# The stack of Yang-Mills fields on Lorentzian manifolds

## TL;DR

This paper constructs an abstract and explicit framework for the stack of non-Abelian Yang-Mills fields on Lorentzian manifolds, linking the well-posedness of the Yang-Mills Cauchy problem to parametrized PDEs.

## Contribution

It introduces a homotopy theoretical approach to define and analyze the stack of Yang-Mills fields, extending existing methods to clarify the structure of classifying stacks.

## Key findings

- Established an explicit construction of the Yang-Mills stack on Lorentzian manifolds.
- Formulated a stacky version of the Yang-Mills Cauchy problem and proved its well-posedness.
- Connected the problem to a family of parametrized PDEs.

## Abstract

We provide an abstract definition and an explicit construction of the stack of non-Abelian Yang-Mills fields on globally hyperbolic Lorentzian manifolds. We also formulate a stacky version of the Yang-Mills Cauchy problem and show that its well-posedness is equivalent to a whole family of parametrized PDE problems. Our work is based on the homotopy theoretical approach to stacks proposed in [S. Hollander, Israel J. Math. 163, 93-124 (2008)], which we shall extend by further constructions that are relevant for our purposes. In particular, we will clarify the concretification of mapping stacks to classifying stacks such as $\mathrm{B}G_\mathrm{con}$.