# A Criterion for Categories on Which Every Grothendieck Topology is Rigid

## TL;DR

This paper explores conditions under which all Grothendieck topologies on a category are rigid, offering new characterizations for such categories.

## Contribution

The paper introduces two equivalent characterizations for stably universally rigid categories.

## Key findings

- Universally rigid categories include Cauchy-complete finite categories and Artinian posets.
- Stable universal rigidity is characterized by a winning strategy in a two-player game and local properties of the category.

## Abstract

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\begin{document}$$\mathbf{C}$$\end{document} be a small category. The subtoposes of \documentclass[12pt]{minimal}
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\begin{document}$$[\mathbf{C}^\textrm{op},\mathbf{Set}]$$\end{document} are sometimes all of the form \documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{D}$$\end{document} is a full subcategory of \documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{C}$$\end{document}. This is the case for instance when \documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{C}$$\end{document} is Cauchy-complete and finite, an Artinian poset, or the simplex category. We call such a category universally rigid. A universally rigid category whose slices are also universally rigid, such as the aforementioned examples, is called stably universally rigid. We provide two equivalent characterizations of such categories. The first one stipulates the existence of a winning strategy in a two-player game, and the second one combines two “local” properties of \documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{C}$$\end{document} involving respectively the poset reflections of its slices and its endomorphism monoids.