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nLab > Latest Changes: Goodwillie derivative of the identity functor

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2018

    am starting something. Nothing to be seen yet, but need to save for the moment

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 27th 2018
    • (edited Oct 27th 2018)

    We have a couple more relevant references at Goodwillie calculus:

    • Gregory Arone, Marja Kankaanrinta, The Goodwillie tower of the identity is a logarithm, 1995 (pdf, web)

    • Greg 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)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2018

    Sure, there is loads to be written into this entry. As I said, I was being interrupted. Have to quit for tonight.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2018
    • (edited Nov 23rd 2018)

    expanded the Idea-section and added brief statement of relation to configuration spaces and little nn-disk operads:

    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 is fails one such rule, in an interesting way:

    While in ordinary differential calculus the derivative of the identity function ℝ 1→idℝ 1\mathbb{R}^1 \overset{id}{\to} \mathbb{R}^1 on the real line is itself the identity, this is not the case for the Goodwillie derivative of the identity functor Top */→IdTop */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 •IdD_\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 XX a parallelizable manifold of dimension dim(X)∈ℕdim(X) \in \mathbb{N} (without boundary) and for n∈ℕn \in \mathbb{N} any natural number, the n⋅dim(X)n \cdot dim(X)-shifted suspension spectrum of the configuration space Conf n(X)Conf_n(X) of nn points in XX (whose elements are such configurations, and a basepoint is freely adjoined) is canonically an algebra over the operad of Goodwillie derivatives

    labelConfigurationSpaceAsAlgebraOverGoddwillieDerivativesOfIdentityΣ −ndim(X)Σ ∞Conf n(X)∈Alg(D •Id). \label{ConfigurationSpaceAsAlgebraOverGoddwillieDerivativesOfIdentity} \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 nn-disk operad

    For each natural number d∈ℕd \in \mathbb{N} is a canonical homomorphism of operads from the Goodwillie derivatives of the identity functor to the the dd-fold looping of the little d-disk operad, incarnated as the Fulton-MacPherson operad

    ϕ:D •Id⟶Σ −dE d \phi \;\colon\; D_\bullet Id \longrightarrow \Sigma^{-d} E_d

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

    (Σ −dE d)(k)≔Ω d(k−1)(E n(k)). \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)Con_n(X) from above is also canonically an algebra over the little n-disk operad (Markl 99):

    Σ −ndim(X)Σ ∞Conf n(X)∈Alg(Σ −dim(X)E dim(X)). \Sigma^{-n dim(X)}\Sigma^{\infty} Conf_n(X) \;\in\; Alg\left( \Sigma^{-dim(X)}E_{dim(X)} \right) \,.

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

    ϕ *:Alg(Σ −dim(X)E dim(X))⟶Alg(D •Id) \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 (eq:ConfigurationSpaceAsAlgebraOverGoddwillieDerivativesOfIdentity) and from (eq:ConfigurationSpaceAsAlgebraOverLittleNDiskOperad)

    Σ −ndim(X)Σ ∞Conf n(X)∈Alg(Σ −dim(X)E dim(X))⟶ϕ *Alg(D •Id) \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)

    diff, v2, current

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 23rd 2018

    Where it says there

    While in ordinary differential calculus the derivative of the identity function ℝ 1→idℝ 1\mathbb{R}^1 \overset{id}{\to} \mathbb{R}^1 on the real line is itself the identity

    why isn’t the derivative the constant function with value 11?

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 23rd 2018
    • (edited Nov 23rd 2018)

    Oh, because you’re taking the derivative d(id)d(id) as acting between tangent spaces, as explained here.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2018
    • (edited Nov 23rd 2018)

    Yes!

    Of course it doesn’t matter too much which perspective one takes, as both have their analogue in Goodwillie calculus.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 23rd 2018

    Ok, so I added a word on that.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2018

    made page name singular

    diff, v4, current

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