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 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.

40 pages. v2: New authors added; somewhat awkward case-based system for the operad structures on effective Burnside infinity-categories from the previous version now streamlined, thanks to the (new) notion of symmetric promonoidal infinity-category

References in corpus (2)

• On the algebraic K-theory of higher categories
• From operator categories to topological operads

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