Schwarz Modular Operads Revisited: $\mathcal{SM}=\mathcal{S}\circ\mathcal{M}$

TL;DR

This paper proves that a specific Feynman category related to Schwarz's modular operads is Koszul, using a generalized distributive law theory for groupoid-colored settings.

Contribution

It introduces a generalized approach to distributive laws in groupoid-colored contexts and establishes the Koszul property for Schwarz's modular operad Feynman category.

Findings

The Feynman category for Schwarz's modular operads is Koszul.

Generalization of distributive laws to groupoid-colored settings.

Provides foundational results for algebraic structures related to modular operads.

Abstract

We prove that the Feynman category encoding Schwarz's variant of modular operads is Koszul. Our proof uses a generalization of the theory of distributive laws to the groupoid colored setting.