Derived division functors and mapping spaces

The normalized cochain complex of a simplicial set N(Y ) is endowed with the structure of an E∞ algebra. More specifically, we prove in a previous article that N (Y ) is an algebra over the Barratt-Eccles operad (cf. [2]). According to M. Mandell, under reasonable completeness assumptions, this algebra structure determines the homotopy type of Y (cf. [14]). 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 (cf. [11]) in the framework of E∞ algebras. Precisely, in the category of algebras over the Barratt-Eccles operad, we have a division functor −⊘N∗(X) which is left adjoint to the functor HomF(N∗(X),−). We prove that the associated left derived functor −⊘ N∗(X) is endowed with a quasi-isomorphism N(Y )⊘ N∗(X) ∼ −−→N Map(X, Y ).