Cocomplete toposes whose exact completions are toposes

Let image be a cocomplete topos. We show that if the exact completion of image is a topos then every indecomposable object in image is an atom. As a corollary we characterize 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. We also show that for any topological space X, the exact completion of image is a topos if and only if X is discrete. The corollary in this case characterizes the