2-Segal objects and the Waldhausen construction

TL;DR

This paper extends the discrete $S_ullet$-construction equivalence to a homotopical setting, establishing a Quillen equivalence between model categories of unital 2-Segal and augmented stable double Segal objects.

Contribution

It generalizes the discrete $S_ullet$-construction equivalence to the homotopical context via a Quillen equivalence between relevant model categories.

Findings

Established a Quillen equivalence in the homotopical setting.

Unified discrete and homotopical $S_ullet$-constructions.

Connected the new results with existing known $S_ullet$-constructions.

Abstract

In a previous paper, we showed that a discrete version of the $S_\bullet$-construction gives an equivalence of categories between unital 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 unital 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.