Homotopy characters as a homotopy limit

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.