Algebraic Theories: A Categorical Introduction to General AlgebraCambridge University Press, 18 nov 2010 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. |
Índice
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 |
| 245 | |
| 247 | |
Términos y frases comunes
2-category 2-cells abelian groups Adámek AlgT analogous biequivalence canonical category with finite Chapter cocomplete codomain colim commutes concrete category concrete equivalence concrete functor congruence consider coproducts Corollary define Definition denote directed unions endofunctor equational category equivalence relation exact category Example exists fact filtered colimits finitary monads forgetful functor free algebras free completion full subcategory functor F given H-Alg H-algebra homomorphism idempotent idempotent complete kernel pair left adjoint left covering locally finitely presentable monad monoid monomorphism Morita equivalent morphism f natural isomorphism natural numbers natural transformations one-sorted algebraic categories one-sorted algebraic theory one-sorted theory parallel pair perfectly presentable objects phism precisely preserves finite products preserves sifted colimits Proof Proposition prove pseudoequivalence pullback reflexive coequalizers regular epimorphism regular quotient Remark representable algebras representable functors Rosický S-sorted algebraic S-sorted set small category subalgebra terminal object Theorem Ypop