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Algebraic Theories : A Categorical Introduction to General Algebra - Jiri Adamek

Algebraic Theories

A Categorical Introduction to General Algebra


Published: 18th November 2010
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Algebraic theories, introduced as a concept in the 1960s, have been a fundamental step towards a categorical view of general algebra. Moreover, they have proved very useful in various areas of mathematics and computer science. This carefully developed book gives a systematic introduction to algebra based on algebraic theories that is accessible to both graduate students and researchers. It will facilitate interactions of general algebra, category theory and computer science. A central concept is that of sifted colimits - that is, those commuting with finite products in sets. The authors prove the duality between algebraic categories and algebraic theories and discuss Morita equivalence between algebraic theories. They also pay special attention to one-sorted algebraic theories and the corresponding concrete algebraic categories over sets, and to S-sorted algebraic theories, which are important in program semantics. The final chapter is devoted to finitary localizations of algebraic categories, a recent research area.

'The book is very well written and made as self-contained as it is reasonable for the intended audience of graduate students and researchers.' Zentralblatt MATH

Forewordp. ix
Prefacep. xv
Abstract Algebraic Categories
Preliminariesp. 3
Algebraic theories and algebraic categoriesp. 10
Sifted and filtered colimitsp. 21
Reflexive coequalizersp. 30
Algebraic categories as free completionsp. 38
Properties of algebrasp. 46
A characterization of algebraic categoriesp. 54
From filtered to siftedp. 65
Canonical theoriesp. 74
Algebraic functorsp. 80
Birkhoff's variety theoremp. 89
Concrete Algebraic Categories
One-sorted algebraic categoriesp. 103
Algebras for an endofunctorp. 117
Equational categories of -algebrasp. 127
S-sorted algebraic categoriesp. 139
Special Topics
Morita equivalencep. 153
Free exact categoriesp. 163
Exact completion and reflexive-coequalizer completionp. 182
Finitary localizations of algebraic categoriesp. 195
Postscriptp. 204
Monadsp. 207
Abelian categoriesp. 227
More about dualities for one-sorted algebraic categoriesp. 232
Referencesp. 241
List of symbolsp. 245
Indexp. 247
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780521119221
ISBN-10: 0521119227
Series: Cambridge Tracts in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 268
Published: 18th November 2010
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 23.5 x 15.9  x 2.3
Weight (kg): 0.53