We show that the 2-Segal spaces (also called decomposition spaces) of Dyckerhoff and
Kapranov, and Gálvez-Carrillo, Kock, and Tonks have a natural analogue within
simplicial sets, which we call quasi-2-Segal sets, and that the two ideas enjoy a
similar relationship as the one Segal spaces have with quasicategories. In particular,
we construct 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 our notion of completeness comes from one of the
equivalent characterizations of completeness for Segal spaces). We also prove 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.
Keywords
2-Segal, quasi-2-Segal, complete 2-Segal, decomposition
space, simplicial set, model category, quasicategories,
Segal spaces, Cisinski model structure, path space
criterion
