Tangent ∞-categories and Goodwillie calculus

Duration: 1 hour 5 mins

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Description:	Ching, M Tuesday 9th October 2018 - 14:00 to 15:00


Created:	2018-10-16 12:45
Collection:	Higher structures in homotopy theory
Publisher:	Isaac Newton Institute
Copyright:	Ching, M
Language:	eng (English)


Abstract:	Goodwillie calculus is a set of tools in homotopy theory developed, to some extent, by analogy with ordinary differential calculus. The goal of this talk is to make that analogy precise by describing a common category-theoretic framework that includes both the calculus of smooth maps between manifolds, and Goodwillie calculus of functors, as examples. This framework is based on the notion of "tangent category" introduced first by Rosicky and recently developed by Cockett and Cruttwell in connection with models of differential calculus in logic, with the category of smooth manifolds as the motivating example. In joint work with Kristine Bauer and Matthew Burke (both at Calgary) we generalize to tangent structures on an (∞,2)-category and show that the (∞,2)-category of presentable ∞-categories possesses such a structure. This allows us to make precise, for example, the intuition that the ∞-category of spectra plays the role of the real line in Goodwillie calculus. As an application we show that Goodwillie's definition of n-excisive functor can be recovered purely from the tangent structure in the same way that n-jets of smooth maps are in ordinary calculus. If time permits, I will suggest how other concepts from differential geometry, such as connections, may play out into the context of functor calculus.

Abstract:

Goodwillie calculus is a set of tools in homotopy theory developed, to some extent, by analogy with ordinary differential calculus. The goal of this talk is to make that analogy precise by describing a common category-theoretic framework that includes both the calculus of smooth maps between manifolds, and Goodwillie calculus of functors, as examples. This framework is based on the notion of "tangent category" introduced first by Rosicky and recently developed by Cockett and Cruttwell in connection with models of differential calculus in logic, with the category of smooth manifolds as the motivating example. In joint work with Kristine Bauer and Matthew Burke (both at Calgary) we generalize to tangent structures on an (∞,2)-category and show that the (∞,2)-category of presentable ∞-categories possesses such a structure. This allows us to make precise, for example, the intuition that the ∞-category of spectra plays the role of the real line in Goodwillie calculus. As an application we show that Goodwillie's definition of n-excisive functor can be recovered purely from the tangent structure in the same way that n-jets of smooth maps are in ordinary calculus. If time permits, I will suggest how other concepts from differential geometry, such as connections, may play out into the context of functor calculus.

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