Derived division functors and mapping spaces

by Benoit Fresse
preprint (2002). Under revision (a new version should appear in April).

Abstract: The normalized cochain complex of a simplicial set $N^*(Y)$ is endowed with the structure of an $E_{\infty}$ 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's $T$ functor in the framework of $E_{\infty}$ 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 $N^*(Y)\oslash^L N_*(X)\,\rightarrow\,N^* Map(X,Y)$.

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