Squares K-theory and 2-Segal spaces

We define an $S_{\bullet }$-construction for squares categories, and introduce a class of squares categories we call “proto-Waldhausen” which capture the properties required for the $S_{\bullet }$-construction to model the K-theory space. The primary question we investigate is when the $S_{\bullet }$-construction of a squares category produces a 2-Segal space. We show that the answer to this question is affirmative when the squares category satisfies certain “stability” conditions.