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[资源] 【Graduate Texts in Mathematics 26】【Algebraic Theories】可复制文字

【Graduate Texts in Mathematics 26】【Algebraic Theories】可复制文字
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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