ESSENTIAL LOCALIZATIONS AND INFINITARY EXACT COMPLETION.pdf

Theory and Applications of Categories, Vol. 8, No. 17, 2001, pp. 465–480. ESSENTIAL LOCALIZATIONS AND INFINITARY EXACT COMPLETION ENRICO M. VITALE ABSTRACT. 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 non- abelian 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( ∏ I Ai → ∏ I A → A) ?→ ∏ I (colimHi) . In a subsequent paper [13], Roos introduced a weaker form of (AB6) : (WAB6) With the same notations as in (AB6), the comparison morphism τ is a regular epimorphism. It is then natural to look for a representation of the abelian categories satisfying the same list of conditions as in Roos’s theorem, but replacing (AB6) with (WAB6). The first aim of this paper is to prove that this set of conditions, which seems weaker, still characterizes essential localizations of module categories. In fact we prove a more general result : Roos’s theorem has recently been generalized by Ada?mek, Rosicky? and the author to a non-additive context [1]. They have characterized essential localizations of (multi-sorted, finitary) varieties as those complete and cocomplete Barr-exact categories Received by the editors 2000 Nove