A Criterion for Categories on Which Every Grothendieck Topology is Rigid

Let [Formula: see text] be a small category. The subtoposes of [Formula: see text] are sometimes all of the form [Formula: see text] where [Formula: see text] is a full subcategory of [Formula: see text]. This is the case for instance when [Formula: see text] is Cauchy-complete and finite, an Artinian poset, or the simplex category. We call such a category <i>universally rigid</i>. A universally rigid category whose slices are also universally rigid, such as the aforementioned examples, is called <i>stably universally rigid</i>. We provide two equivalent characterizations of such categories. The first one stipulates the existence of a winning strategy in a two-player game, and the second one combines two "local" properties of [Formula: see text] involving respectively the poset reflections of its slices and its endomorphism monoids.