Spectral Mackey Functors and Equivariant Algebraic K - Theory ( Ii )

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 ∞-categories and a suitable generalization of the second named author’s Day convolution, we endow the∞-category of Mackey functors with a wellbehaved 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 Ktheory 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.