  The Kerr-Schild double copy relates exact solutions of gauge and gravity
theories. In all previous examples, the gravity solution is associated with an
abelian-like gauge theory object, which linearises the Yang-Mills equations.
This appears to be at odds with the double copy for scattering amplitudes, in
which the non-abelian nature of the gauge theory plays a crucial role.
Furthermore, it is not yet clear whether or not global properties of classical
fields - such as non-trivial topology - can be matched between gauge and
gravity theories. In this paper, we clarify these issues by explicitly
demonstrating how magnetic monopoles associated with arbitrary gauge groups can
be double copied to the same solution (the pure NUT metric) in gravity. We
further describe how to match up topological information on both sides of the
double copy correspondence, independently of the nature of the gauge group.
This information is neatly expressed in terms of Wilson line operators, and we
argue through specific examples that they provide a useful bridge between the
classical double copy and the BCJ double copy for scattering amplitudes.
