Cyclic dendroidal sets and cyclic dendroidal spaces admit model structures for cyclic quasi-operads and complete Segal spaces, both Quillen equivalent to simplicial cyclic operads.
Quasi Modular Operads
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abstract
Modular operads are an extension of operads. In the same way that operads, as dendroidal sets, can be considered as presheaves over the category of trees, so can modular operads be considered as presheaves over a category of graphs. This paper contains a definition of the Kan condition for infinity modular operads, as well as a proof of the Nerve Theorem for modular operads, and the equivalence of the modular Kan and Segal conditions. Appendix A contains the same material for cyclic operads.
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Models for cyclic infinity operads
Cyclic dendroidal sets and cyclic dendroidal spaces admit model structures for cyclic quasi-operads and complete Segal spaces, both Quillen equivalent to simplicial cyclic operads.