  We show that the BRST Lagrangian double copy construction of $\mathcal{N}=0$
supergravity as the `square' of Yang-Mills theory finds a natural
interpretation in terms of homotopy algebras. We significantly expand on our
previous work arguing the validity of the double copy at the loop level, and we
give a detailed derivation of the double copied Lagrangian and BRST operator.
Our constructions are very general and can be applied to a vast set of
examples.
