Localizations of locally presentable categories and exact topologies (Q916774)

If A is a locally finitely presentable category, then A is equivalent to Lex \({\mathcal C}\), the category of left exact contravariant functors on the category \({\mathcal C}\) of all finitely presentable objects. This embeds A as an reflective subcategory of the topos, \(Sets^{{\mathcal C}^{op}}\), of presheaves on C. Any localization \({\mathcal L}\) of A \((=\) full reflective subcategory of A having a left exact reflector) is given by a topology, J, as the category of sheaves, \(Sh_ J{\mathcal C}\cap Lex {\mathcal C}\). However not every topology determines a localization. Results in this area are summarised (cf. \textit{F. Borceux} and \textit{B. Veit} [J. Algebra 112, 306-314 (1988; Zbl 0636.18003); Ann. Soc. Sci. Brux., Ser. I 100, 31-42 (1986; Zbl 0612.18005)], (INVALID INPUT)B. Day and \textit{R. Street} [J. Pure Appl. Algebra 63, 225-229 (1990)]) and factorization techniques (cf. \textit{C. Cassidy}, \textit{M.Hébert} and \textit{G. M. Kelly} [J. Aust. Math. Soc., Ser. A 38, 287-329 (1985; Zbl 0573.18002)]) are used to prove a correspondence between localizations and certain ``exact'' topologies.