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Quasi-2-Segal sets

Matt Feller

Vol. 5 (2023), No. 2, 327–367
Abstract

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.

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Keywords
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
Milestones
Received: 15 April 2022
Revised: 26 October 2022
Accepted: 26 November 2022
Published: 4 June 2023
Authors
Matt Feller
Max Planck Institute for Mathematics
Bonn
Germany