Algebraic Theories: A Categorical Introduction to General Algebra
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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2-category 2-cells abelian groups Ad´amek analogous biequivalence canonical category of algebras category with ﬁnite Chapter cocomplete codomain colim colimit cocone commutative concrete category concrete equivalence congruence consider coproducts Corollary deﬁne Deﬁnition denote directed unions Eilenberg–Moore algebra endofunctor equational category equivalence relation exact category Example exists f f f fact ﬁrst forgetful functor free algebras free completion full subcategory functor F given graph h h h H-Alg H-algebra homomorphism idempotent idempotent complete kernel pair left adjoint left covering locally ﬁnitely presentable monad monoid monomorphism Morita equivalent natural isomorphism natural numbers natural transformations one-sorted algebraic categories one-sorted algebraic theories one-sorted theory parallel pair perfectly presentable objects phism precisely preserves ﬁnite products preserves sifted colimits Proof Proposition prove pullback q q q reﬂective reﬂexive coequalizers regular epimorphism regular quotient Remark representable algebras representable functors S-sorted algebraic S-sorted set small category subalgebra terminal object Theorem