Categorified Cyclic Operads

  • Pierre-Louis Curien
  • Jovana ObradovićEmail author


In this paper, we introduce a notion of categorified cyclic operad for set-based cyclic operads with symmetries. Our categorification is obtained by relaxing defining axioms of cyclic operads to isomorphisms and by formulating coherence conditions for these isomorphisms. The coherence theorem that we prove has the form “all diagrams of canonical isomorphisms commute”. Our coherence results come in two flavours, corresponding to the “entries-only” and “exchangeable-output” definitions of cyclic operads. Our proof of coherence in the entries-only style is of syntactic nature and relies on the coherence of categorified non-symmetric operads established by Došen and Petrić. We obtain the coherence in the exchangeable-output style by “lifting” the equivalence between entries-only and exchangeable-output cyclic operads, set up by the second author. Finally, we show that a generalization of the structure of profunctors of Bénabou provides an example of categorified cyclic operad, and we exploit the coherence of categorified cyclic operads in proving that the Feynman category for cyclic operads, due to Kaufmann and Ward, admits an odd version.


Cyclic operads Categorification Coherence 


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.IRIF, InriaUniversité Paris Diderot and πr² TeamParisFrance
  2. 2.Institute of MathematicsCzech Academy of SciencesPragueCzech Republic

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