Koszul Feynman Categories

Abstract

A cubical Feynman category, introduced by the authors in previous work, is a category whose functors to a base category C behave like operads in 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 ∞ versions of generalizations of operads for both graph based and non-graph based examples.

Koszul duality for operadic categories

“…which commutes up to multiplication by −1. Combining it with diagram (12) for the composition laws of the free operad Ω( M ), we obtain the diagram…”

“…Koszulity of the groupoid-colored operad whose algebras are modular operads was proved by Ward [25]. A general approach to graph-based operadic structures in the language of Feynman categories was suggested by Kaufmann and Ward in [12].…”