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

''Algebraic theories, brought as an idea within the Nineteen Sixties, were a primary step in the direction of a specific view of basic algebra. in addition, they've got proved very beneficial in quite a few parts of arithmetic and computing device technology. This conscientiously constructed e-book offers a scientific creation to algebra in keeping with algebraic theories that's obtainable to either graduate scholars and researchers. it is going to facilitateRead more...

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**Example text**

22 Remark In Chapter 7, we study functors preserving filtered and sifted colimits. 26). We use the following terminology. 23 Definition A functor is called finitary if it preserves filtered colimits. 24 Example Here we mention some endofunctors of Set that are finitary. 1. The functor Hn: Set → Set Hn X = Xn is finitary for every natural number n since finite products commute in Set with filtered colimits. 2. A coproduct of finitary functors is finitary. 3. 9). We define the corresponding polynomial functor H : Set → Set as the coproduct of the functors Har(σ ) for σ ∈ H X= n .

3. 13 for S-sorted algebraic categories). The category of algebras of an algebraic theory is quite rich. We already know that every object t of an algebraic theory T yields the representable algebra YT (t) = T (t, −). Other examples of algebras can be obtained, for example, by the formation of limits and colimits. We will now show that limits always exist and are built up at the level of sets. Also, colimits always exist, but they are seldom built up at the level of sets. We will study colimits in subsequent chapters.

The interested reader can find expositions of various aspects of algebraic theories in the following literature: finitary theories and their algebras in general categories (Barr & Wells, 1985; Borceux, 1994; Hyland & Power, 2007; Pareigis, 1970; Pedicchio & Rovatti, 2004; Schubert, 1972). infinitary theories (Linton, 1966; Wraith, 1970). applications of theories in computer science (Barr & Wells, 1990; Wechler, 1992). Manes (1976) is, in spite of its title, devoted to monads, not theories; an introduction to monads can be found in Appendix A.