2-Segal sets and the Waldhausen construction
arXiv:1609.02853
Abstract
It is known by results of Dyckerhoff-Kapranov and of Gálvez--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.
48 pages. Final version. Will appear in Proceedings of WIT