Tunisian Journal of Mathematics

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

Clark Barwick, Saul Glasman, and Jay Shah

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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

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

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

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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References

  • C. Barwick, “Multiplicative structures on algebraic $K$-theory”, Doc. Math. 20 (2015), 859–878.
  • C. Barwick, “On the algebraic $K$-theory of higher categories”, J. Topol. 9:1 (2016), 245–347.
  • C. Barwick, “Spectral Mackey functors and equivariant algebraic $K$-theory (I)”, Adv. Math. 304 (2017), 646–727.
  • C. Barwick, “From operator categories to higher operads”, Geom. Topol. 22:4 (2018), 1893–1959.
  • C. Barwick and S. Glasman, “Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin”, preprint.
  • C. Barwick, E. Dotto, S. Glasman, D. Nardin, and J. Shah, “Parametrized higher algebra”. To appear.
  • C. Barwick, E. Dotto, S. Glasman, D. Nardin, and J. Shah, “Parametrized higher category theory”. To appear.
  • B. Day, Construction of biclosed categories, Ph.D. thesis, University of New South Wales, 1970.
  • A. W. M. Dress, Notes on the theory of representations of finite groups, I: The Burnside ring of a finite group and some AGN-applications, Universität Bielefeld, 1971.
  • S. Glasman, “Day convolution for $\infty$-categories”, Math. Res. Lett. 23:5 (2016), 1369–1385.
  • B. Guillou and J. P. May, “Models of $G$-spectra as presheaves of spectra”, preprint, 2013.
  • B. Guillou and J. P. May, “Permutative $G$-categories in equivariant infinite loop space”, preprint, 2014.
  • J. L. G. Lewis, “The theory of Green functors”, unpublished manuscript.
  • L. G. Lewis, Jr. and M. A. Mandell, “Equivariant universal coefficient and Künneth spectral sequences”, Proc. London Math. Soc. $(3)$ 92:2 (2006), 505–544.
  • J. Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton University Press, Princeton, NJ, 2009.
  • J. Lurie, “On the classification of topological field theories”, pp. 129–280 in Current developments in mathematics, 2008, International Press, Somerville, MA, 2009.
  • J. Lurie, “Derived algebraic geometry, XII: Proper morphisms, completions, and the Grothendieck existence theorem”, preprint, 2011. Available on the web page of the author.
  • J. Lurie, “Higher algebra”, preprint, 2014. Available on the web page of the author.
  • S. Schwede and B. Shipley, “Stable model categories are categories of modules”, Topology 42:1 (2003), 103–153.
  • A. Grothendieck, P. Berthelot, and L. Illusie, Théorie des intersections et théorème de Riemann–Roch (Séminaire de Géométrie Algébrique du Bois Marie 1966–1967), Lecture Notes in Math. 225, Springer, 1971.