#### Volume 21, issue 3 (2021)

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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
##### Abstract

In a previous paper, we showed that a discrete version of the ${S}_{\bullet }$–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 ${S}_{\bullet }$–construction. We show that this equivalence fits together with the result in the discrete case and briefly discuss how it encompasses other known ${S}_{\bullet }$–constructions.

##### Keywords
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