## Tunisian Journal of Mathematics

### 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 $∞$-categories and a suitable generalization of the second named author’s Day convolution, we endow the $∞$-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.

#### Article information

Source
Tunisian J. Math., Volume 2, Number 1 (2020), 97-146.

Dates
Received: 30 July 2018
Revised: 11 December 2018
Accepted: 27 December 2018
First available in Project Euclid: 2 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.tunis/1554170463

Digital Object Identifier
doi:10.2140/tunis.2020.2.97

Mathematical Reviews number (MathSciNet)
MR3933393

Zentralblatt MATH identifier
07074072

#### Citation

Barwick, Clark; Glasman, Saul; Shah, Jay. Spectral Mackey functors and equivariant algebraic $K$-theory, II. Tunisian J. Math. 2 (2020), no. 1, 97--146. doi:10.2140/tunis.2020.2.97. https://projecteuclid.org/euclid.tunis/1554170463

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