By J. Adámek, J. Rosický, E. M. Vitale, F. W. Lawvere
Algebraic theories, brought as an idea within the Nineteen Sixties, were a basic step in the direction of a specific view of normal algebra. additionally, they've got proved very helpful in a variety of components of arithmetic and machine technological know-how. This conscientiously constructed e-book offers a scientific advent to algebra according to algebraic theories that's obtainable to either graduate scholars and researchers. it is going to facilitate interactions of basic algebra, classification concept and machine technology. A critical idea is that of sifted colimits - that's, these commuting with finite items in units. The authors turn out the duality among algebraic different types and algebraic theories and speak about Morita equivalence among algebraic theories. in addition they pay specific cognizance to one-sorted algebraic theories and the corresponding concrete algebraic different types over units, and to S-sorted algebraic theories, that are very important in application semantics. the ultimate bankruptcy is dedicated to finitary localizations of algebraic different types, a contemporary learn sector.
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Extra info for Algebraic Theories: A Categorical Introduction to General Algebra
Algebras are then functors from T to the category of sets that preserve finite products. Homomorphisms of algebras are represented by natural transformations. We now introduce a more general definition of an algebraic theory and its algebras. See Chapter 11 for Lawvere’s original concept of a one-sorted theory and Chapter 14 for S-sorted theories. In the present chapter, we also study basic concepts such as limits of algebras and representable algebras, and we introduce some of the main examples of algebraic categories.
Let Colim(Set C , B) be the full subcategory of B Set of all functors preserving colimits. Then composition with YC op defines a functor − · YC op : Colim(Set C , B) → B C . op The preceding universal property tells us that this functor is an equivalence. 12 Example 1. A famous classical example is the free completion under filtered colimits denoted by EInd : C → Ind C. Algebraic categories as free completions 43 For a small category C, Ind C can be described as the category of all filtered op colimits of representable functors in Set C , and the functor EInd is the codomain restriction of the Yoneda embedding.
10: both sides represent cocones of the (sifted) diagram F · A with codomain B. 12. 14 Corollary A category A is algebraic iff it is a free completion of a small category with finite coproducts under sifted colimits. 15 Remark Let T be an algebraic theory. If B is cocomplete and the functor F: T op → B preserves finite coproducts, then its extension F ∗: Alg T → B preserving sifted colimits has a right adjoint. In fact, since F preserves finite coproducts, the functor B → B(F −, B) factorizes through Alg T , and the resulting functor R: B → Alg T , B → B(F −, B) is a right adjoint to F ∗ .
Algebraic Theories: A Categorical Introduction to General Algebra by J. Adámek, J. Rosický, E. M. Vitale, F. W. Lawvere