Algebraic Theories
A Categorical Introduction to General Algebra

Up-to-date categorical view of sets with extra algebraic structure (data types), with applications in mathematics and theoretical computer science.

J. Adámek (Author), J. Rosický (Author), E. M. Vitale (Author), F. W. Lawvere (Foreword by)

9780521119221, Cambridge University Press

Hardback, published 18 November 2010

268 pages, 50 b/w illus.
23.1 x 15.5 x 2.3 cm, 0.57 kg

'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

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 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 ?-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.

Subject Areas: Algebra [PBF], Discrete mathematics [PBD]