2-Segal maps associated to a category with cofibrations

Waldhausen's S_-construction gives a way to define the algebraic K-theory space of a category with cofibrations. Specifically, the K-theory space of a category with cofibrations C can be defined as the loop space of the realization of the simplicial topological space |iS_ C |. Dyckerhoff and Kapranov observed that if C is chosen to be a proto-exact category, then this simplicial topological space is 2-Segal. A natural question is then what variants of this S_-construction give 2-Segal spaces. We find that for |iS_ C|, S_, wS_, and the simplicial set whose nth level is the set of isomorphism classes of S_, there are certain 2-Segal maps which are always equivalences. However for all of these simplicial objects, none of the rest of the 2-Segal maps have to be equivalences. We also reduce the question of whether |wS_ C| is 2-Segal in nice cases to the question of whether a simpler simplicial space is 2-Segal. Additionally, we give a sufficient condition for S_ C to be 2-Segal. Along the way we introduce the notion of a generated category with cofibrations and provide an example where the levelwise realization of a simplicial category which is not 2-Segal is 2-Segal.