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 infinity-categories and a suitable generalization of the second named author's Day convolution, we endow the infinity-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.