Spectral Mackey functors and equivariant algebraic K-theory (II)

TL;DR

This paper advances the theory of spectral Mackey functors by establishing a symmetric monoidal structure, enabling the study of spectral Green functors, and providing new proofs and examples in equivariant algebraic K-theory.

Contribution

It introduces a new symmetric monoidal structure on Mackey functors, generalizes Day convolution, and proves that algebraic K-theory of group actions is lax symmetric monoidal.

Findings

Spectral Green functors can be defined for any operad.

Algebraic K-theory of group actions is lax symmetric monoidal.

New proof of the equivariant Barratt-Priddy-Quillen theorem.

Abstract

We study the "higher algebra" of spectral Mackey functors, which the first named author introduced in Part I of this paper. In particular, armed with our new theory of symmetric promonoidal $\infty$-categories and a suitable generalization of the second named author's Day convolution, we endow the $\infty$-category of Mackey functors with a well-behaved symmetric monoidal structure. This makes it possible to speak of spectral Green functors for any operad $O$. We also answer a question of A. Mathew, proving that the algebraic $K$-theory of group actions is lax symmetric monoidal. We also show that the algebraic $K$-theory of derived stacks provides an example. Finally, we give a very short, new proof of the equivariant Barratt-Priddy-Quillen theorem, which states that the algebraic $K$-theory of the category of finite $G$-sets is simply the $G$-equivariant sphere spectrum.