2025/03/05 by Hora, Ryuya, Kamio, Yuhi, Maehara, Yuki
#18B25 #18F10 #Category Theory (math.CT) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.03439
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 E in place of 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≥ 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.