  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 $\mathrm{B}G_\mathrm{con}$.
