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arXiv · 2503.03439

Lawvere's fourth open problem: Levels in the topos of symmetric simplicial sets

Ryuya Hora·Yuhi Kamio·Yuki Maehara
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Abstract

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.

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Ryuya Hora, Yuhi Kamio, Yuki Maehara. 2025-03-25. Lawvere's fourth open problem: Levels in the topos of symmetric simplicial sets. https://arxiv.org/abs/2503.03439

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