Site characterizations for geometric invariants of toposes

We discuss the problem of characterizing the property of a Grothendieck topos to satisfy a given
'geometric' invariant as a property of its sites of definition, and indicate a set of general techniques for
establishing such criteria. We then apply our methodologies to specific invariants, notably including the
property of a Grothendieck topos to be localic (resp. atomic, locally connected, equivalent to a presheaf
topos), obtaining explicit site characterizations for them.