Chain Realization of Differential Modules with $\infty$-Simplicial Faces and the $B$-Construction over $A_\infty$-Algebras

On the basis of the colored version of Koszul duality, the notion of a differential module with $\infty$-simplicial faces is introduced. By using the homotopy technique of differential Lie modules over colored coalgebras, the homotopy invariance of the structure of a differential module with $\infty$-simplicial faces is proved. A relationship between differential modules with $\infty$-simplicial faces and $A_\infty$-algebras is described. The notions of the chain realization of a differential module with $\infty$-simplicial faces and the tensor product of differential modules with $\infty$-simplicial faces are introduced. It is shown that the chain realization of a tensor differential module with $\infty$-simplicial faces constructed from an $A_\infty$-algebra and the $B$-construction over this $A_\infty$-algebra are isomorphic differential coalgebras.