2-Segal Maps associated to a category with cofibrations

Waldhausen's $S_\bullet$-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 $\mathcal{C}$ can be defined as the loop space of the realization of the simplicial topological space $|iS_\bullet \mathcal{C} |$. Dyckerhoff and Kapranov observed that if $\mathcal{C}$ is chosen to be a proto-exact category, then this simplicial topological space is 2-Segal. We say that $X$ is $2$-Segal if its $2$-Segal maps, indexed by decompositions of polygons, are weak equivalences. A natural question is then what variants of this $S_\bullet$-construction give 2-Segal spaces. We find that for $|iS_\bullet \mathcal{C}|$, $S_\bullet\mathcal{C}$, $wS_\bullet\mathcal{C}$, and the simplicial set whose $n$th level is the set of isomorphism classes of $S_\bullet\mathcal{C}$, 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 $|wS_\bullet \mathcal{C}|$ 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 $S_\bullet \mathcal{C}$ 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