# Modular operads and the nerve theorem

## TL;DR

This paper establishes a nerve theorem for modular operads using a category of undirected graphs, characterizing modular operads via a Segal condition and connecting to prior work by Joyal and Kock.

## Contribution

It introduces a category of undirected graphs with a faithful functor into modular operads and characterizes the image via a Segal condition, extending previous results.

## Key findings

- The singular functor from modular operads to presheaves is fully faithful.
- The essential image of this functor is characterized by a Segal condition.
- A connection to Joyal and Kock's larger graph category is established.

## Abstract

We describe a category of undirected graphs which comes equipped with a faithful functor into the category of (colored) modular operads. The associated singular functor from modular operads to presheaves is fully faithful, and its essential image can be classified by a Segal condition. This theorem can be used to recover a related statement, due to Andr\'e Joyal and Joachim Kock, concerning a larger category of undirected graphs whose functor to modular operads is not just faithful but also full.