Quasi-varieties of presheaves

In analogy with the varietal case, we give an abstract characterization of those categories occurring as regular epireflective subcategories of presheaf categories such that the inclusion functor preserves small sums. MSC 2000 : 18A40, 18F20. The aim of this short note is to add one step to the nice parallelism between presheaf categories and algebraic categories. Presheaf categories can be abstractly characterized as exact and extensive categories with a small set of indecomposable, regular projective regular generators (Bunge [1]) (here extensive means with disjoint and universal small sums, and indecomposable means that each morphism from a generator to a small sum of generators factors through one of the canonical injections). Bearing in mind this characterization, Giraud’s axioms for Grothendieck toposes become transparent: drop the assumptions which are not stable under localization (observe that the extensivity assumption, which is a little bit redundant in Bunge’s theorem, is essential in Giraud’s theorem because generators are no longer indecomposable). On the algebraic side of the world, we have Lawvere’s theorem which states that a category is equivalent to an algebraic one if and only if it is exact, and has a finitely presentable regular projective regular generator [6]. Lawvere’s theorem is the non-additive extension of Gabriel-Mitchell’s characterization of module categories. Localizations of module categories are exactly Grothendieck categories (Gabriel-Popescu [3]), and the non-additive version of Gabriel-Popescu’s theorem states that exact categories with a regular generator, and with exact filtered colimits are the localizations of algebraic categories [8].