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nLab > Latest Changes: Goodwillie derivative of the identity functor

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeOct 27th 2018
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Author: Urs
Format: MarkdownItexam starting something. Nothing to be seen yet, but need to save for the moment <a href="https://ncatlab.org/nlab/revision/Goodwillie+derivatives+of+the+identity+functor/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Goodwillie+derivatives+of+the+identity+functor">current</a>

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)
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Author: David_Corfield
Format: MarkdownItexWe have a couple more relevant references at [[Goodwillie calculus]]: * [[Gregory Arone]], Marja Kankaanrinta, _The Goodwillie tower of the identity is a logarithm_, 1995 ([pdf](https://ncatlab.org/nlab/files/Arone95.pdf), [web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.8306)) * [[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](http://hopf.math.purdue.edu/Arone-Mahowald/ArMahowald.pdf))
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
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Author: Urs
Format: MarkdownItexSure, there is loads to be written into this entry. As I said, I was being interrupted. Have to quit for tonight.

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)
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Author: Urs
Format: MarkdownItexexpanded the Idea-section and added brief statement of relation to configuration spaces and little $n$-disk operads: *** ## Idea While the [[Goodwillie calculus of functors]] is an accurate [[(infinity,1)-category theory|(∞,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]] $\mathbb{R}^1 \overset{id}{\to} \mathbb{R}^1$ on the [[real line]] is itself the identity, this is not the case for the [[Goodwillie derivative]] of the [[identity functor]] $Top^{\ast/} \overset{Id}{\to} Top^{\ast/}$ 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](#Ching05a)). ## Properties ### Action on configuration spaces of points For $X$ a [[parallelizable manifold]] of [[dimension]] $dim(X) \in \mathbb{N}$ (without [[manifold with boundary|boundary]]) and for $n \in \mathbb{N}$ any [[natural number]], the $n \cdot dim(X)$-shifted [[suspension spectrum]] of the [[configuration space of points|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 an operad|algebra]] over the operad of Goodwillie derivatives \[ \label{ConfigurationSpaceAsAlgebraOverGoddwillieDerivativesOfIdentity} \Sigma^{-n dim(X)}\Sigma^{\infty} Conf_n(X) \;\in\; Alg\big( D_\bullet Id \big) \,. \] ([Ching 05b, Prop. 3.1](#Ching05b)) ### 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 n-disk operad|little d-disk operad]], incarnated as the [[Fulton-MacPherson operad]] \[ \phi \;\colon\; D_\bullet Id \longrightarrow \Sigma^{-d} E_d \] where the shifted operad on the right has component space in degree $k$ given by the iterated [[loop space]] \[ \left( \Sigma^{-d}E_d \right)(k) \;\coloneqq\; \Omega^{d(k-1)} \left( E_n(k) \right) \,. \] ([Ching 05b, below Lemma 4.1](#Ching05b)) Now the [[configuration space]] $Con_n(X)$ from [above](#ActionOnConfigurationSpacesOfPoints) is _also_ canonically an [[algebra over an operad|algebra]] over the [[little n-disk operad]] ([Markl 99](configuration+space+of+points#Markl99)): \[ \Sigma^{-n dim(X)}\Sigma^{\infty} Conf_n(X) \;\in\; Alg\left( \Sigma^{-dim(X)}E_{dim(X)} \right) \,. \] Conjecturally, the [[operad]]-[[homomorphism]] (eq:HomomorphismToLittleNDiskOperad), via its induced [[functor]] on [[algebras over an operad]] $$ \phi^\ast \;\colon\; Alg\left( \Sigma^{-dim(X)}E_{dim(X)} \right) \longrightarrow Alg\big( D_\bullet Id \big) $$ identifies these two [[algebra over an operad]]-[[structures]] on shifted [[configuration spaces of points]] from (eq:ConfigurationSpaceAsAlgebraOverGoddwillieDerivativesOfIdentity) and from (eq:ConfigurationSpaceAsAlgebraOverLittleNDiskOperad) $$ \Sigma^{-n dim(X)}\Sigma^{\infty} Conf_n(X) \;\in\; Alg\left( \Sigma^{-dim(X)}E_{dim(X)} \right) \overset{ \phi^\ast }{\longrightarrow} Alg\big( D_\bullet Id \big) $$ ([Ching 05b, Conjecture 4.2](#Ching05b)) <a href="https://ncatlab.org/nlab/revision/diff/Goodwillie+derivatives+of+the+identity+functor/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Goodwillie+derivatives+of+the+identity+functor/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Goodwillie+derivatives+of+the+identity+functor">current</a>

expanded the Idea-section and added brief statement of relation to configuration spaces and little $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ -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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ℝ</mi> <mn>1</mn></msup><mover><mo>→</mo><mi>id</mi></mover><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^1 \overset{id}{\to} \mathbb{R}^1</annotation></semantics></math>$ on the real line is itself the identity, this is not the case for the Goodwillie derivative of the identity functor $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mover><mo>→</mo><mi>Id</mi></mover><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Top^{\ast/} \overset{Id}{\to} Top^{\ast/}</annotation></semantics></math>$ on the category of pointed topological spaces.

But the Goodwillie chain rule implies then that the nontrivial Goodwillie derivatives $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi> <mo>•</mo></msub><mi>Id</mi></mrow><annotation encoding="application/x-tex">D_\bullet Id</annotation></semantics></math>$ 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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ a parallelizable manifold of dimension $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">dim(X) \in \mathbb{N}</annotation></semantics></math>$ (without boundary) and for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>$ any natural number, the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>⋅</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">n \cdot dim(X)</annotation></semantics></math>$ -shifted suspension spectrum of the configuration space $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Conf_n(X)</annotation></semantics></math>$ of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ points in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ (whose elements are such configurations, and a basepoint is freely adjoined) is canonically an algebra over the operad of Goodwillie derivatives
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo lspace="0em" rspace="0.16667em">label</mo><mi>ConfigurationSpaceAsAlgebraOverGoddwillieDerivativesOfIdentity</mi><msup><mi>Σ</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mi>n</mi><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msup><msup><mi>Σ</mi> <mn>∞</mn></msup><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>∈</mo><mspace width="0.27778em"/><mi>Alg</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>D</mi> <mo>•</mo></msub><mi>Id</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> \label{ConfigurationSpaceAsAlgebraOverGoddwillieDerivativesOfIdentity} \Sigma^{-n dim(X)}\Sigma^{\infty} Conf_n(X) \;\in\; Alg\big( D_\bullet Id \big) \,. </annotation></semantics></math>$
(Ching 05b, Prop. 3.1)

Relation to the little $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ -disk operad

For each natural number $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">d \in \mathbb{N}</annotation></semantics></math>$ is a canonical homomorphism of operads from the Goodwillie derivatives of the identity functor to the the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>$ -fold looping of the little d-disk operad, incarnated as the Fulton-MacPherson operad
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>ϕ</mi><mspace width="0.27778em"/><mo lspace="0.11111em">:</mo><mspace width="0.27778em"/><msub><mi>D</mi> <mo>•</mo></msub><mi>Id</mi><mo>⟶</mo><msup><mi>Σ</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mi>d</mi></mrow></msup><msub><mi>E</mi> <mi>d</mi></msub></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; D_\bullet Id \longrightarrow \Sigma^{-d} E_d </annotation></semantics></math>$
where the shifted operad on the right has component space in degree $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>$ given by the iterated loop space
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>(</mo><msup><mi>Σ</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mi>d</mi></mrow></msup><msub><mi>E</mi> <mi>d</mi></msub><mo>)</mo></mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>≔</mo><mspace width="0.27778em"/><msup><mi>Ω</mi> <mrow><mi>d</mi><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mrow><mo>(</mo><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \Sigma^{-d}E_d \right)(k) \;\coloneqq\; \Omega^{d(k-1)} \left( E_n(k) \right) \,. </annotation></semantics></math>$
(Ching 05b, below Lemma 4.1)

Now the configuration space $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Con</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Con_n(X)</annotation></semantics></math>$ from above is also canonically an algebra over the little n-disk operad (Markl 99):
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>Σ</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mi>n</mi><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msup><msup><mi>Σ</mi> <mn>∞</mn></msup><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>∈</mo><mspace width="0.27778em"/><mi>Alg</mi><mrow><mo>(</mo><msup><mi>Σ</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msup><msub><mi>E</mi> <mrow><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo>)</mo></mrow><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Sigma^{-n dim(X)}\Sigma^{\infty} Conf_n(X) \;\in\; Alg\left( \Sigma^{-dim(X)}E_{dim(X)} \right) \,. </annotation></semantics></math>$
Conjecturally, the operad-homomorphism (eq:HomomorphismToLittleNDiskOperad), via its induced functor on algebras over an operad
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mspace width="0.27778em"/><mo lspace="0.11111em">:</mo><mspace width="0.27778em"/><mi>Alg</mi><mrow><mo>(</mo><msup><mi>Σ</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msup><msub><mi>E</mi> <mrow><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo>)</mo></mrow><mo>⟶</mo><mi>Alg</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>D</mi> <mo>•</mo></msub><mi>Id</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> \phi^\ast \;\colon\; Alg\left( \Sigma^{-dim(X)}E_{dim(X)} \right) \longrightarrow Alg\big( D_\bullet Id \big) </annotation></semantics></math>$
identifies these two algebra over an operad-structures on shifted configuration spaces of points from (eq:ConfigurationSpaceAsAlgebraOverGoddwillieDerivativesOfIdentity) and from (eq:ConfigurationSpaceAsAlgebraOverLittleNDiskOperad)
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>Σ</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mi>n</mi><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msup><msup><mi>Σ</mi> <mn>∞</mn></msup><msub><mi>Conf</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>∈</mo><mspace width="0.27778em"/><mi>Alg</mi><mrow><mo>(</mo><msup><mi>Σ</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msup><msub><mi>E</mi> <mrow><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo>)</mo></mrow><mover><mo>⟶</mo><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup></mrow></mover><mi>Alg</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>D</mi> <mo>•</mo></msub><mi>Id</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> \Sigma^{-n dim(X)}\Sigma^{\infty} Conf_n(X) \;\in\; Alg\left( \Sigma^{-dim(X)}E_{dim(X)} \right) \overset{ \phi^\ast }{\longrightarrow} Alg\big( D_\bullet Id \big) </annotation></semantics></math>$
(Ching 05b, Conjecture 4.2)

diff, v2, current
- CommentRowNumber5.
- CommentAuthorDavid_Corfield
- CommentTimeNov 23rd 2018
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Author: David_Corfield
Format: MarkdownItexWhere it says there >While in ordinary [[differential calculus]] the [[derivative]] of the [[identity function]] $\mathbb{R}^1 \overset{id}{\to} \mathbb{R}^1$ on the [[real line]] is itself the identity why isn't the derivative the constant function with value $1$?

Where it says there

While in ordinary differential calculus the derivative of the identity function $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ℝ</mi> <mn>1</mn></msup><mover><mo>→</mo><mi>id</mi></mover><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^1 \overset{id}{\to} \mathbb{R}^1</annotation></semantics></math>$ on the real line is itself the identity

why isn’t the derivative the constant function with value $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>$ ?
- CommentRowNumber6.
- CommentAuthorDavid_Corfield
- CommentTimeNov 23rd 2018
- (edited Nov 23rd 2018)
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Author: David_Corfield
Format: MarkdownItexOh, because you're taking the derivative $d(id)$ as acting between tangent spaces, as explained [here](https://ncatlab.org/nlab/show/chain+rule#elementary_calculus).

Oh, because you’re taking the derivative $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>id</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(id)</annotation></semantics></math>$ as acting between tangent spaces, as explained here.
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeNov 23rd 2018
- (edited Nov 23rd 2018)
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Author: Urs
Format: MarkdownItexYes! Of course it doesn't matter too much which perspective one takes, as both have their analogue in Goodwillie calculus.

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
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Author: David_Corfield
Format: MarkdownItexOk, so I added a word on that.

Ok, so I added a word on that.
- CommentRowNumber9.
- CommentAuthorUrs
- CommentTimeNov 23rd 2018
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Author: Urs
Format: MarkdownItexmade page name singular <a href="https://ncatlab.org/nlab/revision/diff/Goodwillie+derivative+of+the+identity+functor/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Goodwillie+derivative+of+the+identity+functor/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Goodwillie+derivative+of+the+identity+functor">current</a>

made page name singular

diff, v4, current

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nLab > Latest Changes: Goodwillie derivative of the identity functor

Idea

Properties

Action on configuration spaces of points

Relation to the little n&lt;math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding="application/x-tex"&gt;n&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;-disk operad

Relation to the little $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ -disk operad