Algebraic & Geometric Topology

An algebraic model for commutative $H\mskip-1mu\mathbb{Z}$–algebras

Birgit Richter and Brooke Shipley

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We show that the homotopy category of commutative algebra spectra over the Eilenberg–Mac Lane spectrum of an arbitrary commutative ring R is equivalent to the homotopy category of E–monoids in unbounded chain complexes over R. We do this by establishing a chain of Quillen equivalences between the corresponding model categories. We also provide a Quillen equivalence to commutative monoids in the category of functors from the category of finite sets and injections to unbounded chain complexes.

Article information

Algebr. Geom. Topol., Volume 17, Number 4 (2017), 2013-2038.

Received: 29 September 2015
Revised: 9 December 2016
Accepted: 11 January 2017
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)

Eilenberg–Mac Lane spectra symmetric spectra $E_\infty$–differential graded algebras Dold–Kan correspondence


Richter, Birgit; Shipley, Brooke. An algebraic model for commutative $H\mskip-1mu\mathbb{Z}$–algebras. Algebr. Geom. Topol. 17 (2017), no. 4, 2013--2038. doi:10.2140/agt.2017.17.2013.

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  • C Berger, I Moerdijk, Axiomatic homotopy theory for operads, Comment. Math. Helv. 78 (2003) 805–831
  • C Berger, I Moerdijk, Resolution of coloured operads and rectification of homotopy algebras, from “Categories in algebra, geometry and mathematical physics” (A Davydov, M Batanin, M Johnson, S Lack, A Neeman, editors), Contemp. Math. 431, Amer. Math. Soc. (2007) 31–58
  • B Cenkl, Cohomology operations from higher products in the de Rham complex, Pacific J. Math. 140 (1989) 21–33
  • S,G Chadwick, M,A Mandell, $E_n$ genera, Geom. Topol. 19 (2015) 3193–3232
  • J,D Christensen, M Hovey, Quillen model structures for relative homological algebra, Math. Proc. Cambridge Philos. Soc. 133 (2002) 261–293
  • T Church, J,S Ellenberg, B Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015) 1833–1910
  • D Dugger, Replacing model categories with simplicial ones, Trans. Amer. Math. Soc. 353 (2001) 5003–5027
  • P,G Goerss, M,J Hopkins, Moduli spaces of commutative ring spectra, from “Structured ring spectra” (A Baker, B Richter, editors), London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 151–200
  • J,E Harper, Homotopy theory of modules over operads in symmetric spectra, Algebr. Geom. Topol. 9 (2009) 1637–1680
  • P,S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, Amer. Math. Soc. (2003)
  • M Hovey, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 165 (2001) 63–127
  • M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149–208
  • A Joyal, The theory of quasi-categories and its applications, lecture notes (2008) Available at \setbox0\makeatletter\@url {\unhbox0
  • J Lurie, Higher algebra, unpublished manuscript (2014) Available at \setbox0\makeatletter\@url {\unhbox0
  • M,A Mandell, Flatness for the $E_\infty$ tensor product, from “Homotopy methods in algebraic topology” (J,P,C Greenlees, R,R Bruner, N Kuhn, editors), Contemp. Math. 271, Amer. Math. Soc. (2001) 285–309
  • M,A Mandell, Topological André–Quillen cohomology and $E_\infty$ André–Quillen cohomology, Adv. Math. 177 (2003) 227–279
  • M,A Mandell, J,P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82 (2001) 441–512
  • J,P May, K Ponto, More concise algebraic topology: localization, completion, and model categories, Univ. of Chicago Press (2012)
  • D Pavlov, J Scholbach, Admissibility and rectification of colored symmetric operads, preprint (2014)
  • D Pavlov, J Scholbach, Symmetric operads in abstract symmetric spectra, preprint (2014)
  • B Richter, On the homology and homotopy of commutative shuffle algebras, Israel J. Math. 209 (2015) 651–682
  • S Sagave, C Schlichtkrull, Diagram spaces and symmetric spectra, Adv. Math. 231 (2012) 2116–2193
  • C Schlichtkrull, The homotopy infinite symmetric product represents stable homotopy, Algebr. Geom. Topol. 7 (2007) 1963–1977
  • R Schwänzl, R,M Vogt, Strong cofibrations and fibrations in enriched categories, Arch. Math. $($Basel$)$ 79 (2002) 449–462
  • S Schwede, An untitled book project about symmetric spectra Available at \setbox0\makeatletter\@url {\unhbox0
  • S Schwede, B,E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000) 491–511
  • S Schwede, B Shipley, Equivalences of monoidal model categories, Algebr. Geom. Topol. 3 (2003) 287–334
  • B Shipley, A convenient model category for commutative ring spectra, from “Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$–theory” (P Goerss, S Priddy, editors), Contemp. Math. 346, Amer. Math. Soc. (2004) 473–483
  • B Shipley, $H\mathbb Z$–algebra spectra are differential graded algebras, Amer. J. Math. 129 (2007) 351–379
  • M Spitzweck, Operads, algebras and modules in general model categories, preprint (2001)
  • D White, Model structures on commutative monoids in general model categories, preprint (2014)