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


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.


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.


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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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