Astérisque - Parutions - 338 (2011) < Parutions < 2011

Operads and chain rules for the calculus of functors
Greg Arone, Michael Ching
Astérisque 338 (2011), 158 pages
Acheter l'ouvrage

Résumé :
Opérades et règles de la chaîne pour le calcul fonctoriel
Nous étudions la structure des dérivées de Goodwillie d'un foncteur d'homotopie pointé d'espaces topologiques possédant une base. Ces dérivées forment, de manière naturelle, un bimodule au-dessus de l'opérade, celui des dérivées du foncteur identité. Nous utilisons ces structures de bimodule pour donner une règle de la chaîne pour les dérivées supérieures en calcul fonctoriel, étendant celle de Klein et Rognes. La règle de la chaîne exprime les dérivées de en tant que produit de composition des dérivées de et de au-dessus des dérivées de l'identité. Il y a deux ingrédients principaux dans nos preuves. Premièrement, nous construisons des nouveaux modèles pour les dérivées de Goodwillie des foncteurs de spectres. Ces modèles fournissent des applications de composition naturelles avec des structure de module et d'opérade. Ensuite, nous utilisons une construction de cobarre cosimplicielle pour porter cette structure aux foncteurs d'espaces topologiques. Une forme de la dualité de Koszul pour les opérades de spectres joue un rôle-clé dans cette preuve.

Abstract:
We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We then use these bimodule structures to give a chain rule for higher derivatives in the calculus of functors, extending that of Klein and Rognes. This chain rule expresses the derivatives of as a derived composition product of the derivatives of and over the derivatives of the identity. There are two main ingredients in our proofs. Firstly, we construct new models for the Goodwillie derivatives of functors of spectra. These models allow for natural composition maps that yield operad and module structures. Then, we use a cosimplicial cobar construction to transfer this structure to functors of topological spaces. A form of Koszul duality for operads of spectra plays a key role in this.

Class. math. : 55P65

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