A classification of geometric morphisms and localizations for presheaf categories and algebraic categories

We give necessary and sufficient conditions on a functor k : C -> epsilon, where C is an algebraic theory, in for the induced functor epsilon(k-,-) : epsilon -> Alg(C) to be a geometric morphism or a localization. We apply our techniques also to the particular case of module categories and to the case of presheaf categories. (c) 2006 Elsevier Inc. All rights reserved