2.5 The free V-category on a Set-category . . . . . . . . . . . . . . . . . . . . 35 2.6 Universe-enlargement V ⊂ 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 . . . . . . . . . . . . . . . . . . . . . . . 39 3.3 Limits in functor categories; double limits and iterated limits . . . . . . . . 40 3.4 The connexion with classical conical limits when V = Set . . . . . . . . . . 42 3.5 Full subcategories and limits; the closure of a full subcategory . . . . . . . 44 3.6 Strongly generating functors . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.7 Tensor and cotensor products . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.8 Conical limits in a V-category . . . . . . . . . . . . . . . . . . . . . . . . . 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 V ⊂ V . . . . . . 54 v vi CONTENTS 3.12 The existence of a limit-- and colimit-- preserving universe--enlargement V ⊂ V.. 56 4 Kan extensions 59 4.1 The definition of Kan extensions; their expressibility by limits and colimits 59 4.2 Elementary properties and examples . . . . . . . . . . . . . . . . . . . . . 61 4.3 A universal property of LanK G; its inadequacy as a definition . . . . . . . 64 4.4 Iterated Kan extensions; Kan adjoints; [Aop, V] as the free cocompletion . . 66 4.5 Initial diagrams as the left Kan extensions into V . . . . . . . . . . . . . . 68 4.6 Filtered categories when V = Set . . . . . . . . . . . . . . . . . . . . . . . 71 4.7 Factorization into an initial functor and a discrete op--fibration . . . . . . . 73 4.8 The general representability and adjoint--functor theorems . . . . . . . . . . 75 4.9 Representability and adjoint-functor theorems when V = Set . . . . . . . . 77 4.10 Existence and characterization of the left Kan extension . . . . . . . . . . 80 5 Density 85 5.1 Definition of density, and equivalent formulations . . . . . . . . . . . . . . 85 5.2 Composability and cancellability properties of dense functors . . . . . . . . 87 5.3 Examples of density; comparison with strong generation . . . . . . . . . . 89 5.4 Density of a fully-faithful K in terms of K--absolute colimits . . . . . . . . 92 5.5 Functor categories; small projectives; Morita equivalence . . . . . . . . . . 94 5.6 Left Kan extensions and adjoint functor theorems involving density . . . . 96 5.7 The free completion of A with respect to ... . . . . . . . . . . . . . . . . . 98 5.8 Cauchy completion as the idempotent-splitting completion when V = Set . 100 5.9 The finite--colimit completion when V = Set . . . . . . . . . . . . . . . . . 101 5.10 The filtered-colimit completion when V = Set . . . . . . . . . . . . . . . . 102 5.11 The image under [K, 1] : [C,B]--> [A,B] . . . . . . . . . . . . . . . . . . 104 5.12 Description of the image in terms of K-comodels . . . . . . . . . . . . . . . 106 5.13 A dense K : A-->C as an essentially-algebraic theory . . . . . . . . . . . 109 5.14 The image under [K, 1] : [C, B]---> [A, B] . . . . . . . . . . . . . . . . . . . 110 6 Essentially-algebraic theories defined by reguli and by sketches 113 6.1 Locally-bounded categories; the local boundedness of V . . . . . . . . . . . 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 Republished in: Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1-136 http://www.tac.mta.ca/tac/reprints/articles/10/tr10.dvi http://www.tac.mta.ca/tac/reprints/articles/10/tr10.ps http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf TAC Reprints Home Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. 2000 Mathematics Subject Classification: 18-02, 18D10, 18D20. Key words and phrases: enriched categories, monoidal categories. Originally published by Cambridge University Press, London Mathematical Society Lecture Notes Series 64, 1982 @G.M. Kelly, 2005. Permission to copy for private use granted."

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Originally published as: Cambridge University Press, Lecture Notes in Mathematics 64, 1982. Republished in: Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1--136 http://www.tac.mta.ca/tac/reprints/articles/10/tr10.dvi http://www.tac.mta.ca/tac/reprints/articles/10/tr10.ps @Kelly, 2005. http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf TAC Reprints Home

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