Diagram spaces, diagram spectra and spectra of units

Lind, John A¹

¹Department of Mathematics, The Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218, USA

Algebraic and Geometric Topology, Tome 13 (2013) no. 4, pp. 1857-1935

Résumé

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.

Détail

DOI : 10.2140/agt.2013.13.1857

Classification : 55P42, 55P43, 55P47, 55U35, 55U40, 55P48, 18G55
Keywords: $E_\infty$–spaces, infinite loop spaces, structured ring spectra, symmetric spectra, orthogonal spectra, EKMM, spectra of units

Bibliographie

[1] M Ando, A J Blumberg, D J Gepner, M J Hopkins, C Rezk, Units of ring spectra and Thom spectra (2009)

[2] M Ando, M J Hopkins, C Rezk, Multiplicative Orientations of $KO$–theory and of the spectrum of topological modular forms, preprint (2010)

[3] A J Blumberg, Progress towards the calculation of the $K$–theory of Thom spectra, PhD thesis, University of Chicago (2005)

[4] A J Blumberg, R L Cohen, C Schlichtkrull, Topological Hochschild homology of Thom spectra and the free loop space, Geom. Topol. 14 (2010) 1165

[5] M Bökstedt, Topological Hochschild homology, Bielefeld (1985)

[6] M Brun, Topological Hochschild homology of $\mathbf{Z}/p^n$, J. Pure Appl. Algebra 148 (2000) 29

[7] 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)

[8] A D Elmendorf, M A Mandell, Rings, modules, and algebras in infinite loop space theory, Adv. Math. 205 (2006) 163

[9] J Hollender, R M Vogt, Modules of topological spaces, applications to homotopy limits and $E_\infty$ structures, Arch. Math. $($Basel$)$ 59 (1992) 115

[10] M Hovey, Model categories, Math. Surv. Monogr. 63, Amer. Math. Soc. (1999)

[11] M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149

[12] L G Lewis Jr., J P May, M Steinberger, J E Mcclure, Equivariant stable homotopy theory, Lecture Notes in Mathematics 1213, Springer (1986)

[13] M A Mandell, J P May, Equivariant orthogonal spectra and $S$-modules, Mem. Amer. Math. Soc. 755, Amer. Math. Soc. (2002)

[14] M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82 (2001) 441

[15] M A Mandell, B Shipley, A telescope comparison lemma for THH, Topology Appl. 117 (2002) 161

[16] J P May, $E_{\infty }$ spaces, group completions, and permutative categories, from: "New developments in topology" (editor G Segal), London Math. Soc. Lecture Note Ser. 11, Cambridge Univ. Press (1974) 61

[17] J P May, $E_{\infty }$ ring spaces and $E_{\infty }$ ring spectra, Lecture Notes in Mathematics 577, Springer (1977) 268

[18] J P May, The spectra associated to $\mathcal{I}$-monoids, Math. Proc. Cambridge Philos. Soc. 84 (1978) 313

[19] J P May, The spectra associated to permutative categories, Topology 17 (1978) 225

[20] J P May, What are $E_\infty$ ring spaces good for?, from: "New topological contexts for Galois theory and algebraic geometry" (editors A Baker, B Richter), Geom. Topol. Monogr. 16 (2009) 331

[21] J P May, What precisely are $E_\infty$ ring spaces and $E_\infty$ ring spectra?, from: "New topological contexts for Galois theory and algebraic geometry" (editors A Baker, B Richter), Geom. Topol. Monogr. 16 (2009) 215

[22] J P May, J Sigurdsson, Parametrized homotopy theory, Math. Surv. Monogr. 132, Amer. Math. Soc. (2006)

[23] J P May, R Thomason, The uniqueness of infinite loop space machines, Topology 17 (1978) 205

[24] J P Meyer, Bar and cobar constructions, II, J. Pure Appl. Algebra 43 (1986) 179

[25] C Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006) 969

[26] J Rognes, Topological logarithmic structures, from: "New topological contexts for Galois theory and algebraic geometry" (editors A Baker, B Richter), Geom. Topol. Monogr. 16 (2009) 401

[27] S Sagave, C Schlichtkrull, Diagram spaces and symmetric spectra, Adv. Math. 231 (2012) 2116

[28] C Schlichtkrull, Units of ring spectra and their traces in algebraic $K$–theory, Geom. Topol. 8 (2004) 645

[29] C Schlichtkrull, Thom spectra that are symmetric spectra, Doc. Math. 14 (2009) 699

[30] C Schlichtkrull, Higher topological Hochschild homology of Thom spectra, J. Topol. 4 (2011) 161

[31] M Schulman, Homotopy limits and colimits and enriched homotopy theory

[32] S Schwede, $S$-modules and symmetric spectra, Math. Ann. 319 (2001) 517

[33] S Schwede, On the homotopy groups of symmetric spectra, Geom. Topol. 12 (2008) 1313

[34] S Schwede, B E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000) 491

[35] B Shipley, Symmetric spectra and topological Hochschild homology, $K$–Theory 19 (2000) 155

[36] M Shulman, Comparing composites of left and right derived functors, New York J. Math. 17 (2011) 75