Lawvere's fourth open problem: Levels in the topos of symmetric simplicial sets
Ryuya Hora, Yuhi Kamio, Yuki Maehara
TL;DR
This work resolves Lawvere's fourth open problem for the presheaf topos $\mathbf{PSh}(\mathbf{F})$ by determining the Aufhebung values for levels in the topos of symmetric sets. The authors develop a graph-theoretic framework based on EZ-decomposition and EZ-congruence, introducing reduction graphs and a propagativity criterion to convert local cycle-fillers into global coskeletality. They prove the sharp bounds that establish $a_{-\infty}=0$, $a_{0}=1$, $a_{1}=2$, $a_{2}=4$, $a_{l}=2l-1$ for $l\ge 3$, and $a_{\infty}=\infty$, matching the known simplicial case for $l\ge 3$. A key innovation is translating combinatorial constraints into the propagation-graph setting, and handling a special case $(n,k)=(1,3)$ separately. The results provide a complete solution to Lawvere's Problem 4 for symmetric sets and deepen the understanding of level-based dimensions in topos theory.
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.