nLab Goodwillie-Taylor tower

Idea

In the context of Goodwillie calculus the Taylor tower of an (∞,1)-functor $F$ to a Goodwillie-differentiable (∞,1)-category is its stagewise approximation by n-excisive (∞,1)-functors $P_n F$ (its n-excisive projections)

$$\cdots \to P_{n+1} F \to P_n F \to \cdots \to P_1 F \to P_0 F \,.$$

which is analogous to the approximation of a smooth function by its Taylor series.

If this tower converges to $F$, then $F$ is analogous to an analytic function and is called an analytic (∞,1)-functor.

The spectral sequence induced by the filtration given by the Goodwillie-Taylor tower is the Goodwillie spectral sequence.

Properties

Convergence

convergence for $\rho$-analytic (∞,1)-functors on $\rho$-connective objects (…) (Goodwillie 03, theorem 1.13, see also Munson-Volic 15, theorem 10.1.51 and section 10.2.4)

Examples

Stable splitting of mapping spaces

Under some conditions, the Goodwillie-Taylor tower of mapping spaces into sufficiently high suspensions is a direct sum of suspension spectra of configuration spaces. This stable splitting of mapping spaces is originally due to (Snaith 74, Bödigheimer 87) and was understood as a Goodwillie-Taylor tower in Arone 99, see also Ching 05.

Related concepts

• n-excisive (∞,1)-functor

• Postnikov tower, chromatic tower

References

General

The concept originates in

• Thomas Goodwillie, Calculus III: Taylor Series, Geom. Topol. 7(2003) 645-711 (arXiv:math/0310481)

Review includes

• Brian Munson, Ismar Volic, Cubical homotopy theory, Cambridge University Press, 2015 pdf

See also

• Gregory Arone, Michael Ching, A classification of Taylor towers of functors of spaces and spectra (arXiv:1209.5661)

Stable splitting of mapping spaces

The stable splitting of mapping spaces is originally due to

• Victor Snaith, A stable decomposition of $\Omega^n S^n X$, Journal of the London Mathematical Society 7 (1974), 577 - 583 (pdf)

Review and generalization is due to

• Carl-Friedrich Bödigheimer, Stable splittings of mapping spaces, Algebraic topology. Springer 1987. 174-187 (pdf)

Further generalization and interpretation in terms of the Goodwillie-Taylor tower of mapping spaces is due to

• Greg Arone, A generalization of Snaith-type filtration, Transactions of the American Mathematical Society 351.3 (1999): 1123-1150. (pdf)

• Michael Ching, Calculus of Functors and Configuration Spaces, Conference on Pure and Applied Topology Isle of Skye, Scotland, 21-25 June, 2005 (pdf)