Derived division functors and mapping spaces

TL;DR

This paper develops a new model for mapping spaces between topological spaces using E-infinity algebra structures and an extension of Lannes' T functor, providing a homotopical algebra framework for understanding these spaces.

Contribution

It extends the formalism of Lannes' T functor to E-infinity algebras and constructs a model for mapping spaces via derived division functors in this setting.

Findings

Constructed a model of Map(X,Y) using E-infinity algebra techniques.

Extended Lannes' T functor formalism to the category of E-infinity algebras.

Proved the derived division functor yields a quasi-isomorphism to the cochains of the mapping space.

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).