2-Segal spaces and the Waldhausen construction

In order to provide a common framework to model several flavors of Hall algebras, Dyckerhoff-Kapranov and Galvez-Carrillo-Kock-Tonks independently introduced the notion of a 2-Segal space, which is a generalization of an ordinary Segal space. The leading example of 2-Segal space was the Waldhausen construction of an exact category, which encodes into a simplicial space the main homological properties of the given exact category. The Waldhausen construction makes sense for a more general input, and the goal of the talk is to explain that in fact all 2-Segal spaces arise as the Waldhausen construction of a suitable categorical structure. More precisely, the Waldhausen construction induces an equivalence of homotopy theories between the category of stable augmented double Segal spaces and the category of 2-Segal spaces. This is joint work with Bergner, Osorno, Ozornova and Scheimbauer.