Derived division functors and mapping spaces
Abstract
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).
Cite
@article{arxiv.math/0208091,
title = {Derived division functors and mapping spaces},
author = {Benoit Fresse},
journal= {arXiv preprint arXiv:math/0208091},
year = {2007}
}
Comments
34 pages. I have simplified a demonstration (cf. section 5) and removed unnecessary arguments in this revised version. I have added lemma 2.4.3 which occurs in several demonstrations