Quasi-2-Segal sets (Q6113270)

The author shows that the \(2\)-Segal spaces (also called decomposition spaces) of \textit{T. Dyckerhoff} and \textit{M. Kapranov} [Higher Segal spaces. Cham: Springer (2019; Zbl 1459.18001)], and \textit{I. Gálvez-Carrillo} et al. [Adv. Math. 331, 952--1015 (2018; Zbl 1403.00023); Adv. Math. 333, 1242--1292 (2018; Zbl 1403.18016)] have a natural analogue within simplicial sets, which is called quasi-\(2\)-Segal sets, and that the two ideas enjoy a similar relationship as the one of Segal spaces have with quasicategories. In particular, the author constructs a model structure on the category of simplicial sets whose fibrant objects are the quasi-\(2\)-Segal sets which is Quillen equivalent to a model structure for complete \(2\)-Segal spaces (where the notion of completeness comes from one of the equivalent characterizations of completeness for Segal spaces). The author also establishes a path space criterion, which says that a simplicial set is a quasi-\(2\)-Segal set if and only if its path spaces (also called décalage) are quasicategories, as well as an edgewise subdivision criterion.