Categorified cyclic operads

The purpose of this paper is to establish a notion of categorified cyclic operad for set-based cyclic operads with symmetries, based on individual composition operations. The categorifications we introduce are obtained by replacing sets (of operations of the same arity) with categories, by relaxing certain defining axioms (like associativity and commutativity) to isomorphisms, while leaving the equivariance strict, and by formulating coherence conditions for these isomorphisms. The coherence theorem that we prove has the form " all diagrams of canonical isomorphisms commute ". For entries-only categorified cyclic operads, our proof is of syntactic nature and relies on the coherence of categorified non-symmetric operads established by Došen and Petric. We prove the coherence of exchangeable-output categorified cyclic operads by " lifting to the categorified setting " the equivalence between entries-only and exchangeable-output cyclic operads, set up by the second author