Israel Journal of Mathematics

, Volume 202, Issue 1, pp 423–460 | Cite as

Group actions on Segal operads

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Abstract

We give a Quillen equivalence between model structures for simplicial operads, described via the theory of operads, and Segal operads, thought of as certain reduced dendroidal spaces. We then extend this result to give a Quillen equivalence between the model structures for simplicial operads equipped with a group action and the corresponding Segal operads.

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References

  1. [1]
    B. Badzioch, Algebraic theories in homotopy theory, Annals of Mathematics 155 (2002), 895–913.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    C. Berger and I. Moerdijk, On an extension of the notion of Reedy category, Mathematische Zeitschrift 269 (2011), 977–1004.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    C. Berger and I. Moerdijk, Resolution of coloured operads and rectification of homotopy algebras, in Categories in Algebra, Geometry and Mathematical Physics, Contemporary Mathematics, Vol. 431, American Mathematical Society, Providence, RI, 2007, pp. 31–58.CrossRefGoogle Scholar
  4. [4]
    J. E. Bergner, Correction to “Simplicial monoids and Segal categories”, arXiv:0806.1767 [math.AT].Google Scholar
  5. [5]
    J. E. Bergner, Rigidification of algebras over multi-sorted theories, Algebraic & Geometric Topology 6 (2006), 1925–1955.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    J. E. Bergner, Simplicial monoids and Segal categories, in Categories in Algebra, Geometry and Mathematical Physics, Contemporary Mathematics, Vol. 431, American Mathematical Society, Providence, RI, 2007, pp. 59–83.CrossRefGoogle Scholar
  7. [7]
    J. E. Bergner, Three models for the homotopy theory of homotopy theories, Topology 46 (2007), 397–436.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    J. E. Bergner, Adding inverses to diagrams encoding algebraic structures, Homology, Homotopy and Applications 10 (2008), 149–174.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    J. E. Bergner and P. Hackney, Reedy categories which encode the notion of category actions, (2012), preprint, arXiv:1207.3467 [math.AT].Google Scholar
  10. [10]
    J. M. Boardman and R. M. Vogt, Homotopy Invariant Algebraic Structures on Topological Spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin, 1973.MATHGoogle Scholar
  11. [11]
    F. Borceux, Handbook of Categorical Algebra. 1, Encyclopedia of Mathematics and its Applications, Vol. 50, Cambridge University Press, Cambridge, 1994.CrossRefGoogle Scholar
  12. [12]
    F. Borceux, Handbook of Categorical Algebra. 2, Encyclopedia of Mathematics and its Applications, Vol. 51, Cambridge University Press, Cambridge, 1994.CrossRefMATHGoogle Scholar
  13. [13]
    D.-C. Cisinski and I. Moerdijk, Dendroidal Segal spaces and ∞-operads, Journal of Topology 6 (2013), 675–704.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    D.-C. Cisinski and I. Moerdijk, Dendroidal sets and simplicial operads, (2011), Journal of Topology 6 (2013), 705–756.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    D.-C. Cisinski and I. Moerdijk, Dendroidal sets as models for homotopy operads, Journal of Topology 4 (2011), 257–299.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    W. G. Dwyer and D. M. Kan, Function complexes in homotopical algebra, Topology 19 (1980), 427–440.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    P. G. Goerss and J. F. Jardine, Simplicial Homotopy Theory, Progress in Mathematics, Vol. 174, Birkhäuser, Basel, 1999.CrossRefMATHGoogle Scholar
  18. [18]
    M. Hall, Jr., The Theory of Groups, The Macmillan Co., New York, 1959.MATHGoogle Scholar
  19. [19]
    P. S. Hirschhorn, Model Categories and their Localizations, Mathematical Surveys and Monographs, Vol. 99, American Mathematical Society, Providence, RI, 2003.MATHGoogle Scholar
  20. [20]
    M. Hovey, Model Categories, Mathematical Surveys and Monographs, Vol. 63, American Mathematical Society, Providence, RI, 1999.MATHGoogle Scholar
  21. [21]
    S. Mac Lane, Categories for the Working Mathematician, Second Edition, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1998.MATHGoogle Scholar
  22. [22]
    M. Markl, S. Shnider and J. Stasheff, Operads in Algebra, Topology and Physics, Mathematical Surveys and Monographs, Vol. 96, American Mathematical Society, Providence, RI, 2002.MATHGoogle Scholar
  23. [23]
    J. P. May, The Geometry of Iterated Loop Spaces, Lecture Notes in Mathematics, Vol. 271, Springer-Verlag, Berlin, 1972.MATHGoogle Scholar
  24. [24]
    I. Moerdijk and I. Weiss, Dendroidal sets, Algebraic & Geometric Topology 7 (2007), 1441–1470.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    D. Quillen, Homotopical Algebra, Lecture Notes in Mathematics, Vol. 43, Springer-Verlag, berlin, 1967.MATHGoogle Scholar
  26. [26]
    C. W. Rezk, Spaces of algebra structures and cohomology of operads, Thesis (Ph.D.), Massachusetts Institute of Technology, available at http://www.math.uiuc.edu/~rezk/rezk-thesis.dvi.
  27. [27]
    W. Zhang, Group Operads and Homotopy Theory, (2011), preprint, arXiv:1111.7090v2 [math.AT].Google Scholar

Copyright information

© Hebrew University of Jerusalem 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, RiversideRiversideUSA
  2. 2.Department of MathematicsUniversity of California RiversideRiversideUSA

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