Squares K-theory and 2-Segal spaces

TL;DR

This paper introduces a new $S_ullet$-construction for squares categories, defining 'proto-Waldhausen' categories, and investigates conditions under which this construction yields a 2-Segal space, advancing the understanding of K-theory models.

Contribution

It defines proto-Waldhausen squares categories and establishes conditions for the $S_ullet$-construction to produce 2-Segal spaces, linking category properties to K-theory modeling.

Findings

$S_ullet$-construction for squares categories is effective under stability conditions.

Proto-Waldhausen categories capture necessary properties for K-theory modeling.

The $S_ullet$-construction produces 2-Segal spaces when certain stability conditions are met.

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.