Invariant classes for families of complexes

We consider families of chain-cochain infinite complexes $\mathcal{C}$ of spaces with elements depending on a number of parameters, and endowed with a converging associative multiple product. The existence of left/right local/non-local square-vanishing ideals is assumed for subspaces of $\mathcal{C}$-spaces. We show that a set of differential and orthogonality relations together with coherence conditions on indices of a chain-cochain complex $\mathcal{C}$ elements generates families of graded differential algebras. With the appropriate orthogonality conditions on completions of $\mathcal{C}$ elements in the multiple product, we define the equivalence classes of cohomology invariants.