# Comparison of Waldhausen constructions

## TL;DR

This paper compares various Waldhausen $S_{ullet}$-constructions, showing that a generalized version recovers classical and relative constructions across different categorical contexts.

## Contribution

It introduces a unifying generalized Waldhausen $S_{ullet}$-construction that encompasses known constructions for exact categories and stable $(
olinebreak ext{infinity,1})$-categories.

## Key findings

- The generalized construction recovers classical $S_{ullet}$-constructions.
- It applies to exact categories, stable and exact $( ext{infinity,1})$-categories.
- It also includes the relative $S_{ullet}$-construction for exact functors.

## Abstract

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