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.
Centazzo, C., & Vitale, E. (2006). A classification of geometric morphisms and localizations for presheaf categories and algebraic categories. Journal of Algebra, 303(1), 77-96. https://doi.org/10.1016/j.jalgebra.2006.06.013 (Original work published 2006)