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$2$–Segal objects and the Waldhausen construction

Julia E Bergner, Angélica M Osorno, Viktoriya Ozornova, Martina Rovelli and Claudia I Scheimbauer

Algebraic & Geometric Topology 21 (2021) 1267–1326

In a previous paper, we showed that a discrete version of the S–construction gives an equivalence of categories between 2–Segal sets and augmented stable double categories. Here, we generalize this result to the homotopical setting, by showing that there is a Quillen equivalence between a model category for 2–Segal objects and a model category for augmented stable double Segal objects which is given by an S–construction. We show that this equivalence fits together with the result in the discrete case and briefly discuss how it encompasses other known S–constructions.

2-Segal space, Waldhausen $S_{\bullet}$–construction, double Segal space, model category
Mathematical Subject Classification 2010
Primary: 18D05, 18G55, 19D10, 55U35, 55U40
Secondary: 18G30, 55U10
Received: 25 January 2019
Revised: 3 February 2020
Accepted: 19 June 2020
Published: 11 August 2021
Julia E Bergner
Department of Mathematics
University of Virginia
Charlottesville, VA
United States
Angélica M Osorno
Department of Mathematics
Reed College
Portland, OR
United States
Viktoriya Ozornova
Fakultät für Mathematik
Ruhr-Universität Bochum
Martina Rovelli
Mathematical Sciences Institute
The Australian National University
Canberra, ACT
Department of Mathematics and Statistics
University of Massachusetts
Amherst, MA
United States
Claudia I Scheimbauer
Zentrum Mathematik
Technische Universität München
Garching bei München