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I begin with a self-contained introduction to Homotopy algebras, which
are generalizations of familiar structures such as Lie or associative
algebras that in physics emerged in string theory but that more recently have
begun to be recognized as the underlying structure of general classical and
quantum field theories. This framework allows one, in particular, to formulate two deep connections between gauge theories such as Yang-Mills theory and gravity, as a first step toward a first-principle derivation: these are, first, the so-called double copy relations between the scattering amplitudes of gauge theory and of gravity and, second, the holographic or AdS/CFT relation between a gravity theory on AdS and a dual CFT on the boundary.