Spectral Mackey functors and equivariant algebraic $K$-theory, II

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 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.