# Coloring Operads for Algebraic Field Theory

## TL;DR

This paper formalizes algebraic field theories using operads, constructing operads whose algebras correspond to theories satisfying Einstein causality, and explores adjunctions and quantization within this framework.

## Contribution

It introduces a novel operadic framework for algebraic field theories, including constructions of field theory operads and their algebraic properties.

## Key findings

- Constructed field theory operads satisfying Einstein causality.
- Established adjunctions between different field theory types.
- Developed a compatible quantization functor for linear theories.

## Abstract

In these proceedings we summarize previous work where we formalize a general concept of algebraic field theories using operads. After giving a gentle reminder of algebraic quantum field theory, operads and their algebras, we construct field theory operads, whose algebras are exactly algebraic field theories. Specifically, they satisfy a suitable version of the Einstein causality axiom. From this construction we get adjunctions between different types of field theories, including adjunctions related to local-to-global extensions and the time-slice axiom, and a quantization functor for linear field theories that is compatible with these structures. We also take first steps towards a derived linear quantization functor.