nLab Goodwillie derivative of the identity functor

Idea

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).

Properties

Action on configuration spaces of points

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

(1)$$\Sigma^{-n dim(X)}\Sigma^{\infty} Conf_n(X) \;\in\; Alg\big( D_\bullet Id \big) \,.$$

(Ching 05b, Prop. 3.1)

Relation to the little $n$-disk operad

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

(2)$$\phi \;\colon\; D_\bullet Id \longrightarrow \Sigma^{-d} E_d$$

where the shifted operad on the right has component space in degree $k$ given by the iterated loop space

(3)$$\left( \Sigma^{-d}E_d \right)(k) \;\coloneqq\; \Omega^{d(k-1)} \left( E_n(k) \right) \,.$$

(Ching 05b, below Lemma 4.1)

Now the configuration space $Con_n(X)$ from above is also canonically an algebra over the little n-disk operad (Markl 99):

(4)$$\Sigma^{-n dim(X)}\Sigma^{\infty} Conf_n(X) \;\in\; Alg\left( \Sigma^{-dim(X)}E_{dim(X)} \right) \,.$$

Conjecturally, the operad-homomorphism (3), via its induced functor on algebras over an operad

$$\phi^\ast \;\colon\; Alg\left( \Sigma^{-dim(X)}E_{dim(X)} \right) \longrightarrow Alg\big( D_\bullet Id \big)$$

identifies these two algebra over an operad-structures on shifted configuration spaces of points from (?) and from (4)

$$\Sigma^{-n dim(X)}\Sigma^{\infty} Conf_n(X) \;\in\; Alg\left( \Sigma^{-dim(X)}E_{dim(X)} \right) \overset{ \phi^\ast }{\longrightarrow} Alg\big( D_\bullet Id \big)$$

(Ching 05b, Conjecture 4.2)

References

• 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)