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.