Derived division functors and mapping spaces

The normalized cochain complex of a simplicial set N^(Y) is endowed with the structure of an E_{infinity} algebra. More specifically, we prove in a previous article that N^(Y) is an algebra over the Barratt-Eccles operad. According to M. Mandell, under reasonable completeness assumptions, this algebra structure determines the homotopy type of Y. In this article, we construct a model of the mapping space Map(X,Y). For that purpose, we extend the formalism of Lannes’ T functor in the framework of E_{infinity} algebras. Precisely, in the category of algebras over the Barratt-Eccles operad, we have a division functor -oslash N_(X) which is left adjoint to the functor Hom_F(N_(X),-). We prove that the associated left derived functor -oslash^L N_(X) is endowed with a quasi-isomorphism N^(Y) oslash^L N_(X) --> N^* Map(X,Y).