Essential Localizations and Infinitary Exact Completion

We prove the universal property of the infinitary exact completion of a category with weak small limits. As an application, we slightly weaken the conditions characterizing essential localizations of varieties (in particular, of module categories) and of presheaf categories. Introduction An essential localization is a reflective subcategory such that the reflector has a left adjoint. In [12], Roos gave an abstract characterization of essential localizations of module categories, proving that they are those complete and cocomplete abelian categories with a regular generator, satisfying the following conditions (we write the conditions in a nonabelian style, more convenient for the general framework of this work) (AB4*) Regular epimorphisms are product-stable ; (AB5) Filtered colimits are exact, i.e. commute with finite limits ; (AB6) Given a small family of functors (Hi : Ai → A)I defined on small filtered categories, the canonical comparison τ is an isomorphism τ : colim( ∏