  We give an interpretation of holography in the form of the AdS/CFT
correspondence in terms of homotopy algebras. A field theory such as a bulk
gravity theory can be viewed as a homotopy Lie or $L_{\infty}$ algebra. We
extend this dictionary to theories defined on manifolds with a boundary,
including the conformal boundary of AdS, taking into account the cyclic
structure needed to define an action with the correct boundary terms.
Projecting fields to their boundary values then defines a homotopy retract,
which in turn implies that the cyclic $L_{\infty}$ algebra of the bulk theory
is equivalent, up to homotopy, to a cyclic $L_{\infty}$ algebra on the
boundary. The resulting action is the `on-shell action' conventionally computed
via Witten diagrams that, according to AdS/CFT, yields the generating
functional for the correlation functions of the dual CFT. These results are
established with the help of new techniques regarding the homotopy transfer of
cyclic $L_{\infty}$ algebras.
