Lawvere's fourth open problem: Levels in the topos of symmetric simplicial sets
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2025
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| _version_ | 1866917967680241664 |
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| author | Hora, Ryuya Kamio, Yuhi Maehara, Yuki |
| author_facet | Hora, Ryuya Kamio, Yuhi Maehara, Yuki |
| contents | In the topos of simplicial sets, it makes sense to ask the following question about a given natural number $n$: what is the minimum value $m$ such that $n$-skeletality implies $m$-coskeletality? This is an instance of the Aufhebung relation in the sense of Lawvere, who introduced this notion for an arbitrary Grothendieck topos $\mathcal{E}$ in place of $\mathbf{sSet}$, and levels/essential subtopoi in place of dimensions.
We compute this Aufhebung relation for the topos of symmetric simplicial sets. In particular, we show that it is given by $2l-1$ for the level labelled by $l\geq 3$, which coincides with the previously known case of simplicial sets. This result provides a solution to the fourth of the seven open problems in topos theory posed by Lawvere in 2009. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03439 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lawvere's fourth open problem: Levels in the topos of symmetric simplicial sets Hora, Ryuya Kamio, Yuhi Maehara, Yuki Category Theory Combinatorics 18F10, 18B25 In the topos of simplicial sets, it makes sense to ask the following question about a given natural number $n$: what is the minimum value $m$ such that $n$-skeletality implies $m$-coskeletality? This is an instance of the Aufhebung relation in the sense of Lawvere, who introduced this notion for an arbitrary Grothendieck topos $\mathcal{E}$ in place of $\mathbf{sSet}$, and levels/essential subtopoi in place of dimensions. We compute this Aufhebung relation for the topos of symmetric simplicial sets. In particular, we show that it is given by $2l-1$ for the level labelled by $l\geq 3$, which coincides with the previously known case of simplicial sets. This result provides a solution to the fourth of the seven open problems in topos theory posed by Lawvere in 2009. |
| title | Lawvere's fourth open problem: Levels in the topos of symmetric simplicial sets |
| topic | Category Theory Combinatorics 18F10, 18B25 |
| url | https://arxiv.org/abs/2503.03439 |