  Asymptotic observables in quantum field theory beyond the familiar $S$-matrix
have recently attracted much interest, for instance in the context of gravity
waveforms. Such observables can be understood in terms of
Schwinger-Keldysh-type 'amplitudes' computed by a set of modified Feynman rules
involving cut internal legs and external legs labelled by time-folds.
  In parallel, a homotopy-algebraic understanding of perturbative quantum field
theory has emerged in recent years. In particular, passing through homotopy
transfer, the $S$-matrix of a perturbative quantum field theory can be
understood as the minimal model of an associated (quantum) $L_\infty$-algebra.
  Here we bring these two developments together. In particular, we show that
Schwinger-Keldysh amplitudes are naturally encoded in an $L_\infty$-algebra,
similar to ordinary scattering amplitudes. As before, they are computed via
homotopy transfer, but using deformation-retract data that are not canonical
(in contrast to the conventional $S$-matrix). We further show that the
$L_\infty$-algebras encoding Schwinger-Keldysh amplitudes and ordinary
amplitudes are quasi-isomorphic (meaning, in a suitable sense, equivalent).
This entails a set of recursion relations that enable one to compute
Schwinger-Keldysh amplitudes in terms of ordinary amplitudes or vice versa.
