# Lectures on Feynman categories

## TL;DR

This paper introduces Feynman categories, a universal categorical framework for encoding various operad-like structures and their applications across mathematics and physics, including connections to moduli spaces and Hopf algebras.

## Contribution

It provides a comprehensive introduction to Feynman categories, detailing their theoretical foundations and diverse applications in operad theory, algebra, topology, and quantum field theory.

## Key findings

- Feynman categories unify operad-like theories such as PROPs, modular operads, and properads.
- They relate to algebraic structures over operads and geometric moduli spaces.
- The framework leads to constructions of Hopf- and bi-algebras with applications in multiple fields.

## Abstract

These are expanded lecture notes from lectures given at the Workshop on higher structures at MATRIX Melbourne. These notes give an introduction to Feynman categories and their applications. Feynman categories give a universal categorical way to encode operations and relations. This includes the aspects of operad--like theories such as PROPs, modular operads, twisted (modular) operads, properads, hyperoperads and their colored versions. There is more depth to the general theory as it applies as well to algebras over operads and an abundance of other related structures, such as crossed simplicial groups, the augmented simplicial category or FI--modules. Through decorations and transformations the theory is also related to the geometry of moduli spaces. Furthermore the morphisms in a Feynman category give rise to Hopf-- and bi--algebras with examples coming from topology, number theory and quantum field theory. All these aspects are covered.