Feynman graphs, and nerve theorem for compact symmetric multicategories (extended abstract)

TL;DR

This paper introduces a category of Feynman graphs and establishes a nerve theorem linking compact symmetric multicategories to presheaves on these graphs, providing a new categorical characterization.

Contribution

It defines a category of Feynman graphs and proves a nerve theorem that characterizes compact symmetric multicategories via presheaves satisfying a Segal condition.

Findings

Feynman graphs form a category related to compact symmetric multicategories

Compact symmetric multicategories are characterized as presheaves on Feynman graphs

A Segal condition is used to characterize these presheaves

Abstract

We describe a category of Feynman graphs and show how it relates to compact symmetric multicategories (coloured modular operads) just as linear orders relate to categories and rooted trees relate to multicategories. More specifically we obtain the following nerve theorem: compact symmetric multicategories can be characterised as presheaves on the category of Feynman graphs subject to a Segal condition. This text is a write-up of the second-named author's QPL6 talk; a more detailed account of this material will appear elsewhere.