By Jiří Adámek, ing.; Jiří Rosický; E M Vitale

ISBN-10: 0521119227

ISBN-13: 9780521119221

''Algebraic theories, brought as an idea within the Sixties, were a basic step in the direction of a express view of basic algebra. furthermore, they've got proved very invaluable in a number of components of arithmetic and laptop technology. This conscientiously built booklet offers a scientific creation to algebra according to algebraic theories that's obtainable to either graduate scholars and researchers. it is going to facilitate Read more...

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They create an obvious zig-zag for (x, x ) ≈ (y, y ). From this it follows that the map a1 ×a1 A×A GG c×c B ×B G (B/ ∼) × (B / ∼ ) a2 ×a2 is a coequalizer, as required. 3 Corollary For every algebraic theory T , the category Alg T is closed in Set T under reflexive coequalizers. 2. 4 Example In a category with kernel pairs, every regular epimorphism is a reflexive coequalizer. In fact, if r1 , r2 is a kernel pair of a regular epimorphism e: A → B, A id ~~ ~~ ~  ~~ r1 A dd dd dd d1 e  R d id dd dd dd d1  r2 A ~ ~ ~ ~~ e ~ ~ B then e is a coequalizer of r1 , r2 .

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 ∗ . 16 Remark Let T be a finitely complete small category, and Lex T denote the full subcategory of Set T of finite limits preserving functors. 1. YT : T op → Lex T preserves finite colimits. 2. The embedding Lex T → Set T preserves limits and filtered colimits. 3. Lex T is cocomplete. 5. 5. 17 Theorem For every finitely complete small category T , the Yoneda embedding YT : T op → Lex T is a free completion of T op under filtered colimits.

An algebra for the theory T is a functor A: T → Set preserving finite products. We denote by Alg T the category of algebras of T . Morphisms, called homomorphisms, are the natural transformations; that is, Alg T is a full subcategory of the functor category Set T . 2 Definition A category is algebraic if it is equivalent to Alg T for some algebraic theory T . 3 Remark An algebraic theory is by definition a small category. However, throughout the book, we do not take care of the difference between small and essentially small: a category is essentially small if it is equivalent to a small category.

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Algebraic theories : a categorical introduction to general algebra by Jiří Adámek, ing.; Jiří Rosický; E M Vitale

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