Waldhausen's $S_\bullet$-construction

TL;DR

This paper surveys Waldhausen's S-construction, explaining its different versions and their importance in defining algebraic K-theory, with a focus on their 2-Segality properties.

Contribution

It provides an expository overview of Waldhausen's S-construction and discusses its various forms and their roles in algebraic K-theory.

Findings

Different versions of Waldhausen's S-construction are surveyed.

The role of S-construction in defining K-theory is clarified.

Discussion of 2-Segality properties of the constructions.

Abstract

This note is an expository contribution for a proceedings volume of the workshop "Higher Segal Spaces and their Applications to Algebraic K-theory, Hall Algebras, and Combinatorics". We survey various versions of Waldhausen's S-construction and the role they play in defining K-theory, and we discuss their 2-Segality properties.