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303 pages, Reprints in Theory and Applications of Categories, No.1, 2005, pp. 1--289, reprint of 1985 book

Abstract:

Version 1.1, Reprinted by Theory and Applications of Categories: Categories (functors, natural transformations, limits, colimits, adjoint functors,filtered colimits), topoi (sheaves on a space; the Beck conditions), triples (The Kleisli and Eilenberg-Moore Categories, morphisms of triples, adjoint triples), theories (sketches,The Ehresmann--Kennison Theorem, Finite--Product Theories, Left Exact theories). Logical functors, topoi properties, sheaves form a topos, sheaf for a topology, Grothendieck topologies, Giraud's Theorem, categories in a topos, Representation Theorems: Freyd's Representation Theorems, Morphisms of Sites, Natural Number Objects,Countable Toposes and Separable Toposes, Barr's theorem. Cocone Theories: Geometric Theories, Properties of Model Categories. Colimits of Triple Algebras: Free Triples, Distributive Laws. Bobliography. Authors cited correction and comment : "GENERAL COMMENT Our text is intended primarily as an exposition of the mathematics, not a historical treatment of it. In particular, if we state a theorem without attribution we do not in any way intend to claim that it is original with this book. We note specifically that most of the material in Chapters 4 and 8 is an extensive reformulation of ideas and theorems due to C. Ehresmann, J. Benabou, C. Lair and their students, to Y. Diers, and to A. Grothendieck and his students. We learned most of this material second hand or recreated it, and so generally do not know who did it first. We will happily correct mistaken attributions when they come to our attention."..." p. 9 (Peter Johnstone). Exercise (SGRPOID) is incorrect as it stands; a semilattice without identity satisfies (i) through (iii) but is not a category. Condition (iii) must be strengthened...p. 26 (D.Cubric). Property (ii) of Exercise (SUBF); p. 27 (Dwight Spencer), second line from bottom: T : S--->T should be t : S--> T. pp. 39-40 (Peter Johnstone). It should be noted that the product of an empty collection of objects in a category must be a terminal object. Then the phrase after the comma on line 4 of p. 40 should read, which by an obvious inductive argument is equivalent to requiring that the category have a terminal object and that any two objects have a product. p. 43 (Peter Johnstone). Exercise (PROD)(b) should read: \Show that if a category has a terminal object and all products of pairs of objects, then it has all finite products. p. 49 (Peter Johnstone). Exercise (FCR) uses the concept of small category without defining it. It is used in the main body of the text on page 66 and later, and small sketch occurs on p. 146. A graph or a category is small if its arrows constitute a set. A sketch is small if its graph is small and its cones and coconesconstitute a set. In connection with the discussion of foundations on page ix of the Preface, no matter what set theory is used, one is going to have to deal with categories and graphs whose arrows do not constitute sets. On p. 49: Closing parenthesis missing at end of Exercise (EAPL)(a)." Additional corrections available at the URL: http://www.cwru.edu/artsci/math/wells/pub/ttt.html

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Open access: Copyright@2000 by Michael Barr and Charles Frederick Wells. "This version may be downloaded and printed in unmodified form for private use only." http://www.cwru.edu/artsci/math/wells/pub/ttt.html

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