The combinatorics of modular operads are governed by undirected graphs of arbitrary genus, and are therefore significantly more complex than those of ordinary operads. I'll discuss a very general notion of modular operad - essentially the compact symmetric multicategories of Joyal and Kock - for which the combination of the contraction operation with a unital operadic composition presents particular challenges for constructing a suitable nerve.
In this talk, I'll break down the issues and sketch a construction of a composite monad for modular operads. The decomposition enables us to apply Weber's nerve machinery to obtain a fully faithful nerve. Perhaps more importantly, the combinatorics of modular operads - and especially the tricky bits - are made fully explicit. This provides a roadmap for generalising results to this setting, and for further generalising the setting itself, and I'll indicate some of these new directions.