Graphical combinatorics and a distributive law for modular operads
arXiv:1911.05914 · doi:10.1016/j.aim.2021.108011
Abstract
This work presents a detailed analysis of the combinatorics of modular operads. These are operad-like structures that admit a contraction operation as well as an operadic multiplication. Their combinatorics are governed by graphs that admit cycles, and are known for their complexity. In 2011, Joyal and Kock introduced a powerful graphical formalism for modular operads. This paper extends that work. A monad for modular operads is constructed and a corresponding nerve theorem is proved, using Weber's abstract nerve theory, in the terms originally stated by Joyal and Kock. This is achieved using a distributive law that sheds new light on the combinatorics of modular operads.
A mistake in the published version, where I incorrectly referred to certain graph morphisms as `monomorphisms' has been corrected, and the terminology updated throughout. See Rmk 0.6 and Section 4.1. Some other tidying up and minor corrections have also been made