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On the duality between varieties and algebraic theories. (English) Zbl 1090.18004

A variety \(\mathcal V\) of algebras can be described via an algebraic theory \(\mathcal T\) as follows: \(\mathcal T\) is a small category with finite products such that algebras of \(\mathcal V\) correspond to models of \(\mathcal T\) and homomorphisms of algebras correspond to natural transformations of models. The authors introduce the category of all varieties whose morphisms are called algebraically exact functors, i.e., functors between varieties which are induced by morphisms of their theories. A full characterization of these functors is presented: they are right adjoints preserving filtered colimits and regular epimorphisms. The authors prove a duality between the category of all varieties and the category of all Cauchy-complete algebraic theories. The duality closely follows the known Gabriel-Ulmer duality between locally finitely presentable categories and left exact theories.

MSC:

18C10 Theories (e.g., algebraic theories), structure, and semantics
08B25 Products, amalgamated products, and other kinds of limits and colimits
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