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\begin{document}

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\section*{n-excisive (∞,1)-functor}

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{cocartesian_cubical_diagrams}{(Co-)Cartesian cubical diagrams}\dotfill \pageref*{cocartesian_cubical_diagrams} \linebreak
\noindent\hyperlink{excisive_functors}{$n$-Excisive functors}\dotfill \pageref*{excisive_functors} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{nExcisiveApproximation}{$n$-Excisive approximation and reflection}\dotfill \pageref*{nExcisiveApproximation} \linebreak
\noindent\hyperlink{homogeneous_pieces}{Homogeneous pieces}\dotfill \pageref*{homogeneous_pieces} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{GoodwillienJets}{Goodwillie $n$-jets}\dotfill \pageref*{GoodwillienJets} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{cocartesian_cubical_diagrams}{}\subsubsection*{{(Co-)Cartesian cubical diagrams}}\label{cocartesian_cubical_diagrams}

Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[finite (∞,1)-colimits]].

\begin{defn}
\label{StronglyCoCartesian}\hypertarget{StronglyCoCartesian}{}
An $n$-[[cube]] in $\mathcal{C}$, hence an [[(∞,1)-functor]] $\Box^n \longrightarrow \mathcal{C}$, is called \emph{strongly homotopy co-cartesian} or just \emph{strongly co-cartesian}, if all its 2-dimensional square faces are [[homotopy pushout]] [[diagrams]] in $\mathcal{C}$.

\end{defn}
\begin{defn}
\label{Cartesian}\hypertarget{Cartesian}{}
An $n$-[[cube]] in $\mathcal{D}$, hence an [[(∞,1)-functor]] $\Box^n \longrightarrow \mathcal{D}$, is called \emph{homotopy cartesian} or just \emph{cartesian}, if its ``first'' object exhibits a [[homotopy limit]]-[[cone]] over the remaining objects.

\end{defn}
(e.g. \hyperlink{HigherAlg}{Lurie, def. 6.1.1.2 with prop. 6.1.1.15})

\hypertarget{excisive_functors}{}\subsubsection*{{$n$-Excisive functors}}\label{excisive_functors}

\begin{defn}
\label{}\hypertarget{}{}
An [[(∞,1)-functor]] $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ is \textbf{$n$-excisive} for $n \in \mathbb{N}$ (or \emph{polynomial} of degree at most $n$) if whenever $X$ is a strongly cocartesian $(n + 1)$-[[cube]] in $\mathcal{C}$, def. \ref{StronglyCoCartesian}, then $F(X)$ is a cartesian cube in $\mathcal{D}$, def. \ref{Cartesian}.

A 1-excisive (∞,1)-functor is often just called \emph{[[excisive (∞,1)-functor]]} for short.

\end{defn}
An $(\infty,1)$-functor which is $n$-excisive for some $n \in \mathbb{N}$ is also called a \textbf{polynomial (∞,1)-functor} (not to be confused with [[polynomial (∞,1)-functor|other concepts having the same name]]). It has \textbf{degree $k$} when the smallest value of $n$ for which it is $n$-excisive is $k$.

\begin{remark}
\label{}\hypertarget{}{}
This notion is comparable to how a [[polynomial]] of degree at most $n$ is determined by its values on $n + 1$ distinct points.

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

\hypertarget{nExcisiveApproximation}{}\subsubsection*{{$n$-Excisive approximation and reflection}}\label{nExcisiveApproximation}

Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[finite (∞,1)-colimits]] and a [[terminal object in an (∞,1)-category|terminal object]], and let $\mathcal{D}$ be a [[Goodwillie-differentiable (∞,1)-category]].

\begin{defn}
\label{InclusionOfNExcisive}\hypertarget{InclusionOfNExcisive}{}
For $n \in \mathbb{N}$

\begin{displaymath}
Exc^n(\mathcal{C}, \mathcal{D})
  \hookrightarrow
  Func(\mathcal{C}, \mathcal{D})
\end{displaymath}
for the [[full sub-(∞,1)-category]] of the [[(∞,1)-functor (∞,1)-category]] on those [[(∞,1)-functors]] which are $n$-excisive.

\end{defn}
\begin{prop}
\label{nExcisiveApproximations}\hypertarget{nExcisiveApproximations}{}
The inclusion of def. \ref{InclusionOfNExcisive} is [[left exact (∞,1)-functor|lex]] [[reflective sub-(∞,1)-category|reflective]], hence the inclusion functor has a [[left adjoint|left]] [[adjoint (∞,1)-functor]]

\begin{displaymath}
P_n \;\colon\; Func(\mathcal{C}, \mathcal{D}) \longrightarrow
  Exc^n(\mathcal{C}, \mathcal{D})
\end{displaymath}
which moreover is [[left exact (infinity,1)-functor|left exact]] (preserves [[finite (∞,1)-limits]]).

\end{prop}
This is essentially the statement of (\hyperlink{Goodwillie03}{Goodwillie 03, theorem 1.8}). In the above form it appears explicitly as (\hyperlink{HigherAlg}{Lurie, theorem 6.1.1.10}). The construction of the reflector $P_n$ is in (\hyperlink{HigherAlg}{Lurie, constrution 6.1.1.27}).

For $n = 1$ this reflection is \emph{[[spectrification]]}.

\begin{cor}
\label{}\hypertarget{}{}
For $\mathcal{D} = \mathbf{H}$ an [[(∞,1)-topos]], then for all $n \in \mathbb{N}$ we have that

\begin{displaymath}
Exc^n(\mathcal{C}, \mathbf{H}) \in (\infty,1)Topos
\end{displaymath}
is an [[(∞,1)-topos]]. (For $n  \gt 1$ this is in general \emph{not} a [[hypercomplete (∞,1)-topos]], even if $\mathbf{H}$ is.)

\end{cor}
This observation is due to [[Charles Rezk]]. It appears as (\hyperlink{HigherAlg}{Lurie, remark 6.1.1.11}).

\begin{remark}
\label{}\hypertarget{}{}
A [[site]] of definition of $Exc^n(\mathcal{C}, \mathbf{H}) \hookrightarrow PSh(\mathcal{C}^{op}, \mathbf{H})$ is the \emph{[[Weiss topology]]} on $\mathcal{C}^{op}$.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
As $n$ ranges, the tower of $n$-excisive approximations of an $(\infty,1)$-functor, accordding to prop. \ref{nExcisiveApproximations}, forms a tower [[analogy|analogous]] to the the [[Taylor series]] of a [[smooth function]]. This is called the [[Goodwillie-Taylor tower]]

\begin{displaymath}
\cdots \to P_{n+1} F \to P_n F \to \cdots \to P_1 F \to P_0 F
  \,.
\end{displaymath}
If this converges to $F$, then $F$ is analogous to an [[analytic function]] and is called an \emph{[[analytic (∞,1)-functor]]}.

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
In the situation of def. \ref{nExcisiveApproximations}, the functors $F$ for which $P_{n-1}F \simeq \ast$ (hence the $P_{n-1}$ [[anti-modal types]]) are called $n$-[[reduced (∞,1)-functors]].

\end{defn}
(e.g. \hyperlink{HigherAlg}{Lurie, def. 6.1.2.1})

\hypertarget{homogeneous_pieces}{}\subsubsection*{{Homogeneous pieces}}\label{homogeneous_pieces}

A polynomial $\infty$-functor of degree $k$ --- that is, a $k$-excisive functor which is not $n$-excisive for any $n\lt k$ --- is a \textbf{homogeneous} polynomial if its approximation by an $(k-1)$ degree polynomial is trivial.

(\ldots{})

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[finite (∞,1)-colimits]] and with [[terminal object in an (∞,1)-category|terninal object]]. Let $\mathcal{D}$ be a [[pointed category|pointed]] [[Goodwillie-differentiable (∞,1)-category]]. Write $\mathcal{C}^{\ast/}$ for the [[pointed objects]] in $\mathcal{C}$.

Then for all natural numbers $n \geq 1$ composition with the [[forgetful functor]] $\mathcal{C}^{\ast/} \to \mathcal{C}$ induces an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
Homog^n(\mathcal{C},\mathcal{D}) \stackrel{\simeq}{\longrightarrow} Homog^n(\mathcal{C}^{\ast/}, \mathcal{D})
\end{displaymath}
\end{prop}
(\hyperlink{HigherAlg}{Lurie, prop. 6.1.2.11})

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

\hypertarget{GoodwillienJets}{}\subsubsection*{{Goodwillie $n$-jets}}\label{GoodwillienJets}

Write $\infty Grpd_{fin}$ for the [[(∞,1)-category]] of [[finite homotopy types]], hence those freely generated by [[finite (∞,1)-colimits]] from the point. Write $\infty Grpd_{fin}^{\ast/}$ for the [[pointed object|pointed]] finite homotopy types.

\begin{example}
\label{}\hypertarget{}{}
For $\mathbf{H}$ an [[(∞,1)-topos]] we have that

\begin{itemize}%
\item $Exc^0(\infty Grpd_{fin}^{\ast/},\mathbf{H}) \simeq \mathbf{H}$ is the collection of constant functors, hence the original [[(∞,1)-topos]] itself;


\item $Exc^1(\infty Grpd_{fin}^{\ast/},\mathbf{H}) \simeq T\mathbf{H}$ is the collection of [[parameterized spectra]] in $\mathbf{H}$, hence the [[tangent (∞,1)-topos]] of $\mathbf{H}$.



\end{itemize}
Hence one might refer to the tower of toposes

\begin{displaymath}
\cdots \to J^n \mathbf{H} \to \cdots \to J^2 \mathbf{H} \to T \mathbf{H} \to \mathbf{H}
\end{displaymath}
with

\begin{displaymath}
J^n \mathbf{H} \coloneqq Exc^n(\infty Grpd^{\ast/}, \mathbf{H})
\end{displaymath}
the tower of ``[[Goodwillie calculus|Goodwillie]] [[jet (∞,1)-categories]]'' of $\mathbf{H}$.

\end{example}
see (\hyperlink{HigherAlg}{Lurie, def. 1.4.2.8 and around p. 823})

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

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


\item [[Goodwillie calculus]]

\begin{itemize}%
\item [[Goodwillie-Taylor tower]]


\item [[analytic (∞,1)-functor]]



\end{itemize}

\item [[polynomial (∞,1)-functor]]


\item [[model structure for n-excisive functors]]



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

The notion of $n$-excisive functors was introduced in

\begin{itemize}%
\item [[Thomas Goodwillie]], \emph{Calculus II, Analytic functors}, K-Theory 5 (1991/92), no. 4, 295-332

\end{itemize}
The [[Taylor tower]] formed by $n$-excisive functors was then studied 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}
See also

\begin{itemize}%
\item [[Charles Rezk]], \emph{A streamlined proof of Goodwillie's $n$-excisive approximation} (\href{http://arxiv.org/abs/0812.1324}{arXiv:0812.1324})

\end{itemize}
A discussion in the general abstract context of [[(∞,1)-category theory]] is in

\begin{itemize}%
\item [[Jacob Lurie]], section 6.1.1 of \emph{[[Higher Algebra]]}

\end{itemize}
A [[model structure 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}, Advances in Mathematics 214 (2007) 92-115 (\href{http://arxiv.org/abs/math/0601221}{arXiv:math/0601221})


\item [[Georg Biedermann]], Oliver R\"o{}ndigs, \emph{Calculus of functors and model categories II} (\href{http://arxiv.org/abs/1305.2834v2}{arXiv:1305.2834v2})



\end{itemize}
Relation to [[Mackey functors]] is discussed in

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

\end{itemize}
[[!redirects n-excisive (∞,1)-functors]]

[[!redirects n-excisive (infinity,1)-functor]] [[!redirects n-excisive (infinity,1)-functors]]

[[!redirects n-excisive functor]] [[!redirects n-excisive functors]]

[[!redirects n-excisive approximation]] [[!redirects n-excisive approximations]]



\end{document}