Cocomplete toposes whose exact completions are toposes
The problem of characterization of the toposes whose exact completions are toposes is stated as an open problem in the last paragraph of p.131 of \textit{A. Carboni} [J. Pure Appl. Algebra 103, 117--148 (1995; Zbl 0839.18002)]. In this paper the author shows that if \(C\) is a cocomplete topos, then if the exact completion of \(C\) is a topos, then every indecomposable object in \(C\) is an atom. As a corollary the author characterizes the locally connected Grothendieck toposes whose exact completions are toposes. This result strengthens both the Lawvere-Schanuel characterization of Boolean presheaf toposes and Hofstra's characterization of the locally connected Grothendieck toposes whose exact completion is a Grothendieck topos. Also, the author shows that for any topological space \(X,\) the exact completion of \(SH(X)\) is a topos if and only if \(X\) is discrete. The corollary in this case characterizes the Grothendieck toposes with enough points whose exact completions are toposes.
scientific article; zbMATH DE number 5167560