**Title:** 2-Segality and the S.-construction

**Abstract:** 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 $C$ can be defined as the loop space of the realization of the simplicial topological space $|iS_\bullet C |$. 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 $S_\bullet$-construction give 2-Segal spaces. In this talk, we give the necessary background in this area and discuss work in progress that aims to address the preceding question.