Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: Goodwillie derivative of the identity functor

Bottom of Page

1 to 9 of 9

  1.  
    • 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

1 to 9 of 9

  1.