Feynman categories and Representation Theory

TL;DR

This paper presents a new algebraic perspective on Feynman categories, viewing them as monoidal categories and their representations as monoidal functors, expanding their applications in various mathematical fields.

Contribution

It introduces a new algebraic approach to Feynman categories, providing additional examples, details, and applications in representation theory and related fields.

Findings

New algebraic framework for Feynman categories

Extended examples and constructions

Broader applications in mathematics and physics

Abstract

We give a presentation of Feynman categories from a representation--theoretical viewpoint. Feynman categories are a special type of monoidal categories and their representations are monoidal functors. They can be viewed as a far reaching generalization of groups, algebras and modules. Taking a new algebraic approach, we provide more examples and more details for several key constructions. This leads to new applications and results. The text is intended to be a self--contained basis for a crossover of more elevated constructions and results in the fields of representation theory and Feynman categories, whose applications so far include number theory, geometry, topology and physics.