A Criterion for Categories on which every Grothendieck Topology is Rigid

Abstract

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 Cauchy-complete 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 give two formulations of a sufficient condition. The first formulation involves a two-player game, and the second formulation combines two "local" properties of $\mathbf{C}$.