Properadic Graphical Sets and Infinity Properads

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


We define the category \(\mathtt{Set}^{\Gamma ^{\mathop{\mathrm{op}}\nolimits } }\) of graphical sets. There is an adjoint pair Open image in new window in which the right adjoint N is called the properadic nerve. The symmetric monoidal product of properads in Chap.  4 induces, via the properadic nerve, a symmetric monoidal closed structure on \(\mathtt{Set}^{\Gamma ^{\mathop{\mathrm{op}}\nolimits } }\). Then we define an -properad as a graphical set in which every inner horn has a filler. If, furthermore, every inner horn filler is unique, then it is called a strict -properad. The rest of this chapter contains two alternative descriptions of a strict -properad. One description is in terms of the graphical analogs of the Segal maps, and the other is in terms of the properadic nerve.


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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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