# A graphical category for higher modular operads

## TL;DR

This paper develops a homotopy theory for a weak version of modular operads using a Quillen model structure on simplicial presheaves over a new category of undirected graphs, extending operad theory.

## Contribution

Introduces a novel category of undirected graphs, $ extbf{U}$, and establishes a homotopy theory for weak modular operads via a Quillen model structure.

## Key findings

- Defined the category $ extbf{U}$ for undirected graphs
- Established a Quillen model structure on simplicial presheaves over $ extbf{U}$
- Extended the framework to cyclic and stable modular operads

## Abstract

We present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy. This homotopy theory takes the form of a Quillen model structure on the collection of simplicial presheaves for a certain category of undirected graphs. This new category of undirected graphs, denoted $\mathbf{U}$, plays a similar role for modular operads that the dendroidal category $\Omega$ plays for operads. We carefully study properties of $\mathbf{U}$, including the existence of certain factorization systems. Related structures, such as cyclic operads and stable modular operads, can be similarly treated using categories derived from $\mathbf{U}$.