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\begin{document}

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\section*{Goodwillie-Taylor tower}

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

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{convergence}{Convergence}\dotfill \pageref*{convergence} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{StableSplittingOfMappingSpaces}{Stable splitting of mapping spaces}\dotfill \pageref*{StableSplittingOfMappingSpaces} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{stable_splitting_of_mapping_spaces_2}{Stable splitting of mapping spaces}\dotfill \pageref*{stable_splitting_of_mapping_spaces_2} \linebreak


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

In the context of [[Goodwillie calculus]] the \emph{Taylor tower} of an [[(∞,1)-functor]] $F$ to a [[Goodwillie-differentiable (∞,1)-category]] is its stagewise approximation by [[n-excisive (∞,1)-functors]] $P_n F$ (its \href{n-excisive+%28?%2C1%29-functor#nExcisiveApproximation}{n-excisive projections})

\begin{displaymath}
\cdots \to P_{n+1} F \to P_n F \to \cdots \to P_1 F \to P_0 F
  \,.
\end{displaymath}
which is [[analogy|analogous]] to the approximation of a [[smooth function]] by its [[Taylor series]].

If this tower converges to $F$, then $F$ is analogous to an [[analytic function]] and is called an \emph{[[analytic (∞,1)-functor]]}.

The [[spectral sequence]] induced by the [[filtration]] given by the Goodwillie-Taylor tower is the \emph{[[Goodwillie spectral sequence]]}.

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

\hypertarget{convergence}{}\subsubsection*{{Convergence}}\label{convergence}

convergence for $\rho$-[[analytic (∞,1)-functors]] on $\rho$-[[n-connected object of an (infinity,1)-topos|connective objects]] (\ldots{}) (\hyperlink{Goodwillie03}{Goodwillie 03, theorem 1.13}, see also \hyperlink{MunsonVolic15}{Munson-Volic 15, theorem 10.1.51 and section 10.2.4})

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

\hypertarget{StableSplittingOfMappingSpaces}{}\subsubsection*{{Stable splitting of mapping spaces}}\label{StableSplittingOfMappingSpaces}

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 \emph{[[stable splitting of mapping spaces]]} is originally due to (\hyperlink{Snaith74}{Snaith 74}, \hyperlink{Boedigheimer87}{Bödigheimer 87}) and was understood as a Goodwillie-Taylor tower in \hyperlink{Arone99}{Arone 99}, see also \hyperlink{Ching05}{Ching 05}.

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

\begin{itemize}%
\item [[n-excisive (∞,1)-functor]]


\item [[Postnikov tower]], [[chromatic tower]]



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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The concept originates in

\begin{itemize}%
\item [[Thomas Goodwillie]], \emph{Calculus III: Taylor Series}, Geom. Topol. 7(2003) 645-711 (\href{http://arxiv.org/abs/math/0310481}{arXiv:math/0310481})

\end{itemize}
Review includes

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

\end{itemize}
See also

\begin{itemize}%
\item [[Gregory Arone]], [[Michael Ching]], \emph{A classification of Taylor towers of functors of spaces and spectra} (\href{http://arxiv.org/abs/1209.5661}{arXiv:1209.5661})

\end{itemize}
\hypertarget{stable_splitting_of_mapping_spaces_2}{}\subsubsection*{{Stable splitting of mapping spaces}}\label{stable_splitting_of_mapping_spaces_2}

The [[stable splitting of mapping spaces]] is originally due to

\begin{itemize}%
\item [[Victor Snaith]], \emph{A stable decomposition of $\Omega^n S^n X$}, Journal of the London Mathematical Society 7 (1974), 577 - 583 (\href{https://www.maths.ed.ac.uk/~v1ranick/papers/snaiths.pdf}{pdf})

\end{itemize}
Review and generalization is due to

\begin{itemize}%
\item [[Carl-Friedrich Bödigheimer]], \emph{Stable splittings of mapping spaces}, Algebraic topology. Springer 1987. 174-187 (\href{http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf}{pdf})

\end{itemize}
Further generalization and interpretation in terms of the Goodwillie-Taylor tower of mapping spaces is due to

\begin{itemize}%
\item [[Greg Arone]], \emph{A generalization of Snaith-type filtration}, Transactions of the American Mathematical Society 351.3 (1999): 1123-1150. (\href{https://www.ams.org/journals/tran/1999-351-03/S0002-9947-99-02405-8/S0002-9947-99-02405-8.pdf}{pdf})


\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}
[[!redirects Goodwillie-Taylor towers]]

[[!redirects Taylor tower]] [[!redirects Taylor towers]]

[[!redirects Goodwillie-Taylor series]]



\end{document}