Let $\mathbf{ C}$ be a Cauchy-complete category. The subtoposes of $[\mathbf{ C}^{ \mathrm{ op}},\mathbf{ Set}]$ are sometimes all of the form $[\mathbf{ D}^{ \mathrm{ op}},\mathbf{ Set}]$ 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 $[\mathbf{ C}^{ \mathrm{ op}},\mathbf{ Set}]$ is induced by a subcategory of $\mathbf{ C}_{ /X}$. 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.
Categories: math.CT