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Localisation homotopique et foncteurs entre espaces vectoriels

Abstract : We study categories of functors from a small additive category to a category of modules. For this purpose, we use localization technics in the homotopy categoriy of simplicial objects. Define the degree of a polynomial functor through cross-effects. We first justify the existence of a localization where local objects are those simplicial functors having homotopy groups of degree n or less. The category of functors is filtered by the sequence of sub-categories of functors of degree n. This filtration gives raise to a tower of homotopical localizations. Next we describe the n-th fiber of this tower. It uses the category of symetric n-variables functors and localizations in this setting. Here, local functors are those having linear homotopy groups in each variable. In the case of functors from the category of free modules of finite rank, we obtain another description, using simplicial modules on a simplicial ring. For vector spaces on the field with two elements, the homotopy of the simplicial ring in the n-th tensor product of the dual Steenrod algebra. We end with a calculationof homotopy groups of the simplicial modules obtained from functors associated to the symetric, the exterior and the divided power algebra.
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Submitted on : Monday, October 25, 2004 - 4:44:13 PM
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  • HAL Id : tel-00007207, version 1



Olivier Renaudin. Localisation homotopique et foncteurs entre espaces vectoriels. Mathématiques [math]. Université de Nantes, 2000. Français. ⟨tel-00007207⟩



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