Differential Lie Modules over Curved Colored Coalgebras and $\infty$-Simplicial Modules

The notion of differential Lie module over a curved colored coalgebra is introduced. The homotopy invariance of the structure of differential Lie module over a curved colored coalgebra is proved. The notion of $\infty$-simplicial module is introduced using the construction of a differential Lie module over a curved colored coalgebra and the Koszul duality theory for quadratic-scalar colored algebras. The homotopy invariance of the structure of a $\infty$-simplicial module is proved.