Koszul Feynman Categories

TL;DR

This paper proves that all cubical Feynman categories are Koszul, providing explicit minimal resolutions that facilitate modeling of infinity operads and their generalizations in various mathematical contexts.

Contribution

It establishes the Koszul property for cubical Feynman categories and constructs explicit minimal cofibrant resolutions for them.

Findings

All cubical Feynman categories are Koszul.

Provides explicit minimal cofibrant resolutions.

Enables modeling of infinity operads and their generalizations.

Abstract

A cubical Feynman category, introduced by the authors in previous work, is a category whose functors to a base category $\mathcal{C}$ behave like operads in $\mathcal{C}$. In this note we show that every cubical Feynman category is Koszul. The upshot is an explicit, minimal cofibrant resolution of any cubical Feynman category, which can be used to model $\infty$ versions of generalizations of operads for both graph based and non-graph based examples.