DG-CATEGORIES AND SIMPLICIAL BAR COMPLEXES

We prove that the DG category KC(A) of DG complexes in 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.