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.