2–Segal objects and the Waldhausen construction

Abstract

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.