Calculus in homotopy theory

Abstract: (joint with M. Anel, E. Finster and A. Joyal) In classical calculus one studies smooth functions via their Taylor series. Its $n$-th homogeneous layer is governed by a single coefficient: the $n$-th derivative. As part of his effort to relate algebraic K-theory to topological cyclic homology Goodwillie during the 90s introduced "Goodwillie calculus" to homotopy theory. A homotopy invariant functor is viewed as an analogue of a smooth function and resolved into a tower whose $n$-th homogeneous layer is governed by a single coefficient: a spectrum (in the sense of homotopy theory) with $\Sigma_n$-action. Goodwillie calculus is now a central tool in homotopy theory. Around the same time (and influenced by Goodwillie) Michael Weiss constructed "orthogonal calculus": space-valued functors from the category of finite dimensional Euclidean vector spaces with morphism given by Stiefel manifolds are resolved into an orthogonal tower whose $n$-th homogeneous layer is governed by a spectrum with an action by $O(n)$. Weiss' theory has found many applications in differential topology. People have wondered for a long time whether both theories have a common description. We can give one. In fact, it turns out that the theory of $\infty$-topoi is the perfect language. For any left exact localization $L$ of an $\infty$-topos we construct a tower $(P_n)_{n\ge 0}$ of left exact localizations such that $P_0=L$. The pointed objects of the layers form stable $\infty$-categories. The tower is analogous to the completion tower of a commutative ring with respect to an ideal. It specializes to Goodwillie's and Weiss' tower.

I am going to tell you a bit about all these towers.

algebraic geometryanalysis of PDEsalgebraic topologycomplex variablesdifferential geometrygeneral topologygeometric topologyK-theory and homologymetric geometrysymplectic geometry

Audience: researchers in the topic

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CRM - Séminaire du CIRGET / Géométrie et Topologie

Series comments: Hybrid seminar of geometry and topology. Laboratory : CIRGET - www.cirget.uqam.ca The homepage of the seminar is www.cirget.uqam.ca/fr/seminaires.html

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