Free Finitary Algebras in a Cocomplete Cartesian Closed Category

Abstract

In [2] Volger proved that the underlying functor of a category of set-valued models of an r-ary theory has a left adjoint. We want to show that his proof remains valid if instead of set valued models of an r-ary theory models of a finitary theory with values in an arbitrary cocomplete cartesian closed category are considered. As Volger for sets we show for any cocomplete cartesian closed category T C that for every finitary theory (S being a skeleton of the full subcategory of finite sets) the restriction of the left adjoint of on C(s) is a functor in ; here brackets around the exponent indicate as usual a restriction to functors which preserve finite products. We are very much indebted to the referee for pointing out that our proof of the last statement is only based on the properties of C mentioned above and the fact that S has and T preserves finite products. With this in mind and retaining only that part of the cartesian closedness which is relevant for the following considerations we can state the following.