Algebraic & Geometric Topology

Diagram spaces, diagram spectra and spectra of units

John A Lind

Full-text: Open access

Abstract

This article compares the infinite loop spaces associated to symmetric spectra, orthogonal spectra and EKMM S–modules. Each of these categories of structured spectra has a corresponding category of structured spaces that receives the infinite loop space functor Ω. We prove that these models for spaces are Quillen equivalent and that the infinite loop space functors Ω agree. This comparison is then used to show that two different constructions of the spectrum of units gl1R of a commutative ring spectrum R agree.

Article information

Source
Algebr. Geom. Topol., Volume 13, Number 4 (2013), 1857-1935.

Dates
Received: 5 October 2009
Revised: 21 November 2012
Accepted: 20 January 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715625

Digital Object Identifier
doi:10.2140/agt.2013.13.1857

Mathematical Reviews number (MathSciNet)
MR3073903

Zentralblatt MATH identifier
1271.55008

Subjects
Primary: 55P42: Stable homotopy theory, spectra 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.) 55P47: Infinite loop spaces 55U35: Abstract and axiomatic homotopy theory 55U40: Topological categories, foundations of homotopy theory
Secondary: 55P48: Loop space machines, operads [See also 18D50] 18G55: Homotopical algebra

Keywords
$E_\infty$–spaces infinite loop spaces structured ring spectra symmetric spectra orthogonal spectra EKMM spectra of units

Citation

Lind, John A. Diagram spaces, diagram spectra and spectra of units. Algebr. Geom. Topol. 13 (2013), no. 4, 1857--1935. doi:10.2140/agt.2013.13.1857. https://projecteuclid.org/euclid.agt/1513715625


Export citation

References

  • M Ando, A J Blumberg, D J Gepner, M J Hopkins, C Rezk, Units of ring spectra and Thom spectra (2009)
  • M Ando, M J Hopkins, C Rezk, Multiplicative Orientations of $KO$–theory and of the spectrum of topological modular forms, preprint (2010) Available at \setbox0\makeatletter\@url http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf {\unhbox0
  • A J Blumberg, Progress towards the calculation of the $K$–theory of Thom spectra, PhD thesis, University of Chicago (2005) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/305416322/ {\unhbox0
  • A J Blumberg, R L Cohen, C Schlichtkrull, Topological Hochschild homology of Thom spectra and the free loop space, Geom. Topol. 14 (2010) 1165–1242
  • M Bökstedt, Topological Hochschild homology, Bielefeld (1985)
  • M Brun, Topological Hochschild homology of ${\bf Z}/p\sp n$, J. Pure Appl. Algebra 148 (2000) 29–76
  • A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Math. Surv. Monogr. 47, Amer. Math. Soc. (1997)
  • A D Elmendorf, M A Mandell, Rings, modules, and algebras in infinite loop space theory, Adv. Math. 205 (2006) 163–228
  • J Hollender, R M Vogt, Modules of topological spaces, applications to homotopy limits and $E\sb \infty$ structures, Arch. Math. $($Basel$)$ 59 (1992) 115–129
  • M Hovey, Model categories, Math. Surv. Monogr. 63, Amer. Math. Soc. (1999)
  • M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149–208
  • L G Lewis, Jr, J P May, M Steinberger, J E McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics 1213, Springer, Berlin (1986)
  • M A Mandell, J P May, Equivariant orthogonal spectra and $S$-modules, Mem. Amer. Math. Soc. 755, Amer. Math. Soc. (2002)
  • M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82 (2001) 441–512
  • M A Mandell, B Shipley, A telescope comparison lemma for THH, Topology Appl. 117 (2002) 161–174
  • J P May, $E\sb{\infty }$ spaces, group completions, and permutative categories, from: “New developments in topology”, (G Segal, editor), London Math. Soc. Lecture Note Ser. 11, Cambridge Univ. Press (1974) 61–93
  • J P May, $E\sb{\infty }$ ring spaces and $E\sb{\infty }$ ring spectra, Lecture Notes in Mathematics 577, Springer, Berlin (1977)
  • J P May, The spectra associated to $\mathcal{I}$-monoids, Math. Proc. Cambridge Philos. Soc. 84 (1978) 313–322
  • J P May, The spectra associated to permutative categories, Topology 17 (1978) 225–228
  • J P May, What are $E\sb \infty$ ring spaces good for?, from: “New topological contexts for Galois theory and algebraic geometry”, (A Baker, B Richter, editors), Geom. Topol. Monogr. 16 (2009) 331–365
  • J P May, What precisely are $E\sb \infty$ ring spaces and $E\sb \infty$ ring spectra?, from: “New topological contexts for Galois theory and algebraic geometry”, (A Baker, B Richter, editors), Geom. Topol. Monogr. 16 (2009) 215–282
  • J P May, J Sigurdsson, Parametrized homotopy theory, Math. Surv. Monogr. 132, Amer. Math. Soc. (2006)
  • J P May, R Thomason, The uniqueness of infinite loop space machines, Topology 17 (1978) 205–224
  • J-P Meyer, Bar and cobar constructions, II, J. Pure Appl. Algebra 43 (1986) 179–210
  • C Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006) 969–1014
  • J Rognes, Topological logarithmic structures, from: “New topological contexts for Galois theory and algebraic geometry”, (A Baker, B Richter, editors), Geom. Topol. Monogr. 16 (2009) 401–544
  • S Sagave, C Schlichtkrull, Diagram spaces and symmetric spectra, Adv. Math. 231 (2012) 2116–2193
  • C Schlichtkrull, Units of ring spectra and their traces in algebraic $K$–theory, Geom. Topol. 8 (2004) 645–673
  • C Schlichtkrull, Thom spectra that are symmetric spectra, Doc. Math. 14 (2009) 699–748
  • C Schlichtkrull, Higher topological Hochschild homology of Thom spectra, J. Topol. 4 (2011) 161–189
  • M Schulman, Homotopy limits and colimits and enriched homotopy theory
  • S Schwede, $S$-modules and symmetric spectra, Math. Ann. 319 (2001) 517–532
  • S Schwede, On the homotopy groups of symmetric spectra, Geom. Topol. 12 (2008) 1313–1344
  • S Schwede, B E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000) 491–511
  • B Shipley, Symmetric spectra and topological Hochschild homology, $K$–Theory 19 (2000) 155–183
  • M Shulman, Comparing composites of left and right derived functors, New York J. Math. 17 (2011) 75–125