Goodwillie calculus – approximation of homotopy theories by stable homotopy theories
symmetric monoidal (∞,1)-category of spectra
While the Goodwillie calculus of functors is an accurate (∞,1)-categorification of ordinary differential calculus, in that it obeys various analogous rules, such as notably the chain rule, it fails one such rule, in an interesting way:
While in ordinary differential calculus the derivative of the identity function $\mathbb{R}^1 \overset{id}{\to} \mathbb{R}^1$ on the real line (considered as a map between tangent spaces) is itself the identity, this is not the case for the Goodwillie derivative of the identity functor $Top^{\ast/} \overset{Id}{\to} Top^{\ast/}$ on the category of pointed topological spaces.
But the Goodwillie chain rule implies then that the nontrivial Goodwillie derivatives $D_\bullet Id$ of the identity functor satisfy a good algebraic compositional law. Indeed, they form an operad in spectra (Ching 05a).
For $X$ a parallelizable manifold of dimension $dim(X) \in \mathbb{N}$ (without boundary) and for $n \in \mathbb{N}$ any natural number, the $n \cdot dim(X)$-shifted suspension spectrum of the configuration space $Conf_n(X)$ of $n$ points in $X$ (whose elements are such configurations, and a basepoint is freely adjoined) is canonically an algebra over the operad of Goodwillie derivatives
For each natural number $d \in \mathbb{N}$ is a canonical homomorphism of operads from the Goodwillie derivatives of the identity functor to the the $d$-fold looping of the little d-disk operad, incarnated as the Fulton-MacPherson operad
where the shifted operad on the right has component space in degree $k$ given by the iterated loop space
Now the configuration space $Con_n(X)$ from above is also canonically an algebra over the little n-disk operad (Markl 99):
Conjecturally, the operad-homomorphism (3), via its induced functor on algebras over an operad
identifies these two algebra over an operad-structures on shifted configuration spaces of points from (?) and from (4)
Gregory Arone, Mark Mahowald, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Inventiones mathematicae February 1999, Volume 135, Issue 3, pp 743-788 (pdf)
Gregory Arone, Marja Kankaanrinta, The Goodwillie tower of the identity is a logarithm, 1995 (pdf, web)
Michael Ching, Bar constructions for topological operads and the Goodwillie derivatives of the identity, Geom. Topol. 9 (2005) 833-934 (arXiv:math/0501429)
Michael Ching, Calculus of Functors and Configuration Spaces, Conference on Pure and Applied Topology Isle of Skye, Scotland, 21-25 June, 2005 (pdf)
Last revised on November 23, 2018 at 10:57:19. See the history of this page for a list of all contributions to it.