Abstract
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.
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References
Bénabou, J.: Les distributeurs. Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée, rapport 33 (1973)
Borceux, F.: Handbook of Categorical Algebra I: Basic Category Theory. Cambridge University Press, Cambridge (1994)
Burroni, A.: Higher-dimensional word problems with applications to equational logic. Theor. Comput. Sci. 115(1), 43–62 (1993)
Chapoton, F.: Anticyclic Operads and Auslander-Reiten Translation, slides (2006)
Cheng, E., Gurski, N., Riehl, E.: Cyclic multicategories, multivariable adjunctions and mates. J. K-Theory 13(2), 337–396 (2014)
Curien, P.-L., Mimram, S.: Coherent presentations of monoidal categories. Log. Methods Comput. Sci. 13(3), 1–38 (2017)
Curien, P.-L., Obradović, J.: A formal language for cyclic operads. High. Struct. 1(1), 22–55 (2017)
Curien, P.-L., Obradovic, J., Ivanovic, J.: Syntactic aspects of hypergraph polytopes. J. Homotopy Relat. Struct. 14(1), 235–279 (2019)
Day, B., Street, R.: Lax monoids, pseudo-operads and convolution. Contemp. Math. 318, 75–96 (2003)
Dehling, M., Vallette, B.: Symmetric homotopy theory for operads. (2015). arXiv:1503.02701
Došen, K., Petrić, Z.: Hypergraph polytopes. Topol. Appl. 158, 1405–1444 (2011)
Došen, K., Petrić, Z.: Weak Cat-operads. Log. Methods Comput. Sci. 11(1), 1–23 (2015)
Doubek, M., Jurčo, B., Markl, M., Sachs, I.: Algebraic structure of String Field theory (2016) (unpublished manuscript)
Getzler, E., Kapranov, M.: Cyclic operads and cyclic homology. In: Bott, R. (ed.) Geometry, Topology and Physics for Raoul Bott, pp. 167–201. International Press, Cambridge (1995)
Guiraud, Y., Malbos, P.: Polygraphs of finite derivation type. Math. Struct. Comput. Sci. 28(2), 155–201 (2018)
Kaufmann, R.M., Ward, B.C.: Feynman categories. In: Astérisque (Société Mathématique de France). Numéro, vol. 387 (2017)
Kelly, G.M.: On the operads of. J. P. May. Repr. Theory Appl. Categ. 13, 1–13 (2005)
Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, Berlin (1998)
Markl, M.: Operads and PROPs. In: Bott, R. (ed.) Handbook for Algebra, vol. 5, pp. 87–140. Elsevier/North-Holland, Amsterdam (2008)
Markl, M.: Modular envelopes, OSFT and nonsymmetric (non-\(\Sigma \)) modular operads. J. Noncommut. Geom. 10, 775–809 (2016)
Markl, M., Schnider, S., Stasheff, J.: Operads in Algebra, Topology and Physics. American Mathematical Society, Providence (2002)
Obradović, J.: Monoid-like definitions of cyclic operads. Theory Appl. Categ. 32(12), 396–436 (2017)
Street, R.: Limits indexed by category-valued 2-functors. J. Pure Appl. Algebra 8(2), 149–181 (1976)
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Communicated by Stephen Lack.
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The work on the final version of this paper was supported by Praemium Academiae of M. Markl and RVO: 67985840. Jovana Obradović was additionally supported by the Grant GA CR P201/12/G028.
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Curien, PL., Obradović, J. Categorified Cyclic Operads. Appl Categor Struct 28, 59–112 (2020). https://doi.org/10.1007/s10485-019-09569-7
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DOI: https://doi.org/10.1007/s10485-019-09569-7