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The cotriple resolution of differential graded algebras

Author: Benoit Fresse
Journal: Proc. Amer. Math. Soc. 144 (2016), 4693-4707
MSC (2010): Primary 18D50; Secondary 18G55, 18G30
Published electronically: May 23, 2016
MathSciNet review: 3544521
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Abstract: We consider the cotriple resolution of algebras over operads in differential graded modules. We focus, to be more precise, on the example of algebras over the differential graded Barratt-Eccles operad and on the example of commutative algebras. We prove that the geometric realization of the cotriple resolution (in the sense of model categories) gives a cofibrant resolution functor on these categories of differential graded algebras.

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  • [1] Clemens Berger and Benoit Fresse, Combinatorial operad actions on cochains, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 1, 135-174. MR 2075046,
  • [2] A. K. Bousfield, Cosimplicial resolutions and homotopy spectral sequences in model categories, Geom. Topol. 7 (2003), 1001-1053 (electronic). MR 2026537,
  • [3] A. K. Bousfield and V. K. A. M. Gugenheim, On $ {\rm PL}$ de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94. MR 0425956
  • [4] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573
  • [5] A. K. Bousfield, C. Peterson, and L. Smith, The rational homology of function spaces, Arch. Math. (Basel) 52 (1989), no. 3, 275-283. MR 989883,
  • [6] David Chataur and Katsuhiko Kuribayashi, An operadic model for a mapping space and its associated spectral sequence, J. Pure Appl. Algebra 210 (2007), no. 2, 321-342. MR 2320001,
  • [7] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997. With an appendix by M. Cole. MR 1417719
  • [8] Benoit Fresse, Koszul duality of operads and homology of partition posets, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $ K$-theory, Contemp. Math., vol. 346, Amer. Math. Soc., Providence, RI, 2004, pp. 115-215. MR 2066499,
  • [9] Benoit Fresse, Modules over operads and functors, Lecture Notes in Mathematics, vol. 1967, Springer-Verlag, Berlin, 2009. MR 2494775
  • [10] Benoit Fresse.
    Homotopy of operads and Grothendieck-Teichmüller groups.
    Book project. Manuscript available at the url:˜ fresse/OperadHomotopyBook, 2015.
  • [11] John E. Harper and Kathryn Hess, Homotopy completion and topological Quillen homology of structured ring spectra, Geom. Topol. 17 (2013), no. 3, 1325-1416. MR 3073927,
  • [12] Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
  • [13] Michael A. Mandell, $ E_\infty $ algebras and $ p$-adic homotopy theory, Topology 40 (2001), no. 1, 43-94. MR 1791268,
  • [14] Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269-331 (1978). MR 0646078

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Additional Information

Benoit Fresse
Affiliation: CNRS, UMR 8524 - Laboratoire Paul Painlevé, University Lille, F-59000 Lille, France

Received by editor(s): April 6, 2015
Received by editor(s) in revised form: February 4, 2016
Published electronically: May 23, 2016
Additional Notes: This research was supported in part by grant ANR-11-BS01-002 “HOGT” and by Labex ANR-11-LABX-0007-01 “CEMPI”. The author is grateful to Paul Goerss, who introduced him to the homotopical constructions studied in this paper during his first visit at Northwestern University a long time ago. The author also thanks Victor Turchin and Thomas Willwacher who provided the author the motivation to write this article. Lastly, the author thanks the referee for instructive comments on analogues of the result of this article in the field of stable homotopy theory
Communicated by: Michael A. Mandell
Article copyright: © Copyright 2016 American Mathematical Society

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