An algebraic model for commutative Hℤ–algebras

Richter, Birgit¹ ; Shipley, Brooke²

¹Department Mathematik, Universität Hamburg, Hamburg, Germany
²Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, United States

Algebraic and Geometric Topology, Tome 17 (2017) no. 4, pp. 2013-2038

Résumé

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.

Détail

DOI : 10.2140/agt.2017.17.2013

Classification : 55P43
Keywords: Eilenberg–Mac Lane spectra, symmetric spectra, $E_\infty$–differential graded algebras, Dold–Kan correspondence

Bibliographie

[1] C Berger, I Moerdijk, Axiomatic homotopy theory for operads, Comment. Math. Helv. 78 (2003) 805 | DOI

[2] C Berger, I Moerdijk, Resolution of coloured operads and rectification of homotopy algebras, from: "Categories in algebra, geometry and mathematical physics" (editors A Davydov, M Batanin, M Johnson, S Lack, A Neeman), Contemp. Math. 431, Amer. Math. Soc. (2007) 31 | DOI

[3] B Cenkl, Cohomology operations from higher products in the de Rham complex, Pacific J. Math. 140 (1989) 21 | DOI

[4] S G Chadwick, M A Mandell, En genera, Geom. Topol. 19 (2015) 3193 | DOI

[5] J D Christensen, M Hovey, Quillen model structures for relative homological algebra, Math. Proc. Cambridge Philos. Soc. 133 (2002) 261 | DOI

[6] T Church, J S Ellenberg, B Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015) 1833 | DOI

[7] D Dugger, Replacing model categories with simplicial ones, Trans. Amer. Math. Soc. 353 (2001) 5003 | DOI

[8] P G Goerss, M J Hopkins, Moduli spaces of commutative ring spectra, from: "Structured ring spectra" (editors A Baker, B Richter), London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 151 | DOI

[9] J E Harper, Homotopy theory of modules over operads in symmetric spectra, Algebr. Geom. Topol. 9 (2009) 1637 | DOI

[10] P S Hirschhorn, Model categories and their localizations, 99, Amer. Math. Soc. (2003)

[11] M Hovey, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 165 (2001) 63 | DOI

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

[13] A Joyal, The theory of quasi-categories and its applications, lecture notes (2008)

[14] J Lurie, Higher algebra, unpublished manuscript (2014)

[15] M A Mandell, Flatness for the E∞ tensor product, from: "Homotopy methods in algebraic topology" (editors J P C Greenlees, R R Bruner, N Kuhn), Contemp. Math. 271, Amer. Math. Soc. (2001) 285 | DOI

[16] M A Mandell, Topological André–Quillen cohomology and E∞ André–Quillen cohomology, Adv. Math. 177 (2003) 227 | DOI

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

[18] J P May, K Ponto, More concise algebraic topology: localization, completion, and model categories, Univ. of Chicago Press (2012)

[19] D Pavlov, J Scholbach, Admissibility and rectification of colored symmetric operads, preprint (2014)

[20] D Pavlov, J Scholbach, Symmetric operads in abstract symmetric spectra, preprint (2014)

[21] B Richter, On the homology and homotopy of commutative shuffle algebras, Israel J. Math. 209 (2015) 651 | DOI

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

[23] C Schlichtkrull, The homotopy infinite symmetric product represents stable homotopy, Algebr. Geom. Topol. 7 (2007) 1963 | DOI

[24] R Schwänzl, R M Vogt, Strong cofibrations and fibrations in enriched categories, Arch. Math. Basel 79 (2002) 449 | DOI

[25] S Schwede, An untitled book project about symmetric spectra,

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

[27] S Schwede, B Shipley, Equivalences of monoidal model categories, Algebr. Geom. Topol. 3 (2003) 287 | DOI

[28] B Shipley, A convenient model category for commutative ring spectra, from: "Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic –theory" (editors P Goerss, S Priddy), Contemp. Math. 346, Amer. Math. Soc. (2004) 473 | DOI

[29] B Shipley, HZ–algebra spectra are differential graded algebras, Amer. J. Math. 129 (2007) 351 | DOI

[30] M Spitzweck, Operads, algebras and modules in general model categories, preprint (2001)

[31] D White, Model structures on commutative monoids in general model categories, preprint (2014)