Graphical combinatorics and a distributive law for modular operads

TL;DR

This paper advances the understanding of modular operads by extending graphical formalisms, constructing a monad, and proving a nerve theorem using a distributive law to clarify their complex combinatorics.

Contribution

It introduces a new distributive law for modular operads, constructs a monad, and proves a nerve theorem, extending Joyal and Kock's graphical formalism.

Findings

A monad for modular operads is constructed.

A nerve theorem for modular operads is proved.

A new distributive law elucidates the combinatorics of modular operads.

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.