Representability and local representability of algebraic theories
A finitary monosorted algebraic theory \( \cal T \) is called locally representable (or representable) in a category \( \cal K \) with finite products if every its initial segment is the domain of a full faithful finite-products-preserving functor into \( \cal K \) (or if \( \cal T \) itself is the domain of such a functor). The question of when local representability implies representability is discussed, and theories \( \cal T \) for which local representability always implies representability are fully characterized.
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