Segal K-theory factors through Waldhausen categories

TL;DR

This paper demonstrates that Segal's K-theory for symmetric monoidal categories can be factored through Waldhausen categories, establishing a connection that allows all connective spectra to be realized via Waldhausen K-theory.

Contribution

It introduces a construction of a Waldhausen category from a symmetric monoidal category that preserves K-theory, bridging two important frameworks in algebraic K-theory.

Findings

Segal's K-theory factors through Waldhausen categories

Every connective spectrum can be obtained via Waldhausen K-theory

The constructed Waldhausen category has K-theory weakly equivalent to Segal's K-theory

Abstract

We show that Segal's K-theory of symmetric monoidal categorizes can be factored through Waldhausen categories. In particular, given a symmetric monoidal category $C$, we produce a Waldhausen category $\Gamma(C)$ whose K-theory is weakly equivalent to the Segal K-theory of $C$. As a consequence, we show that every connective spectrum may be obtained via Waldhausen K-theory.