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. 
Olivia Caramello
math.CT math.AG math.LO