JOURNAL OF ALGEBRA 139, 505-526 (1991) A Characterization of Quasi-Toposes FRANCIS BORCEUX* Unicersitt! Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium AND MARIA CRISTINA PEDICCHIO Universitri degii studi di Trieste, 34000 Trieste, Italy Communicated by Walter Feit Received September 18, 1989 A locally presentable category 6 is the category of models of some theory F which can be described using a-limits, for a regular cardinal a. Our aim is to find conditions which ensure good properties of the category d of models: when is 6 Cartesian closed? When is d a quasi-topos? When is d a topos? and so on. In this paper, we focus our attention on some semantic conditions. A locally a-presentable category 8 can thus be presented as the category of cc-left exact presheaves on a small category $9 with a-colimits. Clearly, each subpresheaf S of an E-left exact presheaf F has an a-left exact closure SC F. This closure operation plays a key role in the theory of a-left exact functors and we study, first, conditions on dz~Lex(%?) which ensure the universality of that closure operation. We prove this universality to be equivalent to the universality of strongly epimorphic families in &, but also to the fact that a-Lex(%‘) is an epireflective subcategory of a Grothendieck topos. This Grothendieck topos will play an important role in the rest of the paper; its construction depends heavily on the fact that, under our assumptions, a-left exact subpresheaves are stable under arbitrary inter- sections, from the point of view of the internal logic of the topos 4 of presheaves. Among the epireflective subcategories of a Grothendieck topos, we find the categories of separated objects for a given topology in this topos. These * Research supported by FNRS Grant 88/89 2/5-WK-E 178, by the CNR, and by NATO grant CRC 900959. 505 0021-8693/91 $3.00 Copyright C> 1991 by Academic Press, Inc. All rights of reproduction in any form reserved