A Criterion for Categories on which every Grothendieck Topology is Rigid

Let $\mathbf{C}$ be a Cauchy-complete category. The subtoposes of $[{\mathbf{C}}^{\mathrm{o}\mathrm{p}},\mathbf{S}\mathbf{e}\mathbf{t}]$ are sometimes all of the form $[{\mathbf{D}}^{\mathrm{o}\mathrm{p}},\mathbf{S}\mathbf{e}\mathbf{t}]$ where $\mathbf{D}$ is a full subcategory of $\mathbf{C}$. This is the case for instance when $\mathbf{C}$ is finite, an Artinian poset, or the simplex category. In order to unify these situations, we characterize the small categories $\mathbf{C}$ such that for every $X\in \mathbf{C}$, every subtopos of $[{\mathbf{C}}^{\mathrm{o}\mathrm{p}},\mathbf{S}\mathbf{e}\mathbf{t}]$ is induced by a subcategory of ${\mathbf{C}}_{\mathrm{/}X}$. We provide two equivalent characterizations. The first one uses a two-player game, and the second one combines two “local” properties of $\mathbf{C}$ involving respectively the poset reflections of its slices and its endomorphism monoids.