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122 pages, PDF, 2004 reprint of the1963 orig, thesis and 1968 seminar paper

Abstract:

Author's Contents and Author' s comments on p.6: ``1. Seven ideas introduced in the 1963 thesis: ******** (1) The category of categories is an accurate and useful framework for algebra, geometry, analysis, and logic, therefore its key features need to be made explicit. (2) The construction of the category whose objects are maps from a value of one given functor to a value of another given functor makes possible an elementary treatment of adjointness free of smallness concerns and also helps to make explicit both the existence theorem for adjoints and the calculation of the specific class of adjoints known as Kan extensions. (3) Algebras (and other structures, models, etc.) are actually functors to a background category from a category which abstractly concentrates the essence of a certain general concept of algebra, and indeed homomorphisms are nothing but natural transformations between such functors. Categories of algebras are very special, and explicit axiomatic characterizations of them can be found, thus providing a general guide to the special features of construction in algebra. (4) The Kan extensions themselves are the key ingredient in the unification of a large class of universal constructions in algebra (as in [Chevalley, 1956])." 1 Seven ideas introduced in the 1963 thesis . . . . . . . . . . . . . . . . . . . 8 2 Delays and Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Comments on the chapters of the 1963 Thesis . . . . . . . . . . . . . . . . 10 4 Some developments related to the problem list in the 1968 Article . . . . . 17 5 Concerning Notation and Terminology . . . . . . . . . . . . . . . . . . . . 18 6 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 References 20 B Functorial Semantics of Algebraic Theories 23 Introduction 24 I The category of categories and adjoint functors 26 1 The category of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Adjoint functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3 Regular epimorphisms and monomorphisms . . . . . . . . . . . . . . . . . 58 II Algebraic theories 61 1 The category of algebraic theories . . . . . . . . . . . . . . . . . . . . . . . 61 2 Presentations of algebraic theories . . . . . . . . . . . . . . . . . . . . . . . 69 III Algebraic categories 74 1 Semantics as a coadjoint functor . . . . . . . . . . . . . . . . . . . . . . . . 74 2 Characterization of algebraic categories . . . . . . . . . . . . . . . . . . . . 81 IV Algebraic functors 90 1 The algebra engendered by a prealgebra . . . . . . . . . . . . . . . . . . . 90 2 Algebraic functors and their adjoints . . . . . . . . . . . . . . . . . . . . . 93 3 CONTENTS 4 V Certain 0-ary and unary extensions of algebraic theories 97 1 Presentations of algebras: polynomial algebras . . . . . . . . . . . . . . . . 97 2 Monoids of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3 Rings of operators (Theories of categories of modules) . . . . . . . . . . . . 105 References 106 C Some Algebraic Problems in the context of Functorial Semantics of Algebraic Theories 108 1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 2 Methodological remarks and examples . . . . . . . . . . . . . . . . . . . . 112 3 Solved problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4 Unsolved problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5 Completion problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 References 120 Originally published as: Ph.D. thesis, Columbia University, 1963 and in **Reports of the Midwest Category Seminar II**, 1968, 41-61, The authors' comments are by F. William Lawvere, 2004. Keywords: Algebraic theories, functorial semantics Maths Classification 2000 MSC: 18.10 Republished in: Reprints in Theory and Applications of Categories, No. 5 (2004) pp 1-121 http://www.tac.mta.ca/tac/reprints/articles/5/tr5.dvi http://www.tac.mta.ca/tac/reprints/articles/5/tr5.ps http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf

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http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html Originally published as: Ph.D. thesis, Columbia University, 1963 and in Reports of the Midwest Category Seminar II, 1968, 41-61, @Springer-Verlag, and used by permission. The authors comments are @ F. William Lawvere, 2004.

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