In the paper under review, the author introduces the notion of a differential Lie module over a curved coalgebra as a generalization of the notion of a differential Lie module over a coalgebra introduced by Smirnov. Then the author describes the basic homotopy and categorical properties of it, and applications of the homotopy technique to studying homotopy properties of differential modules over co-\(B\)-constructions of curved coalgebras Koszul dual to quadratic-scalar algebras.
These quadratic-scalar algebras are not required to be Koszul, and the commutative rings over which they are defined are not required to be fields of characteristic zero. Moreover, the author exemplifies the application of the homotopy constructions to constructing homotopy invariant counterparts of the structure of a differential module over a Clifford algebra and a differential module over an exterior algebra over any commutative unital ring.
The homotopy theory for the homotopy invariant analog of the structure of a differential module over an exterior algebra generalizes the homotopy theory of \(D_\infty\)-differential modules given by the author.