Wheeled Properads and Graphical Wheeled Properads

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


We first recall from Yau and Johnson (A Foundation for PROPs, Algebras, and Modules. Mathematical Surveys and Monographs, vol. 203, Am. Math. Soc., Providence, 2015) the biased and the unbiased definitions of a wheeled properad. There is a symmetric monoidal structure on the category of wheeled properads. Then we define graphical wheeled properads as free wheeled properads generated by connected graphs, possibly with loops and directed cycles. With the exception of the exceptional wheel, a graphical wheeled properad has a finite set of elements precisely when the generating graph is simply connected. So most graphical wheeled properads are infinite. In the rest of this chapter, we discuss wheeled versions of coface maps, codegeneracy maps, and graphical maps, which are used to define the wheeled properadic graphical category \(\Gamma _{\circlearrowright }\). Every wheeled properadic graphical map has a decomposition into codegeneracy maps followed by coface maps.


Connected Graph Monoidal Category Internal Edge Symmetric Monoidal Category Graphical Identity 
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  1. [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
  2. [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

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