A Criterion for Categories on which every Grothendieck Topology is Rigid
Jérémie Marquès
TL;DR
This work addresses the problem of characterizing small Cauchy-complete categories $\mathsf{C}$ for which every Grothendieck topology on each slice $\mathsf{C}_{/X}$ is rigid, i.e., induced by a subcategory. It introduces stable universal rigidity and proves three equivalent characterizations SUR1–SUR3, using a split-epi game, left-idempotent conditions on endomorphism monoids, and the double-negation topology, with the equivalence shown and stability under slices established. Applications unify known cases (finite categories, Artinian posets, and the simplex category) and provide a corollary for the simplex category via a degree function, while also discussing limitations and connections to internal locale theory. The results offer a conceptual framework for understanding rigid subtoposes in presheaf toposes and point toward further exploration of geometric logic and locale-theoretic perspectives.
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 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.