Representability and local representability of algebraic theories

A finitary monosorted algebraic theory is called locally representable (or representable) in a category with finite products if every its initial segment is the domain of a full faithful finite-products-preserving functor into (or if itself is the domain of such a functor). The question of when local representability implies representability is discussed, and theories for which local representability always implies representability are fully characterized. Received December 20, 1996; accepted in final form March 19, 1998.