Comparison of Waldhausen constructions

Abstract

In previous work, we developed a generalized Waldhausen $S_{•}$-construction whose input is an augmented stable double Segal space and whose output is a $2$-Segal space. Here, we prove that this construction recovers the previously known $S_{•}$-constructions for exact categories and for stable and exact $\left(\infty ,1\right)$-categories, as well as the relative $S_{•}$-construction for exact functors.