Advertisement

Introduction

  • Philip Hackney
  • Marcy Robertson
  • Donald Yau
Part of the Lecture Notes in Mathematics book series (LNM, volume 2147)

Abstract

A theory of -properads is developed, extending both the Joyal-Lurie -categories and the Cisinski-Moerdijk-Weiss -operads. Every connected wheel-free graph generates a properad, giving rise to the graphical category \(\Gamma \) of properads. Using graphical analogs of coface maps and the properadic nerve functor, an -properad is defined as an object in the graphical set category \(\mathtt{Set}^{\Gamma ^{\mathop{\mathrm{op}}\nolimits } }\) that satisfies some inner horn extension property. Symmetric monoidal closed structures are constructed in the categories of properads and of graphical sets. Strict -properads, in which inner horns have unique fillers, are given two alternative characterizations, one in terms of graphical analogs of the Segal maps, and the other as images of the properadic nerve. The fundamental properad of an -properad is characterized in terms of homotopy classes of 1-dimensional elements. Using all connected graphs instead of connected wheel-free graphs, a parallel theory of -wheeled properads is also developed.

Keywords

Connected Graph Homotopy Class Unital Tree Directed Cycle Linear Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [BM11]
    C. Berger, I. Moerdijk, On an extension of the notion of Reedy category. Math. Z. 269(3–4), 977–1004 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  2. [BN14]
    M. Bašić, T. Nikolaus, Dendroidal sets as models for connective spectra. J. K-Theory (2014). doi:10.1017/is014005003jkt265Google Scholar
  3. [Ber07]
    J.E. Bergner, A model category structure on the category of simplicial categories. Trans. Am. Math. Soc. 359, 2043–2058 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  4. [Ber10]
    J.E. Bergner, A survey of (, 1)-categories, in IMA Volumes in Mathematics and Its Applications (Springer, Berlin, 2010), pp. 69–83Google Scholar
  5. [BV73]
    J.M. Boardman, R.M. Vogt, Homotopy Invariant Algebraic Structures on Topological Spaces. Lecture Notes in Mathematics, vol. 347 (Springer, Berlin, 1973)Google Scholar
  6. [CM13a]
    D.-C. Cisinski, I. Moerdijk, Dendroidal Segal spaces and -operads. J. Topol. 6, 675–704 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  7. [CM11]
    D.-C. Cisinski, I. Moerdijk, Dendroidal sets as models for homotopy operads. J. Topol. 4, 257–299 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  8. [CM13b]
    D.-C. Cisinski, I. Moerdijk, Dendroidal sets and simplicial operads. J. Topol. 6, 705–756 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  9. [DH13]
    W.G. Dwyer, K. Hess, The Boardman-Vogt tensor product of operadic bimodules, (2013)Google Scholar
  10. [Gan04]
    W.L. Gan, Koszul duality for dioperads. Math. Res. Lett. 10, 109–124 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  11. [Heu11a]
    G. Heuts, Algebras over infinity-operads, (2011)Google Scholar
  12. [Heu11b]
    G. Heuts, An infinite loop space machine for infinity-operads, (2011)Google Scholar
  13. [HHM13]
    G. Heuts, V. Hinich, I. Moerdijk, The equivalence between Lurie’s model and the dendroidal model for infinity-operads, (2013)Google Scholar
  14. [JY]
    M.W. Johnson, D. Yau, Homotopy Theory and Resolutions of Algebraic Structures, (2009)Google Scholar
  15. [Joy02]
    A. Joyal, Quasi-categories and Kan complexes. J. Pure Appl. Algebra 175, 207–222 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  16. [JT06]
    A. Joyal, M. Tierney, Quasi-categories vs Segal spaces. Contemp. Math. 431, 277–326 (2007)MathSciNetCrossRefGoogle Scholar
  17. [KWZ12]
    R.M. Kaufmann, B.C. Ward, J.J. Zúñiga, The odd origin of Gerstenhaber, BV and the master equation, (2012)Google Scholar
  18. [Lei04]
    T. Leinster, Higher Operads, Higher Categories. London Mathematical Society, Lecture Note Series, vol. 298 (Cambridge University Press, Cambridge, 2004)Google Scholar
  19. [Luk13]
    A. Lukács, Dendroidal weak 2-categories, (2013)Google Scholar
  20. [Lur09]
    J. Lurie, Higher Topos Theory. Annals of Mathematics Studies (Princeton University Press, Princeton, 2009)zbMATHGoogle Scholar
  21. [Mar08]
    M. Markl, Operads and PROPs. Handbook of Algebra, vol. 5 (Elsevier, Amsterdam, 2008), pp. 87–140Google Scholar
  22. [MMS09]
    M. Markl, S. Merkulov, S. Shadrin, Wheeled PROPs, graph complexes and the master equation. J. Pure Appl. Algebra 213, 496–535 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  23. [MSS02]
    M. Markl, S. Shnider, J. Stasheff, Operads in Algebra, Topology and Physics. Mathematical Surveys and Monographs, vol. 96 (American Mathematical Society, Providence, 2002)Google Scholar
  24. [May72]
    J.P. May, The Geometry of Iterated Loop Spaces. Lectures Notes in Mathematics, vol. 271 (Springer, Berlin, 1972)Google Scholar
  25. [Mer09]
    S.A. Merkulov, Graph complexes with loops and wheels. Prog. Math. 270, 311–354 (2009)MathSciNetGoogle Scholar
  26. [Mer10a]
    S.A. Merkulov, Wheeled pro(p)file of Batalin-Vilkovisky formalism. Commun. Math. Phys. 295, 585–638 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  27. [Mer10b]
    S.A. Merkulov, Wheeled Props in Algebra, Geometry and Quantization. European Congress of Mathematics (European Mathematical Society, Zürich, 2010), pp. 83–114CrossRefGoogle Scholar
  28. [MW07]
    I. Moerdijk, I. Weiss, Dendroidal sets. Algebraic Geom. Topol. 7, 1441–1470 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  29. [MW09]
    I. Moerdijk, I. Weiss, On inner Kan complexes in the category of dendroidal sets. Adv. Math. 221, 343–389 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  30. [Nik13]
    T. Nikolaus, Algebraic K-theory of infinity-operads, (2013)Google Scholar
  31. [Rez01]
    C. Rezk, A model for the homotopy theory of homotopy theory. Trans. Am. Math. Soc. 353, 973–1007 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  32. [Sim12]
    C. Simpson, Homotopy Theory of Higher Categories, from Segal Categories to n-Categories and Beyond. New Mathematical Monographs, vol. 19 (Cambridge University Press, Cambridge, 2012)Google Scholar
  33. [Val07]
    B. Vallette, A Koszul duality for props. Trans. Am. Math. Soc. 359, 4865–4943 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  34. [YJ15]
    D. Yau, M.W. Johnson, A Foundation for PROPs, Algebras, and Modules. Mathematical Surveys and Monographs, vol. 203 (Am. Math. Soc., Providence, 2015)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Philip Hackney
    • 1
  • Marcy Robertson
    • 2
  • Donald Yau
    • 3
  1. 1.Stockholm UniversityStockholmSweden
  2. 2.University of CaliforniaLos AngelesUSA
  3. 3.Ohio State University, Newark CampusNewarkUSA

Personalised recommendations