2-Segal objects and the Waldhausen construction

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.