Decorated Feynman Categories

TL;DR

This paper extends the concept of Feynman categories by introducing decorated Feynman categories, enabling a unified framework for various operad-like structures and their applications in geometry and number theory.

Contribution

It introduces decorated Feynman categories, providing a systematic way to derive diverse operad-like structures and explaining their underlying categorical foundations.

Findings

Unified framework for non-sigma operads and cyclic operads

Construction of graph complexes from decorated Feynman categories

Functoriality of the decoration procedure

Abstract

In [KW14], the new concept of Feynman categories was introduced to simplify the discussion of operad--like objects. In this present paper, we demonstrate the usefulness of this approach, by introducing the concept of decorated Feynman categories. The procedure takes a Feynman category $\mathfrak F$ and a functor $\mathcal O$ to a monoidal category to produce a new Feynman category ${\mathfrak F}_{dec {\mathcal O}}$. This in one swat explains the existence of non--sigma operads, non--sigma cyclic operads, and the non--sigma--modular operads of Markl as well as all the usual candidates simply from the category $\mathfrak G$, which is a full subcategory of the category of graphs of [BM08]. Moreover, we explain the appearance of terminal objects noted in [Mar15]. We can then easily extend this for instance to the dihedral case. Furthermore, we obtain graph complexes and all other known operadic type notions from decorating and restricting the basic Feynman category $\mathfrak G$ of aggregates of corollas. We additionally show that the construction is functorial. There are further geometric and number theoretic applications, which will follow in a separate preprint.