Algebraic Theories: A Categorical Introduction to General Algebra

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Cambridge University Press, Nov 18, 2010 - Mathematics
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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.
 

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Contents

I Abstract algebraic categories
1
II Concrete algebraic categories
101
III Special topics
151
Postscript
204
Monads
207
Abelian categories
227
More about dualities for onesorted algebraic categories
232
References
241
List of symbols
245
Index
247
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About the author (2010)

J. Adámek is a Professor in the Institute of Theoretical Computer Science at the University of Technology, Braunschweig, Germany.

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

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.

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