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Representability and local representability of algebraic theories. (English) Zbl 0936.18007
A finitary monosorted algebraic theory \(\mathcal T\) is called locally representable (or representable) in a category \(\mathcal K\) with finite products if each of its initial segments is the domain of a full faithful finite-products-preserving functor into \(\mathcal K\) (or if \(\mathcal T\) itself is the domain of such a functor).
The present paper investigates the question when local representability of an algebraic theory, that is, the existence of possibly separate representations of each of its segments, implies its representability.
Theories \(\mathcal T\) that are representable in any category with finite products in which they are locally representable are completely characterized.
Reviewer: R.Halaš (Olomouc)

18C10 Theories (e.g., algebraic theories), structure, and semantics
18B15 Embedding theorems, universal categories
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