1911.05914Nov 14, 2019A 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

Graphical combinatorics and a distributive law for modular operads

Sophie Raynor

Category Theory

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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.

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