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【Graduate Texts in Mathematics 26】【Algebraic Theories】可复制文字
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 Preface In the past decade, category theory has widened its scope and now interacts with many areas of mathematics. This book develops some of the interactions between universal algebra and category theory as well as some of the resulting applications. We begin with an exposition of equationally defineable classes from the point of view of "algebraic theories," but without the use of category theory. This serves to motivate the general treatment of algebraic theories in a category, which is the central concern of the book. (No category theory is presumed; rather, an independent treatment is provided by the second chapter.) Applications abound throughout the text and exercises and in the final chapter in which we pursue problems originating in topological dynamics and in automata theory. This book is a natural outgrowth of the ideas of a small group of mathematicians, many of whom were in residence at the Forschungsinstitut für Mathematik of the Eidgenössische Technische Hochschule in Zürich, Switzerland during the academic year 1966-67. It was in this stimulating atmosphere that the author wrote his doctoral dissertation. The "Zürich School," then, was Michael Barr, Jon Beck, John Gray, Bill Lawvere, Fred Linton, and Myles Tierney (who were there) and (at least) Harry Appelgate, Sammy Eilenberg, John Isbell, and Saunders Mac Lane (whose spiritual presence was tangible.) I am grateful to the National Science Foundation who provided support, under grants GJ 35759 and OCR 72-03733 A01, while I wrote this book. I wish to thank many of my colleagues, particularly Michael Arbib, Michael Barr, Jack Duskin, Hartrnut Ehrig, Walter Felscher, John Isbell, Fred Linton, Saunders Mac Lane, Robert Part\ Michael Pfender, Walter Tholen, Donovan Van Osdol, and Oswald Wyler, whose criticisms and suggestions made it possible to improve many portions of this book; and Saunders Mac Lane, who provided encouragement on many occasions. 目录 Table of Contents Introduction ............................................... . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Chapter 1. Algebraic theories of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1. Finitary Universal Algebra ............................ 7 2. The Clone of an Equational Presentation ................ 15 3. Algebraic Theories .................................... 24 4. The Algebras of a Theory .............................. 32 5. Infinitary Theories .................................... 50 Chapter 2. Trade Secrets of Category Theory .................... 82 1. The Base Category .................................... 82 2. Free Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 115 3. Objects with Structure ................................ 139 Chapter 3. Algebraic Theories in a Category . . . . . . . . . . . . . . . . . . . . .. 161 1. Recognition Theorems ................................ 161 2. Theories as Monoids .................................. 204 3. Abstract Birkhoff Subcategories ........................ 224 4. Regular Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 234 5. Fibre-Complete Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 246 6. Bialgebras............................................ 257 7. Colimits ............................................ 272 Chapter 4. Some Applications and Interactions . . . . . . . . . . . . . . . . . . .. 280 1. Minimal Algebras : Interactions with Topological Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 280 2. Free Algebraic Theories: the Minimal Realization of Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 292 3. Nondeterminism .......................... . . . . . . . . . . .. 309 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 341 Index. . . . . .. . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. 349 |
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