Derived division functors and mapping spaces

الأصل · EN

The normalized cochain complex of a simplicial set N*(Y) is endowed with the structure of an Einfinity 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 Einfinity 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 HomF(N*(X),-). We prove that the associated left derived functor -oslashˡ N*(X) is endowed with a quasi-isomorphism N*(Y) oslashˡ N*(X) --> N* Map(X,Y).