  We define a simplicial enrichment on the category of differential graded Hopf
cooperads (the category of dg Hopf cooperads for short). We prove that our
simplicial enrichment satisfies, in part, the axioms of a simplicial model
category structure on the category of dg Hopf cooperads. We use this simplicial
model structure to define a model of mapping spaces in the category of dg Hopf
cooperads and to upgrade results of the literature about the homotopy
automorphism spaces of dg Hopf cooperads by dealing with simplicial monoid
structures. The rational homotopy theory of operads implies that the homotopy
automorphism spaces of dg Hopf cooperads can be regarded as models for the
homotopy automorphism spaces of the rationalization of operads in topological
spaces (or in simplicial sets). We prove, as a main application, that the
spaces of Maurer--Cartan forms on the Kontsevich graph complex Lie algebras are
homotopy equivalent, in the category of simplicial monoids, to the homotopy
automorphism spaces of the rationalization of the operads of little discs.
