2-Segal sets and the Waldhausen construction
  It is known by results of Dyckerhoff-Kapranov and of G\'alvez--Carrillo-Kock-Tonks that the output of the Waldhausen S.-construction has a unital 2-Segal structure. Here, we prove that a certain S.-functor defines an equivalence between the category of augmented stable double categories and the category of unital 2-Segal sets. The inverse equivalence is described explicitly by a path construction. We illustrate the equivalence for the known examples of partial monoids, cobordism categories with genus constraints and graph coalgebras. 
Julia E. Bergner, Ang\'elica M. Osorno, Viktoriya Ozornova, Martina   Rovelli, Claudia I. Scheimbauer
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