Quasi-2-Segal sets

TL;DR

This paper introduces quasi-2-Segal sets as a simplicial set analogue of 2-Segal spaces, establishing a model structure and equivalences similar to those in Segal space theory, with criteria based on path spaces and subdivisions.

Contribution

It defines quasi-2-Segal sets within simplicial sets, constructs a model structure for them, and proves criteria linking them to quasi-categories, extending the theory of 2-Segal spaces.

Findings

Established a model structure for quasi-2-Segal sets.

Proved a path space criterion linking quasi-2-Segal sets to quasi-categories.

Showed Quillen equivalence with complete 2-Segal spaces.

Abstract

We show that the 2-Segal spaces (also called decomposition spaces) of Dyckerhoff-Kapranov and G\'alvez-Kock-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 quasi-categories. 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 d\'ecalage) are quasi-categories, as well as an edgewise subdivision criterion.