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BIRS Workshop Lecture Videos

The Nilpotence Tower Finster, Eric


Much like the theory of affine schemes and commutative rings, the theory of (higher) topoi leads a dual life: one algebraic and one geometric. In the geometric picture, a topos is a kind of generalized space whose points carry the structure of a category. Dually, in the algebraic point of view, a topos may be thought of as the "ring of continuous functions on a generalized space with values in homotopy types". In this talk, I will explain the connection between Goodwillie's calculus of functors and this algebro-geometric picture of the theory of higher topoi. Specifically, I will describe how one can view the topos of n-excisive functors as an analog of the commutative k-algebra k[x]/x⠿⠺¹, freely generated by a nilpotent element of order n+1. More generally, I will show how every left exact localization E â F of topoi may be extended to a tower of such localizations E â ¯ â Fâ â Fâ â â â â ¯ Fâ = F which we refer to as the Nilpotence Tower, and whose values at an object of E may be seen as a generalized version of the Goodwillie tower of a functor with values in spaces. Under the analogy with scheme theory described above, this construction corresponds to the completion of a commutative ring along an ideal, or, geometrically, to the filtration of the formal neighborhood of a subscheme by it's n-th order sub-neighborhood. I will also explain how, in addition to the homotopy calculus, the orthogonal calculus of Michael Weiss can be seen as an instance of this same construction. This is joint work with M. Anel, G. Biedermann and A. Joyal.

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