DG-categories and simplicial bar complexes

We prove that the DG category $K\mathcal C_A$ of DG complexes in $\mathcal C_A$ assocaited to a DGA $A$, is homotopy equivalent to that of comodules over the bar complex of $A$. We introduce simplicial bar complexes to give the homotopy equivalence. As an application, we show that the category of comodules over the 0-th cohomology of the bar complex of the Deligne algebra is equivalent to that of variations of mixed Tate Hodge structures on an algebraic variety.