Tunisian Journal of Mathematics

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

Clark Barwick, Saul Glasman, and Jay Shah

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 19D99: None of the above, but in this section 55P91: Equivariant homotopy theory [See also 19L47]

spectral Mackey functors spectral Green functors equivariant algebraic $K\mkern-2mu$-theory Day convolution symmetric promonoidal infinity-categories equivariant Barratt–Priddy–Quillen


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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