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Algebraic Theories: A Categorical Introduction to General Algebra
     

Algebraic Theories: A Categorical Introduction to General Algebra

by J. Adamek
 

ISBN-10: 0521119227

ISBN-13: 9780521119221

Pub. Date: 12/31/2010

Publisher: Cambridge University Press

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

Overview

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.

Product Details

ISBN-13:
9780521119221
Publisher:
Cambridge University Press
Publication date:
12/31/2010
Series:
Cambridge Tracts in Mathematics Series , #184
Pages:
268
Product dimensions:
5.90(w) x 9.00(h) x 0.90(d)

Table of Contents

Foreword F. W. Lawvere; Introduction; Preliminaries; Part I. Abstract Algebraic Categories: 1. Algebraic theories and algebraic categories; 2. Sifted and filtered colimits; 3. Reflexive coequalizers; 4. Algebraic categories as free completions; 5. Properties of algebras; 6. A characterization of algebraic categories; 7. From filtered to sifted; 8. Canonical theories; 9. Algebraic functors; 10. Birkhoff's variety theorem; Part II. Concrete Algebraic Categories: 11. One-sorted algebraic categories; 12. Algebras for an endofunctor; 13. Equational categories of Σ-algebras; 14. S-sorted algebraic categories; Part III. Selected Topics: 15. Morita equivalence; 16. Free exact categories; 17. Exact completion and reflexive-coequalizer completion; 18. Finitary localizations of algebraic categories; A. Monads; B. Abelian categories; C. More about dualities for one-sorted algebraic categories; Summary; Bibliography; Index.

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