A Criterion for Categories on which every Grothendieck Topology is Rigid

Let C be a Cauchy-complete category. The subtoposes of [Cᵒᵖ, Set] are sometimes all of the form [Dᵒᵖ, Set] where D is a full Cauchy-complete subcategory of C. This is the case for instance when C is finite, an Artinian poset, or the simplex category. In order to unify these situations, we give two formulations of a sufficient condition. The first formulation involves a two-player game, and the second formulation combines two "local" properties of C.