Algebraic & Geometric Topology

Constructing equivariant spectra via categorical Mackey functors

Anna Marie Bohmann and Angélica Osorno

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Abstract

We give a functorial construction of equivariant spectra from a generalized version of Mackey functors in categories. This construction relies on the recent description of the category of equivariant spectra due to Guillou and May. The key element of our construction is a spectrally enriched functor from a spectrally enriched version of permutative categories to the category of spectra that is built using an appropriate version of K–theory. As applications of our general construction, we produce a new functorial construction of equivariant Eilenberg–Mac Lane spectra for Mackey functors and for suspension spectra for finite G–sets.

Article information

Source
Algebr. Geom. Topol., Volume 15, Number 1 (2015), 537-563.

Dates
Received: 16 June 2014
Revised: 4 August 2014
Accepted: 9 August 2014
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510840922

Digital Object Identifier
doi:10.2140/agt.2015.15.537

Mathematical Reviews number (MathSciNet)
MR3325747

Zentralblatt MATH identifier
1314.55005

Subjects
Primary: 55P42: Stable homotopy theory, spectra 55P91: Equivariant homotopy theory [See also 19L47]
Secondary: 18D20: Enriched categories (over closed or monoidal categories)

Keywords
equivariant stable homotopy theory equivariant spectra Mackey functors permutative categories

Citation

Bohmann, Anna Marie; Osorno, Angélica. Constructing equivariant spectra via categorical Mackey functors. Algebr. Geom. Topol. 15 (2015), no. 1, 537--563. doi:10.2140/agt.2015.15.537. https://projecteuclid.org/euclid.agt/1510840922


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References

  • M Barratt, S Priddy, On the homology of non-connected monoids and their associated groups, Comment. Math. Helv. 47 (1972) 1–14
  • S,R Costenoble, S Waner, Fixed set systems of equivariant infinite loop spaces, Trans. Amer. Math. Soc. 326 (1991) 485–505
  • A,D Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc. 277 (1983) 275–284
  • A,D Elmendorf, M,A Mandell, Rings, modules, and algebras in infinite loop space theory, Adv. Math. 205 (2006) 163–228
  • A,D Elmendorf, M,A Mandell, Permutative categories, multicategories and algebraic $K$–theory, Algebr. Geom. Topol. 9 (2009) 2391–2441
  • B,J Guillou, Strictification of categories weakly enriched in symmetric monoidal categories, Theory Appl. Categ. 24 (2010) 564–579
  • B,J Guillou, J,P May, Models of $G$–spectra as presheaves of spectra (2013)
  • M Hyland, J Power, Pseudo-commutative monads and pseudo-closed $2$–categories, J. Pure Appl. Algebra 175 (2002) 141–185
  • T Leinster, Higher operads, higher categories, London Mathematical Society Lecture Note Series 298, Cambridge Univ. Press (2004)
  • L,G Lewis, Jr, J,P May, M Steinberger, Equivariant stable homotopy theory, Lecture Notes in Mathematics 1213, Springer, Berlin (1986)
  • O Manzyuk, Closed categories vs. closed multicategories, Theory Appl. Categ. 26 (2012) 132–175
  • J,P May, Pairings of categories and spectra, J. Pure Appl. Algebra 19 (1980) 299–346
  • J,P May, Equivariant homotopy and cohomology theory, from: “Symposium on Algebraic Topology in honor of José Adem”, (S Gitler, editor), Contemp. Math. 12, Amer. Math. Soc. (1982) 209–217
  • J,P May, M Merling, A,M Osorno, Equivariant infinite loop space machines
  • R,J Piacenza, Homotopy theory of diagrams and CW–complexes over a category, Canad. J. Math. 43 (1991) 814–824
  • P,F dos Santos, Z Nie, Stable equivariant abelianization, its properties, and applications, Topology Appl. 156 (2009) 979–996
  • K Shimakawa, Infinite loop $G$–spaces associated to monoidal $G$–graded categories, Publ. Res. Inst. Math. Sci. 25 (1989) 239–262