Equivariant algebraic $K$-theory of symmetric monoidal Mackey functors

TL;DR

This work develops a unified framework for equivariant algebraic $K$-theory by organizing symmetric monoidal and Mackey-functor structures across several categorical lenses (permutative, symmetric monoidal, $ ext{∞}$-categorical, and spectral). It shows that every connective genuine $G$-spectrum can be realized as the $K$-theory of a symmetric monoidal Mackey functor, effectively linking Thomason–Mandell-type models with Bohmann–Osorno’s construction through an $ ext{∞}$-categorical approach and a covariant version of the Guillou–May theorem. The results establish equivalences among the homotopy theories of symmetric monoidal Mackey functors, $ ext{∞}$-Mackey functors, and connective genuine $G$-spectra, providing an equivariant analogue of Thomason’s theorem and clarifying the interplay between enrichment, duality, and $K$-theory in the equivariant setting. The paper also discusses multiplicative aspects, showing $K$-theory is lax monoidal but not fully compatible with all $G$-commutative monoids, and outlines directions to connect with Tambara and Green functors in future work.

Abstract

We provide a unifying approach to different constructions of the algebraic $K$-theory of equivariant symmetric monoidal categories. A consequence of our work is that every connective genuine $G$-spectrum is equivalent to the equivariant algebraic $K$-theory of categorical Mackey functors of Bohmann-Osorno.

Theorems & Definitions (104)

• Theorem A: \ref{['cor:BO is all']}
• Theorem B: \ref{['thm:GM no op']}
• Theorem C: \ref{['thm:main']}
• Theorem D: \ref{['lem:Mack ps Perm is infty Mack', 'cor:Mack Mandell', 'cor:str becomes equiv after loc']}
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Thomason, Mandell
• proof
• Corollary 2.5
• proof