  Colour-kinematics duality is a remarkable property of Yang-Mills theory. Its
validity implies a relation between gauge theory and gravity scattering
amplitudes, known as double copy. Albeit fully established at the tree level,
its extension to the loop level is conjectural. Lifting the on-shell,
scattering amplitudes-based description to the level of action functionals, we
argue that a theory that exhibits tree-level colour-kinematics duality can be
reformulated in a way such that its loop integrands manifest a generalised form
of colour-kinematics duality. Moreover, we show how the structures of higher
homotopy theory naturally describe this off-shell reformulation of
colour-kinematics duality.
