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In a previous paper, we showed that a discrete version of the S•-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•-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.

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