Abstract

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.

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