Homotopy Properties of $\infty$-Simplicial Coalgebras and Homotopy Unital Supplemented $A_\infty$-Algebras

The homotopy theory of $\infty$-simplicial coalgebras is developed; in terms of this theory, an additional structure on the tensor bigraded coalgebra of a graded module is described such that endowing the coalgebra with this structure is equivalent to endowing the given graded module with the structure of a homotopy unital $A_\infty$-algebra.