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

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

''Algebraic theories, brought as an idea within the Nineteen Sixties, were a basic step in the direction of a specific view of basic algebra. additionally, they've got proved very helpful in a variety of components of arithmetic and laptop technological know-how. This conscientiously constructed publication offers a scientific creation to algebra according to algebraic theories that's available to either graduate scholars and researchers. it's going to facilitateRead more...

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**Example text**

A is an algebra. 2. El A is a sifted category. 3. A is a sifted colimit of representable algebras. 14. 5). 16, it suffices to prove that (El A)op has finite products. This is obvious: for example, the product of (X, x) and (Z, z) is (X × Z, (x, z)) – recall that (x, z) ∈ AX × AZ = A(X × Z). 3 Remark An analogous result (with a completely analogous proof) holds for small categories T with finite limits: a functor A: T → Set preserves finite limits iff El A is a filtered category iff A is a filtered colimit of representable functors.

4. 15; for point b use, analogously, the equalizer of D(u, −), D(v, −): D(B, −) ⇒ D(A, −), which is the diagram D of all morphisms merging u and v: since colim D = 1, the diagram is nonempty. 15. 5, the diagonal functor : D → DJ is final. Proof 1. Let all such functors be final. We are to show that every finitely generated subcategory J of D has a cocone in D. The inclusion functor d: J → D is an object of the functor category DJ . 3, the slice category d ↓ is connected and thus nonempty. Since d ↓ is precisely the category of cocones of J in D, we obtain the desired cocone.

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.