Download this article
 Download this article For screen
For printing
Recent Issues
Volume 6, Issue 2
Volume 6, Issue 1
Volume 5, Issue 4
Volume 5, Issue 3
Volume 5, Issue 2
Volume 5, Issue 1
Volume 4, Issue 4
Volume 4, Issue 3
Volume 4, Issue 2
Volume 4, Issue 1
Volume 3, Issue 4
Volume 3, Issue 3
Volume 3, Issue 2
Volume 3, Issue 1
Volume 2, Issue 4
Volume 2, Issue 3
Volume 2, Issue 2
Volume 2, Issue 1
Volume 1, Issue 4
Volume 1, Issue 3
Volume 1, Issue 2
Volume 1, Issue 1
The Journal
About the journal
Ethics and policies
Peer-review process
Statement, 2023
Submission guidelines
Submission form
Editorial board
ISSN (electronic): 2576-7666
ISSN (print): 2576-7658
Author index
To appear
Other MSP Journals
Quasi-2-Segal sets

Matt Feller

Vol. 5 (2023), No. 2, 327–367

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.

2-Segal, quasi-2-Segal, complete 2-Segal, decomposition space, simplicial set, model category, quasicategories, Segal spaces, Cisinski model structure, path space criterion
Mathematical Subject Classification
Primary: 18N50, 55U35
Secondary: 18N40, 18N60, 55U10
Received: 15 April 2022
Revised: 26 October 2022
Accepted: 26 November 2022
Published: 4 June 2023
Matt Feller
Max Planck Institute for Mathematics