Segal K-theory factors through Waldhausen categories
Maxine E. Calle, David Chan
TL;DR
This work proves that Segal K-theory of symmetric monoidal categories factors through Waldhausen K-theory by constructing a functor Γ that assigns a Waldhausen category Γ(C) to each symmetric monoidal category C, with a natural weak equivalence K^S(C) ≃ K^W(Γ(C)). The Γ-construction arises from a Grothendieck construction on spans of finite sets and endows Γ(C) with a Waldhausen structure compatible with the monoidal data. Consequently, every connective spectrum can be realized (up to stable equivalence) as Waldhausen K-theory of an appropriate category, yielding an inverse K-theory machinery in the Waldhausen framework and enabling a squares K-theory interpretation via Γ(C). The results unify Segal and Waldhausen approaches, provide explicit models, and extend inverse K-theory to Waldhausen categories, with potential implications for modeling spectra and analyzing K-theoretic invariants through categorical constructions.
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 $Γ(C)$ 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.