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Algebraic Theories
A Categorical Introduction to General Algebra

$80.00 USD

Part of Cambridge Tracts in Mathematics

  • Authors:
  • J. Adámek, Technische Universität Carolo Wilhelmina zu Braunschweig, Germany
  • J. Rosický, Masarykova Univerzita v Brně, Czech Republic
  • E. M. Vitale, Université Catholique de Louvain, Belgium
F. W. Lawvere
View all contributors
  • Date Published: January 2011
  • availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
  • format: Adobe eBook Reader
  • isbn: 9780511984891

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About the Authors
  • 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.

    • Foreword by F. William Lawvere
    • Unifies expositions of various features of localizations in algebra
    • Pedagogical style makes easy reading for graduate students
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    Reviews & endorsements

    '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

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    Product details

    • Date Published: January 2011
    • format: Adobe eBook Reader
    • isbn: 9780511984891
    • contains: 50 b/w illus.
    • availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
  • 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.

  • Authors

    J. Adámek, Technische Universität Carolo Wilhelmina zu Braunschweig, Germany
    J. Adámek is a Professor in the Institute of Theoretical Computer Science at the University of Technology, Braunschweig, Germany.

    J. Rosický, Masarykova Univerzita v Brně, Czech Republic
    J. Rosický is a Professor in the Department of Mathematics and Statistics at Masaryk University, Brno, Czech Republic.

    E. M. Vitale, Université Catholique de Louvain, Belgium
    E. M. Vitale is a Professor in the Institut de Recherche en Mathématique et Physique at the Université Catholique de Louvain, Louvain-la-Neuve, Belgium.

    Foreword

    F. W. Lawvere

    Contributors

    F. W. Lawvere

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