A Criterion for Categories on which every Grothendieck Topology is Rigid

TL;DR

This paper characterizes categories where all subtoposes of a presheaf topos are induced by subcategories, unifying various known cases through new criteria involving local properties and a two-player game.

Contribution

It provides two equivalent characterizations of categories with this property, using a game-theoretic approach and local properties of slices and endomorphism monoids.

Findings

Characterization of categories with all subtoposes induced by subcategories

Two equivalent criteria involving a two-player game and local properties

Unification of known cases such as finite categories and Artinian posets

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.