A Criterion for Categories on which every Grothendieck Topology is Rigid
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911227142209536 |
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| author | Marquès, Jérémie |
| author_facet | Marquès, Jérémie |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18417 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Criterion for Categories on which every Grothendieck Topology is Rigid Marquès, Jérémie Category Theory 18F10 (Primary) 18B25 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. |
| title | A Criterion for Categories on which every Grothendieck Topology is Rigid |
| topic | Category Theory 18F10 (Primary) 18B25 |
| url | https://arxiv.org/abs/2407.18417 |