Vol. 2, No. 1, 2020

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Spectral Mackey functors and equivariant algebraic $K\mkern-2mu$-theory, II

Clark Barwick, Saul Glasman and Jay Shah

Vol. 2 (2020), No. 1, 97–146
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.

Keywords
spectral Mackey functors, spectral Green functors, equivariant algebraic $K\mkern-2mu$-theory, Day convolution, symmetric promonoidal infinity-categories, equivariant Barratt–Priddy–Quillen
Mathematical Subject Classification 2010
Primary: 19D99, 55P91
Milestones
Received: 30 July 2018
Revised: 11 December 2018
Accepted: 27 December 2018
Published: 22 March 2019
Authors
Clark Barwick
School of Mathematics
University of Edinburgh
Edinburgh
United Kingdom
Saul Glasman
School of Mathematics
Institute for Advanced Study
Princeton, NJ
United States
Jay Shah
Department of Mathematics
University of Notre Dame
Notre Dame, IN
United States