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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Tangent &\;infin\;-categories and Goodwillie ca
lculus - Michael Ching (Amherst College)
DTSTART;TZID=Europe/London:20181009T140000
DTEND;TZID=Europe/London:20181009T150000
UID:TALK111844AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/111844
DESCRIPTION:Goodwillie calculus is a set of tools in homotopy
theory developed\, to some extent\, by analogy wit
h ordinary differential calculus. The goal of this
talk is to make that analogy precise by describin
g a common category-theoretic framework that inclu
des both the calculus of smooth maps between manif
olds\, and Goodwillie calculus of functors\, as ex
amples. \; This framework is based on the n
otion of "tangent category" introduced first by Ro
sicky and recently developed by Cockett and Cruttw
ell in connection with models of differential calc
ulus in logic\, with the category of smooth manifo
lds as the motivating example. In joint work with
Kristine Bauer and Matthew Burke (both at Calgary)
we generalize to tangent structures on an (&infin
\;\,2)-category and show that the (&infin\;\,2)-ca
tegory of presentable &infin\;-categories possesse
s such a structure. This allows us to make precise
\, for example\, the intuition that the &infin\;-c
ategory of spectra plays the role of the real line
in Goodwillie calculus. As an application we show
that Goodwillie'\;s definition of n-excisive f
unctor can be recovered purely from the tangent st
ructure in the same way that n-jets of smooth maps
are in ordinary calculus. If time permits\, I wil
l suggest how other concepts from differential geo
metry\, such as connections\, may play out into th
e context of functor calculus.
LOCATION:Seminar Room 2\, Newton Institute
CONTACT:INI IT
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