2–segal objects and the waldhausen construction

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.