Differential perturbations and $D_\infty$-differential modules

In the present paper the notions of a $D_\infty$-differential and a $D_\infty$-differential module are introduced, which are, respectively, homotopically invariant analogues of the differential and the chain complex. Basic homotopic properties of $D_\infty$-differentials and $D_\infty$-differential modules are established. The connection between the Gugenheim–Lambe–Stasheff theory of differential perturbations in homological algebra and the construction of a $D_\infty$-differential module is considered.