  For a Hopf DG-algebra corresponding to a derived algebraic group, we compute
the homotopy limit of the associated cosimplicial system of DG-algebras given
by the classifying space construction. The homotopy limit is taken in the model
category of DG-categories. The objects of the resulting DG-category are
Maurer-Cartan elements of $\operatorname{Cobar}(A)$, or 1-dimensional
$A_\infty$-comodules over $A$. These can be viewed as characters up to homotopy
of the corresponding derived group. Their tensor product is interpreted in
terms of Kadeishvili's multibraces. We also study the coderived category of
DG-modules over this DG-category.
