nLab
Goodwillie-Taylor tower

Contents

Contents

Idea

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

P n+1FP nFP 1FP 0F. \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 FF, then FF 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 ρ\rho-analytic (∞,1)-functors on ρ\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.

References

General

The concept originates in

Review includes

See also

Stable splitting of mapping spaces

The stable splitting of mapping spaces is originally due to

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

Review and generalization is due to

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)

Last revised on November 2, 2018 at 04:33:27. See the history of this page for a list of all contributions to it.