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\section*{Goodwillie calculus}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category theory contents]]

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{goodwillie_calculus}{}\paragraph*{{Goodwillie calculus}}\label{goodwillie_calculus}

[[!include Goodwillie calculus - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{as_an_approximation_to_stabilization_of_a_functor}{As an approximation to stabilization of a functor}\dotfill \pageref*{as_an_approximation_to_stabilization_of_a_functor} \linebreak
\noindent\hyperlink{Analogy}{Analogy between homotopy theory and calculus}\dotfill \pageref*{Analogy} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{toposes_of_polynomial_functors}{$\infty$-Toposes of polynomial $(\infty, 1)$-functors}\dotfill \pageref*{toposes_of_polynomial_functors} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{goodwillie_derivatives_of_the_identity_functor}{Goodwillie derivatives of the identity functor}\dotfill \pageref*{goodwillie_derivatives_of_the_identity_functor} \linebreak
\noindent\hyperlink{goodwillie_derivatives_of_mapping_spaces}{Goodwillie derivatives of mapping spaces}\dotfill \pageref*{goodwillie_derivatives_of_mapping_spaces} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Goodwillie calculus is an [[(∞,1)-category theory|(∞,1)-]][[categorification]] of [[differential calculus]], where [[smooth functions]] between [[Cartesian spaces]] are replaced by [[topological functors]] between [[categories]] of [[pointed topological spaces]].

\hypertarget{as_an_approximation_to_stabilization_of_a_functor}{}\subsubsection*{{As an approximation to stabilization of a functor}}\label{as_an_approximation_to_stabilization_of_a_functor}

The operation of [[stabilization]] that sends an [[(∞,1)-category]] $C$ to the [[stable (∞,1)-category]] $Stab(C)$ does not in general extend to a functor.

We may think of this operation as the analog of \emph{linearizing} a space. Turning an [[(∞,1)-functor]] $F \colon C \to D$ into a functor $Stab(C) \to Stab(D)$ is not unlike performing a first order [[Taylor expansion]] of a function, whence one speaks here of [[Goodwillie-Taylor towers]].

This is what Goodwillie calculus studies.

Let $F: \mathcal{C} \to \mathcal{D}$ (where $\mathcal{C}$ and $\mathcal{D}$ are each either $Top_*$, the category of [[pointed topological spaces]], or $Spec$, the category of [[spectra]]) be a pointed homotopy functor. Associate with $F$ a sequence of spectra, called the derivatives of $F$, denoted by $\partial_1 F, \partial_2 F,\cdots, \partial_n F, \cdots$, or, collectively, by $\partial_* F$. For each $n$ the spectrum $\partial_n F$ has a natural [[action]] of the [[symmetric group]] $\Sigma_n$. Thus, $\partial_\bullet F$ is a [[symmetric sequence]] of spectra.

The Goodwillie-derivatives of $F$ contain substantial information about the homotopy type of $F$. We can form a sequence of `approximations' to $F$ together with natural transformations forming a \emph{[[Goodwillie-Taylor tower]]}. This tower takes the form

\begin{displaymath}
F \to \cdots \to P_n F \to P_{n-1} F \to \cdots\to P_0 F
\end{displaymath}
with $P_n F$ being the universal \emph{n-excisive} approximation to $F$. (A functor is [[n-excisive (∞,1)-functor|n-excisive]] if it takes any $n + 1$-dimensional cube with homotopy pushout squares for faces to a homotopy cartesian cube.) For `[[analytic (∞,1)-functors]]' $F$, this tower converges for sufficiently highly connected $X$, that is

\begin{displaymath}
F(X) \simeq \underset{n}{holim} P_n F(X).
\end{displaymath}
The fibre $D_n F$ of the map $P_n F \to P_{n-1} F$ is an \emph{n-homogeneous} functor in an appropriate sense, and is determined by $\partial_n F$, via the following formula. If $F$ takes values in $Spec$ then

\begin{displaymath}
D_n F(X) \simeq (\partial_n F \wedge X^{\wedge n})_{h \Sigma_n}.
\end{displaymath}
If $F$ takes values in $Top_*$ then one needs to prefix the right hand side with $\Omega^{\infty}$. (Arone \& Ching)

\hypertarget{Analogy}{}\subsubsection*{{Analogy between homotopy theory and calculus}}\label{Analogy}

Here is an overview of the relation between [[homotopy theory]]/[[∞-groupoid]] theory and [[algebraic approaches to differential calculus|differential calculus]] that is the starting point for Goodwillie calculus.

\begin{quote}%
based on a message by [[André Joyal]] to the Category Theory Mailing list, May 12, 2010


\end{quote}
Write $k[ [x] ]$ for the [[ring]] of [[formal power series]] in one variable over a [[field]] $k$. The ring $k[ [x] ]$ bears some resemblances with the category of [[pointed object|pointed]] [[homotopy type]]s (= pointed spaces up to [[weak homotopy equivalence]]s). The category of pointed homotopy types is a ring (the product is the [[smash product]] and the sum is the [[wedge sum]]).

The following dictionary indicates what the correspondence between the two subjects is.

\begin{itemize}%
\item $k$ $\stackrel{\text{corresponds to}}{\mapsto}$ the category of [[pointed set]]s;


\item $k[ [x] ]$ $\mapsto$ the category of pointed [[homotopy type]]s;


\item $x$ $\mapsto$ the pointed circle;


\item the augmentation $(k[ [x] ] \to k)$ $\mapsto$ the [[homotopy group|connected components functor]] $\pi_0$ : pointed homotopy types $\to$ pointed sets


\item the augmentation ideal $J$ $\mapsto$ the [[subcategory]] of pointed [[connected]] spaces;


\item the $n+1$ power of the augmentation ideal $J^{n+1}$ $\mapsto$ the subcategory of pointed $n$-[[connected]] spaces;


\item the product of an element in $J^{n+1}$ with an element of $J^{m+1}$ is an element of $J^{n+m+2}$ $\mapsto$ the [[smash product]] of an $n$-[[connected]] space with a $m$-connected space is $(n+m+1)$-connected;


\item multiplication by $x$ $\mapsto$ the [[suspension]] functor.


\item division by $x$ $\mapsto$ the [[loop space]] functor;

Notice here the difference: the loop functor is [[right adjoint]] to the suspension functor, not its [[inverse]]. Moreover, the loop space of a space has a special structure (it is a [[group object in an (infinity,1)-category|group]]).


\item the ideal $J=x k[ [x] ]$ is [[isomorphism|isomorphic]] to $k[ [x] ]$ via division by $x$ $\mapsto$ similarly, the category of pointed connected spaces is equivalent to the category of topological groups via the loop space functor (it is actually an [[Quillen equivalence]] of [[model categories]]).


\item More generally, the ideal $J^{n+1}$ is isomorphic to $k[ [x] ]$ via division by $x^{n+1}$. $\mapsto$ similarly, the category of $n$-connected spaces is equivalent to the category of $(n+1)$-fold [[topological group]]s (it is actually an [[Quillen equivalence]] of [[model categories]]) via the $(n+1)$-fold loop space functor.


\item the quotient ring $k[ [x] ]/J^{n+1}$ $\mapsto$ the category of $n$-truncated homotopy types (=[[homotopy n-type]]s)


\item The sequence of approximations of a formal power series $f(x)=a_0+a_1x+ \cdots$

$a_0$

$a_0+a_1 x$

$a_0+a_1 x + a_2 x^2$

$\cdots$

$\mapsto$

the [[Postnikov tower]] of a pointed homotopy type $X$:

$[\pi_0(X)]$

$[\pi_0(X); \pi_1(X)]$

$[\pi_0(X); \pi_1(X); \pi_2(X)]$

$\cdots$

Here, $\pi_0(X)$ is the set of connected components of $X$, $[\pi_0(X); \pi_1(X)]$ is the [[fundamental groupoid]] of $X$, $[\pi_0(X); \pi_1(X); \pi_2(X)]$ is the fundamental 2-groupoid of $X$, etc.


\item The differences between $f(x)$ and its successives approximations

\begin{displaymath}
\begin{aligned}
   R_0 = f(x)-a_0  &= a_1 x+a_2 x^2+a_3 x^3+ \cdots
   \\
   R_1 = f(x)-(a_0+a_1 x) &= a_2 x^2 + a_3 x^3 + a_4 x^4+ \cdots
   \\
   R_2 = f(x)-(a_0+a_1x+a_2x^2) &= a_3 x^3 + a_4 x^4 + a_5 x^5 +\cdots
 \end{aligned}
\end{displaymath}
$\mapsto$

the [[Whitehead tower]] of $X$,

$C_0=[0;\pi_1(X), \pi_2(X), \pi_3(X), \cdots]$

$C_1=[0;0, \pi_2(X), \pi_3(X), \cdots]$

$C_2=[0;0, 0, \pi_3(X), \cdots]$

Here, $C_0$ is the connected component of $X$ at the base point, $C_1$ is the [[universal cover]] of $X$ constructed by from paths starting at the base point, $C_2$ is the universal 2-cover of $X$ constructed from paths starting the base point, etc.


\item Division by $x$ is shifting down the coefficients of a power series.

If $f(x)=a_1 x+a_2 x^2 + \cdots$, then $f(x)/x= a_1+a_1 x^2+ \cdots$

Similarly, the loop space functor is shifting down the homotopy groups of a pointed space:

if $X=[a_0,a_1,a_2,...]$ then $\Omega(X)=[a_1,a_2,....]$.

Unfortunately, the [[suspension]] functor does not shift up the [[homotopy group]]s of a space. It is however shifting the first $2n$ homotopy groups of an $n$-connected space $X$ $(n \geq 1)$ by the [[Freudenthal suspension theorem]]

For example, if $X=[0;0,a_2, a_3,...]$ then $\Sigma X=[0;0,0,a_2,a_3...]$, and if $X=[0;0,0, a_3, a_4, a_5,...]$ then $\Sigma X=[0;0,0, 0, a_3, a_4, a_5,...]$.

In other words, the canonical map $X \to \Omega X$ is a $2n$-equivalence if $X$ is $n$-[[connected]] $(n \geq 1)$. If $X[2n]$ denotes the $2n$-type of $X$ (the $2n$-[[truncated|truncation]] of $X$), then we have a [[homotopy equivalence]]

$X[2n]  \to \Omega \Sigma X[2n] \simeq \Omega \Sigma X[2n+1]$.



\end{itemize}
\ldots{}

From a blog \href{http://golem.ph.utexas.edu/category/2008/12/smooth_structures_in_ottawa.html#c020698}{discussion}

\hyperlink{AroneKankaanrinta95}{Arone Kankaanrinta 95} write

\begin{quote}%
The Goodwillie tower of the identity\ldots{}is a tower of functors and natural transformations, which starts with stable homotopy and converges to unstable homotopy. (p. 1)

\ldots{}the Goodwillie tower is an inverse to stable homotopy in the same way as logarithm is an inverse to exponential. (p. 1)

It is the point of this paper that the Goodwillie tower is the homotopy theoretic analog of logarithmic expansion, rather than of Taylor series. (p. 6)


\end{quote}
What's going on, they say, is like finding a function of the form $a^{x - 1}$ which best approximates $x$. This is when $a = e$.

The functor from spaces to spaces which sends $X$ to the [[infinite loop space]] underlying its [[suspension spectrum]]

\begin{displaymath}
\Omega^{\infty}\Sigma^{\infty} X = colim \Omega^n \Sigma^n X
\end{displaymath}
sends coproducts to products and is supposed to be like $e^{x - 1}$. (The ``$-1$'' comes about from issues to do with basepoints.)

A homogeneous linear functor is defined to be one sending coproducts to products, so it is like an [[exponential map|exponential]]. Compared to an exponential, the [[identity functor]] is like a [[logarithm]], so it has a non-trivial Taylor series.

\begin{quote}%
\ldots{}our point of view is that stable homotopy is analogous to the function $e^{x - 1}$ rather than to a linear function, and the Goodwillie tower is an infinite product, rather than an infinite sum, namely it is analogous to the product


\end{quote}
\begin{displaymath}
e^{x - 1} \cdot e^{-\frac{(x - 1)^2}{2}} \cdot e^{\frac{(x - 1)^3}{3}} \ldots = e^{ln(1 + (x - 1))} = x.
\end{displaymath}
\begin{quote}%
(p. 2)


\end{quote}
Gregory Arone in an MO \href{http://mathoverflow.net/questions/32071/is-the-dual-notion-of-a-presheaf-useful}{answer}

\begin{quote}%
Covariant functors from the category of pointed sets to the category of pointed topological spaces are sometimes called $\Gamma$-spaces, and they have been important in algebraic topology. One reason is that $\Gamma$-[[Gamma-space|spaces]] model infinite loop spaces (and therefore connective spectra) and are very helpful for understanding stable homotopy theory.

$\Gamma$-spaces also serve as a model for particularly well-behaved covariant functors from the category of pointed topological spaces to itself. Of course, these functors play an important role in topology as well. I like to think of Goodwillie's Calculus of Homotopy Functors (and also of Michael Weiss's Orthogonal Calculus) as a kind of ``sheaf theory for covariant functors''. In these theories, covariant functors are analogous to presheaves and linear functors are analogous to sheaves (The definition of a linear functor is essentially a homotopy-invariant version of the definition of a sheaf). The process of approximating a general functor by a linear one is analogous to sheafification, and so forth. These theories provide methods for studying certain types of functors, but of course they also tell you something about the category of spaces itself.


\end{quote}
Eric Finster's \href{http://people.virginia.edu/~elf9e/research.pdf}{research statement}

\begin{quote}%
One tantalizing aspect of the Goodwillie calculus is that it suggests the possibility of thinking geometrically about the global structure of homotopy theory. In this interpretation, the category of spectra plays the role of the tangent space to the category of spaces at the one-point space. Moreover, the identity functor from spaces to spaces is not linear\ldots{}and one can interpret this as saying that spaces have some kind of non-trivial curvature.


\end{quote}
However, Goodwillie remarks in the \href{http://www.mfo.de/document/0414/OWR_2004_17.pdf&#8206;}{report} (p. 905) on a Oberwolfach meeting.:

\begin{quote}%
Rhetorical question: If the first derivative of the identity is the identity matrix, why is the second derivative not zero? Answer: Some of the terminology of homotopy calculus works better for functors from spaces to spectra than for functors from spaces to spaces. Specifically, since ``linearity'' means taking pushout squares to pullback squares, the identity functor is not linear and the composition of two linear functors is not linear.

Attempted cryptic remark: Unlike the category of spectra, where pushouts are the same as pullbacks, the category of spaces may be thought of has having nonzero curvature.

Correction: After the talk Boekstedt asked about that remark. We discussed the matter at length and found more than one connection on the category of spaces, but none that was not flat. In fact curvature is the wrong thing to look for. There are in some sense exactly two tangent connections on the category of spaces (or should we say on any model category?). Both are flat and torsion-free. There is a map between them, so it is meaningful to subtract them. As is well-known in differential geometry, the difference between two connections is a 1-form with values in endomorphisms (whereas the curvature is a 2-form with values in endomorphisms). Thus there is a way of discussing the discrepancy between pushouts and pullbacks in the language of differential geometry, but it is a tensor field of a different type from what I had guessed.


\end{quote}
This is from the \href{http://www.mfo.de/programme/schedule/2004/14/OWR_2004_17.pdf}{report} (p. 905) on a Oberwolfach meeting. The table on p. 900 also makes comparisons to differential geometry.

Chris Schommer-Pries

\begin{quote}%
Any linear functor from spaces to spaces is a generalized cohomology theory. More precisely, there is a model category on the functors from spaces to spaces called the model category of W-spaces. Really I should be using pointed spaces here. This model category is one of the standard models for the category spectra and so the fibrant objects can be thought of as the (co)homology theories. The fibrant objects are precisely those functors which are linear in Goodwillie's sense. The example $S P^{\infty}$ corresponds to ordinary cohomology (well there is a $\pi_0$ issue, but let's ignore that). In general evaluating the linear functor $E$ on a space $X$ gives you a space which should be thought of as the smash product of $E$ and $X$.

So now why should spectra/cohomology theories be thought of as linear functors? Well if you think of spectra as analogous to abelian groups, then applying a spectrum to a space (i.e. smashing with it) is a linearization of that space.

Following this analogy, if we now have any old functor from space to spaces we can take its fibrant replacement in $W$-spaces. This is a linear functor which is the best approximation to the original functor. So it is like taking a derivative of a function. Goodwillie's insight was to extend this analogy to encompass the rudiments of calculus. There is in fact a whole series of model categories on functors from spaces to spaces where the fibrant objects are Goodwillie's polynomial functors of degree $n$.


\end{quote}
See also Eric Finster's blog post \href{http://curiousreasoning.wordpress.com/2010/08/02/thoughts-on-the-goodwillie-calculus/}{Thoughts on the Goodwillie Calculus}

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{toposes_of_polynomial_functors}{}\subsubsection*{{$\infty$-Toposes of polynomial $(\infty, 1)$-functors}}\label{toposes_of_polynomial_functors}

For each $n$, the collection of [[n-excisive (∞,1)-functors]] from bare [[homotopy types]] to bare homotopy types is an [[(infinity,1)-topos]], the \emph{\href{jet+category#JetToposes}{jet topos}}.

due to ( \hyperlink{Joyal08}{Joyal 08, 35.5}, with [[Georg Biedermann]]) See also at \emph{[[tangent (infinity,1)-category]]}, and [[Charles Rezk]], appears as (\hyperlink{HigherAlg}{Lurie, remark 6.1.1.11}).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{goodwillie_derivatives_of_the_identity_functor}{}\subsubsection*{{Goodwillie derivatives of the identity functor}}\label{goodwillie_derivatives_of_the_identity_functor}

The Goodwillie derivatives of the [[identity functor]] on [[pointed topological spaces]], form an [[operad]] in [[spectra]] (\hyperlink{Ching05}{Ching 05}).

See at \emph{[[Goodwillie derivatives of the identity functor]]}

\hypertarget{goodwillie_derivatives_of_mapping_spaces}{}\subsubsection*{{Goodwillie derivatives of mapping spaces}}\label{goodwillie_derivatives_of_mapping_spaces}

See at \emph{[[stable splitting of mapping spaces]]}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Goodwillie spectral sequence]]


\item [[Goodwillie-differentiable (∞,1)-category]]


\item [[excisive (∞,1)-functor]], [[n-excisive (∞,1)-functor]]


\item [[Taylor tower]]


\item [[Goodwillie differentiation]], [[Goodwillie chain rule]]


\item [[tangent (∞,1)-category]], [[tangent cohesion]]


\item [[jet (∞,1)-category]]


\item [[orthogonal calculus]]


\item [[Weiss topology]]


\item [[manifold calculus]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Surveys and introductions include

\begin{itemize}%
\item [[Tom Goodwillie]], \emph{The differential calculus of homotopy functors}, Proceedings of the International Congress of Mathematicians. Vol. 1. 1990 (\href{http://math.mit.edu/~nrozen/juvitop/goodwillie-icm.pdf}{pdf})


\item [[Brian Munson]], \emph{Introduction to the manifold calculus of Goodwillie-Weiss} (\href{http://uk.arxiv.org/abs/1005.1698}{arXiv:1005.1698})


\item Nicholas J. Kuhn, \emph{Goodwillie towers and chromatic homotopy: an overview}, Proceedings of the Nishida Fest (Kinosaki 2003), 245--279, Geom. Topol. Monogr., 10, Geom. Topol. Publ., Coventry, 2007.


\item [[Brian Munson]], [[Ismar Volic]], \emph{Cubical homotopy theory}, Cambridge University Press, 2015 \href{http://palmer.wellesley.edu/~ivolic/pdf/Papers/CubicalHomotopyTheory.pdf}{pdf}


\item \href{http://www.birs.ca/events/2011/5-day-workshops/11w5058}{Functor Calculus and Operads}


\item 2012 Talbot Workshop (\href{http://math.mit.edu/conferences/talbot/2012/2012TalbotTalks.pdf}{Talk schedule}, \href{http://math.mit.edu/conferences/talbot/index.php?year=2012&sub=talks}{Notes})



\end{itemize}
Original articles include

\begin{itemize}%
\item [[Thomas Goodwillie]],

\emph{Calculus. I. The first derivative of pseudoisotopy theory}, K-Theory \textbf{4} (1990), no. 1, 1-27. MR 1076523 (92m:57027);

\emph{Calculus. II. Analytic functors}, K-Theory \textbf{5} (1991/92), no. 4, 295-332. MR 1162445 (93i:55015);

\emph{Calculus. III. Taylor series}, Geom. Topol. 7 (2003), 645--711 (\href{http://www.msp.warwick.ac.uk/gt/2003/07/p019.xhtml}{journal}, \href{http://arxiv.org/abs/math/0310481}{arXiv:math/0310481}))


\item Andrew Mauer-Oats, \emph{Algebraic Goodwillie calculus and a cotriple model for the remainder}, Trans. Amer. Math. Soc. \textbf{358} (2006), no. 5, 1869--1895 \href{http://www.ams.org/tran/2006-358-05/S0002-9947-05-03936-X/home.html}{journal}, \href{http://arxiv.org/abs/math/0212095}{math.AT/0212095}



\end{itemize}
A [[model category]] presentation for [[n-excisive functors]] is given in

\begin{itemize}%
\item [[Georg Biedermann]], [[Boris Chorny]], Oliver R\"o{}ndigs, \emph{Calculus of functors and model categories}, Adv. Math., \textbf{214}, n. 1 (2007) 92--115, \href{http://dx.doi.org/10.1016/j.aim.2006.10.009}{doi}, \href{http://arxiv.org/abs/math/0601221}{math.AT/0601221}


\item [[Gregory Arone]], [[Michael Ching]], \emph{Operads And Chain Rules For The Calculus Of Functors}, \href{https://arxiv.org/abs/0902.0399}{arXiv:0902.0399}


\item [[Gregory Arone]], [[Mark Mahowald]], \emph{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 (\href{http://hopf.math.purdue.edu/Arone-Mahowald/ArMahowald.pdf}{pdf})


\item [[Gregory Arone]], Marja Kankaanrinta, \emph{The Goodwillie tower of the identity is a logarithm}, 1995 ([[Arone95.pdf:file]], \href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.8306}{web})



\end{itemize}
A discussion of the theory in light of [[(∞,1)-category]] theory and [[stable (∞,1)-categories]] is in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[(∞,2)-Categories and the Goodwillie Calculus]]}, \href{http://www.math.harvard.edu/~lurie/papers/GoodwillieI.pdf#page=159}{section 5}

\end{itemize}
This is now section 7 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Algebra]]}

\end{itemize}
See also

\begin{itemize}%
\item Luis Pereira, \emph{A general context for Goodwillie Calculus} (\href{http://arxiv.org/abs/1301.2832}{arXiv:1301.2832})


\item [[André Joyal]], \emph{Notes on Logoi}, 2008 (\href{http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf}{pdf})



\end{itemize}
Generalization from [[(infinity,1)-categories]] to [[(infinity,n)-categories]] and relation to [[rational homotopy theory]] is discussed in

\begin{itemize}%
\item [[Gijs Heuts]], \emph{Goodwillie approximations to higher categories} (\href{http://arxiv.org/abs/1510.03304}{arXiv:1510.03304})

\end{itemize}
Relation to [[chromatic homotopy theory]] is discussed in

\begin{itemize}%
\item [[Greg Arone]], [[Mark Mahowald]], \emph{The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres}, Inventiones mathematicae, 1999, Volume 135, Issue 3, pp 743-788 (\href{http://hopf.math.purdue.edu/Arone-Mahowald/ArMahowald.pdf}{pdf})


\item [[Nicholas Kuhn]], \emph{Goodwillie towers and chromatic homotopy: an overview}, Geom. Topol. Monogr. 10 (2007) 245-279 (\href{http://arxiv.org/abs/math/0410342}{arXiv:math/0410342})



\end{itemize}
On the relation to [[configuration space (mathematics)|configuration spaces]]:

\begin{itemize}%
\item [[Michael Ching]], \emph{Bar constructions for topological operads and the Goodwillie derivatives of the identity}, Geom. Topol. 9 (2005) 833-934 (\href{https://arxiv.org/abs/math/0501429}{arXiv:math/0501429})


\item [[Michael Ching]], \emph{Calculus of Functors and Configuration Spaces}, Conference on Pure and Applied Topology Isle of Skye, Scotland, 21-25 June, 2005 (\href{https://www3.amherst.edu/~mching/Work/skye.pdf}{pdf})



\end{itemize}
Discussion in terms of [[spectral Mackey functors]]

\begin{itemize}%
\item [[Saul Glasman]], \emph{Stratified categories, geometric fixed points and a generalized Arone-Ching theorem} (\href{https://arxiv.org/abs/1507.01976}{arXiv:1507.01976}, \href{http://www-users.math.umn.edu/~sglasman/strattalk.pdf}{talk notes pdf})


\item [[Saul Glasman]], \emph{Goodwillie calculus and Mackey functors} (\href{https://arxiv.org/abs/1610.03127}{arXiv:1610.03127})



\end{itemize}
[[!redirects Goodwillie derivative]] [[!redirects Goodwillie derivatives]]

[[!redirects Goodwillie calculus of functors]]



\end{document}