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

Mapping Spaces for DG Hopf Cooperads and Homotopy Automorphisms of the Rationalization of $E_n$-operads