  In the present paper, by using the colored version of the Koszul duality, the
concept of a differential module with $\infty$-simplicial faces is introduced.
The homotopy invariance of the structure of a differential module with
$\infty$-simplicial faces is proved. The relationships between differential
modules with $\infty$-simplicial faces and $A_\infty$-algebras are established.
The notion of a chain realization of a differential module with
$\infty$-simplicial faces and the concept of a tensor product of differential
modules with $\infty$-simplicial faces are introduced. It is proved that for an
arbitrary $A_\infty$-algebra the chain realization of the tensor differential
module with $\infty$-simplicial faces, which corresponds to this
$A_\infty$-algebra, and the $B$-construction of this $A_\infty$-algebra are
isomorphic differential coalgebras.
