  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.
