Quantum field theories on categories fibered in groupoids

TL;DR

This paper develops a flexible, abstract framework for quantum field theories on categories fibered in groupoids, enabling the study of theories with additional geometric structures and their homotopical generalizations.

Contribution

It introduces a novel categorical approach to quantum field theories on structured spacetimes and extends it to homotopy-theoretic models resembling gauge theories.

Findings

Framework for quantum field theories on fibered categories

Use of right Kan extensions to relate to standard QFTs

Homotopy extensions leading to toy models of gauge theories

Abstract

We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as bundles, connections and spin structures. Using right Kan extensions, we can assign to any such theory an ordinary quantum field theory defined on the category of spacetimes and we shall clarify under which conditions it satisfies the axioms of locally covariant quantum field theory. The same constructions can be performed in a homotopy theoretic framework by using homotopy right Kan extensions, which allows us to obtain first toy-models of homotopical quantum field theories resembling some aspects of gauge theories.