Cocomplete toposes whose exact completions are toposes

Let EE be a cocomplete topos. We show that if the exact completion of EE is a topos then every indecomposable object in EE 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 XX, the exact completion of Sh(X) is a topos if and only if XX is discrete. The corollary in this case characterizes the Grothendieck toposes with enough points whose exact completions are toposes.