\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{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{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{SpectrumObjects}{Spectrum objects}\dotfill \pageref*{SpectrumObjects} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{ReflectionAndExcisiveApproximation}{Reflection and excisive approximation}\dotfill \pageref*{ReflectionAndExcisiveApproximation} \linebreak
\noindent\hyperlink{CharacterizationViaAGenericStableObject}{Characterization via a generic stable object}\dotfill \pageref*{CharacterizationViaAGenericStableObject} \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}

An \emph{excisive (∞,1)-functor} $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ is one which sends [[homotopy pushout]] squares to [[homotopy pullback]] squares. If $\mathcal{C}$ is [[pointed object|pointed]] [[finite homotopy types]] and $\mathcal{D}$ is [[spectra]], then this condition is the [[axiom]] of [[excision]] in [[generalized (Eilenberg-Steenrod) cohomology|generalized homotopy]], whence the name.

Moreover, if here $\mathcal{D}$ is instead [[∞Grpd]] (i.e. [[homotopy types]]), or more generally any [[(∞,1)-topos]] $\mathbf{H}$, then excisive functors that send the point to the point (up to equivalence) are still equivalent to [[spectra]] ([[spectrum objects]]) -- essentially by the [[Brown representability theorem]] -- and those without restriction are equivalent to [[parameterized spectra]], hence form the [[tangent (∞,1)-topos]] $T \mathbf{H}$.

As such, excisive functors are the lowest nontrivial stage in the [[Goodwillie-Taylor tower]] that approximates the [[classifying (∞,1)-topos]] $\mathbf{H}[X_\bullet]$ for [[pointed objects]]. The higher stages of this tower are given by the [[n-excisive (∞,1)-functors]].

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

\begin{defn}
\label{ExcisiveFunctor}\hypertarget{ExcisiveFunctor}{}
An [[(∞,1)-functor]] $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ out of an [[(∞,1)-category]] with [[finite (∞,1)-colimits]] is \textbf{excisive} if it takes [[(∞,1)-pushout]] squares in $\mathcal{C}$ to [[(∞,1)-pullback]] squares $\mathcal{D}$.

This is the $n=1$ case of the concept of \emph{[[n-excisive (∞,1)-functor]]}.

\end{defn}
(e.g. \hyperlink{HigherAlg}{HigherAlg, def. 1.4.2.1.})

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

\hypertarget{SpectrumObjects}{}\subsubsection*{{Spectrum objects}}\label{SpectrumObjects}

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{defn}
\label{SpectraAsExcisiveFunctors}\hypertarget{SpectraAsExcisiveFunctors}{}
Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[finite (∞,1)-limits]]. Then \textbf{[[spectrum objects]]} in $\mathcal{C}$ are equivalently [[reduced (infinity,1)-functor|reduced]] [[excisive (∞,1)-functor]] of the form

\begin{displaymath}
\infty Grpd_{fin}^{\ast/} \longrightarrow \mathcal{C}
  \,.
\end{displaymath}
\end{defn}
(Sometimes, e.g. in \hyperlink{HigherAlg}{Lurie, def. 1.4.2.8}, this is taken as the very definition of spectrum objects).

A proof of prop. \ref{SpectraAsExcisiveFunctors} passing through [[model category]] [[presentable (infinity,1)-category|presentations]] for excisive $\infty$-functors and of the [[Bousfield-Friedlander model structure]] for [[sequential spectra]] is due to (\hyperlink{Lydakis98}{Lydakis 98}), see at \emph{[[model structure for excisive functors]]} at \emph{\href{model+structure+for+excisive+functors#ModelStructureForSpectra}{Relation to BF-model structure on sequential spectra}}.

The idea of the equivalence is as follows. Let $E$ be a reduced excisive functor. For each $n \in \mathbb{N}$, write $S^n \in \infty Grpd_{fin}^{\ast/}$ for the [[n-sphere]] and write $E_{n} \coloneqq E(S^n)$. We have the homotopy pushout squares

\begin{displaymath}
\itexarray{
    S^n &\longrightarrow& \ast
    \\
    \downarrow & & \downarrow
    \\
    \ast &\longrightarrow& S^{n+1}
  }
\end{displaymath}
and since $E$ sends them to homotopy pullbacks with the point going to the point, this gives equivalences

\begin{displaymath}
E_n \stackrel{\simeq}{\longrightarrow} \Omega E_{n+1}
  \,.
\end{displaymath}
This makes $E_\bullet$ have the structure of an [[Omega spectrum]]. The idea then is that as such it represents a [[generalized homology theory]] and the value of the excisive functor on any finite homotopy type $X$ is then $\Omega^\infty(E \wedge X)$ (see \hyperlink{HigherAlg}{Lurie, remark 1.4.3.3}).

\begin{remark}
\label{RelationToGeneralizedHomologyTheories}\hypertarget{RelationToGeneralizedHomologyTheories}{}
The traditional definition of a [[generalized homology theory]] is as a functor on (finite) homotopy types with values in [[graded abelian groups]]. The [[Brown representability theorem]] says that these all arise from [[spectra]] $E$ via taking [[stable homotopy groups]] of [[smash products]]: $X \mapsto E_\bullet(X) \coloneqq \pi_\bullet(E \wedge X)$. But due to the existence of [[phantom maps]], this does not quite yield an [[equivalence of (infinity,1)-categories|equivalence]] between spectra and generalized homology theories.

In view of this, the proof of prop. \ref{SpectraAsExcisiveFunctors} may be thought of saying that this mismatch is fixed by refining homotopy groups by full [[homotopy types]] $X \mapsto \Omega^\infty(E\wedge X)$.

Notice also that spectra realized as excisive functors this way are in the spirit of [[coordinate-free spectra]].

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

For the moment we focus on properties of the case of excisive functors $\mathcal{C} \to \mathcal{D}$ for $\mathcal{C} = \infty Grpd_{fin}^{\ast/}$ and $\mathcal{D} = \mathbf{H}$, hence on

\begin{displaymath}
Exc(\infty Grpd_{fin}^{\ast/}, \mathbf{H}) \simeq T \mathbf{H} \hookrightarrow \mathbf{H}[X_\ast]
  \,.
\end{displaymath}
\begin{defn}
\label{ClassifyingToposForPointedObjects}\hypertarget{ClassifyingToposForPointedObjects}{}
Write $\infty Grpd_{fin}$ is the [[(∞,1)-category]] of [[finite homotopy types]]. For $\mathbf{H}$ a given [[base (∞,1)-topos]], write

\begin{displaymath}
\begin{aligned}
    \mathbf{H}[X_\ast]
    & \simeq
    [\infty Grpd_{fin}^{\ast/}, \mathbf{H}]
    \\
    & \simeq
    PSh((\infty Grpd_{fin}^{\ast/})^{op}, \mathbf{H})
  \end{aligned}
\end{displaymath}
for the [[classifying (∞,1)-topos]] (over $\mathbf{H}$) for [[pointed objects]].

\end{defn}
\hypertarget{ReflectionAndExcisiveApproximation}{}\subsubsection*{{Reflection and excisive approximation}}\label{ReflectionAndExcisiveApproximation}

\begin{defn}
\label{P1}\hypertarget{P1}{}
Write

\begin{displaymath}
T_1 \colon \mathbf{H}[X_\ast] \longrightarrow \mathbf{H}[X_\ast]
\end{displaymath}
for the functor given by

\begin{displaymath}
E \mapsto \Omega_{E(\ast)} E(\Sigma(-))
  \,.
\end{displaymath}
Write

\begin{displaymath}
P_1 \coloneqq \underset{\longrightarrow}{\lim}_{n} T_1^{n}
\end{displaymath}
for the [[homotopy colimit]] of the iterations of this functor, with respect to the canonical comparison map.

\end{defn}
Unwinding the definition and using that [[suspension]] is equivalently the \href{join%20of%20topological%20spaces#JoinWith0Sphere}{join with the 0-sphere}, this is indeed the functor of the same name in (\hyperlink{Goodwillie03}{Goodwillie 91, p. 657 (13 of 67)}).

\begin{theorem}
\label{PnLocalizes}\hypertarget{PnLocalizes}{}
The inclusion of excisive functors into $\mathbf{H}[X_*]$ is a [[reflective sub-(∞,1)-category]] with reflector given by $P_1$ from def. \ref{P1}:

\begin{displaymath}
T \mathbf{H} 
   \stackrel{\overset{P_1}{\longleftarrow}}{\hookrightarrow}
  \mathbf{H}[X_\ast]
\end{displaymath}
\end{theorem}
This is due (in the generality of \emph{\href{https://ncatlab.org/nlab/show/n-excisive+functor#nExcisiveApproximation}{n-excisive functor -- n-Excisive Approximation and reflection}}) to (\hyperlink{Goodwillie91}{Goodwillie 91, theorem 1.8}). See also (\hyperlink{HigherAlg}{Lurie, theorem 6.1.1.10, construction 6.1.1.27}).

\begin{prop}
\label{ExcisiveApproximation}\hypertarget{ExcisiveApproximation}{}
Let $\mathcal{C}$ have finite colimits and a terminal object and let $\mathcal{D}$ be differentiable.

The excisive approximation of a [[reduced functor]] $F \colon \mathcal{C} \to \mathcal{D}$ is the [[(infinity,1)-colimit]]

\begin{displaymath}
P_1 F 
    \simeq 
  \underset{\longrightarrow_{\mathrlap{k}}}{\lim} 
  ( \Omega^k \circ F \circ \Sigma^k)
\end{displaymath}
(where $\Omega$ and $\Sigma$ denote [[looping]] and [[suspension]] in $\mathcal{D}$ and in $\mathcal{C}$, respectively).

\end{prop}
(\hyperlink{HigherAlg}{Lurie, example 6.1.1.28})

\begin{remark}
\label{}\hypertarget{}{}
Under the equivalence to [[sequential spectra]] in prop. \ref{SpectraAsExcisiveFunctors}, the formula in prop. \ref{ExcisiveApproximation} is the standard formula for [[spectrification]] of [[prespectra]].

\end{remark}
\hypertarget{CharacterizationViaAGenericStableObject}{}\subsubsection*{{Characterization via a generic stable object}}\label{CharacterizationViaAGenericStableObject}

We discuss a characterization of excisive functors on $\infty Grpd_{fin}^{\ast/}$, hence of [[parameterized spectra]], as the result of [[forcing]] a generic pointed object to become a [[stable homotopy type]]. This general perspective is being highlighted by \hyperlink{AnelFinsterJoyal}{Anel-Finster-Joyal}.

For a slick formulation, we use a generalization of [[powering]] to pointed powers:

\begin{defn}
\label{PointedPower}\hypertarget{PointedPower}{}
For $X$ an object in an [[(∞,1)-category]] $\mathcal{C}$ with [[finite (∞,1)-limits]], and for $S_\ast \in \infty Grpd_{fin}^{\ast/}$ a pointed [[finite ∞-groupoid]], then the \emph{pointed power}

\begin{displaymath}
X^{S_\ast} \in \mathcal{C}
\end{displaymath}
is the object which is the image of $S$ under the essentially unique [[(∞,1)-functor]]

\begin{displaymath}
(\infty Grpd_{fin}^{\ast/})^\op \longrightarrow \mathcal{C}
\end{displaymath}
which preserves [[finite (∞,1)-limits]] and sends $S^0 \leftarrow \ast$ to $X_\ast \to \ast$.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
Excisive functors $\infty Grpd_{fin}^{\ast/} \longrightarrow \mathbf{H}$, def. \ref{ExcisiveFunctor}, are the [[localization of an (∞,1)-category|localization]] of $\mathbf{H}[X_\ast]$, def. \ref{ClassifyingToposForPointedObjects}, at the set of morphisms

\begin{displaymath}
\left\{
    \Sigma \Omega (X_\ast^{S_\ast}) \longrightarrow X_\ast^{S_\ast}
  \right\}_{S \in \infty Grpd_{fin}^{\ast/}}
  \,,
\end{displaymath}
(where $X_{\ast}^{S_\ast}$ is the pointed power, def. \ref{PointedPower}, of the generic pointed object $X_\ast \in \mathbf{H}[X_\ast]$):

\begin{displaymath}
T \mathbf{H}\simeq \mathbf{H}[X_\ast][ (\Sigma \Omega X_\ast^\bullet \to X_\ast^\bullet)^{-1} ]
  \,.
\end{displaymath}
In other words, the [[parameterized spectra]] are those objects in $\mathbf{H}[X_\ast]$ which regard each finite pointed power of the generic pointed object $X_\ast$ as a [[stable homotopy type]].

\end{prop}
\begin{proof}
For $K \in \infty Grpd_{fin}^{\ast/}$, write $R(K) \in (\infty Grpd_{fin}^{\ast/})^{op} \hookrightarrow \mathbf{H}[X_\ast]$ for its [[formal dual]] under [[(∞,1)-Yoneda embedding]]. Since the [[(∞,1)-Yoneda embedding]] preserves [[(∞,1)-limits]], we have

\begin{displaymath}
R(K)^S \simeq R(S \cdot K)
  \,.
\end{displaymath}
Observe that the generic pointed object in $\mathbf{H}[X_\ast]$ is that [[representable functor|represented]] by the [[0-sphere]]:

\begin{displaymath}
X_\ast = R(S^0)
  \,.
\end{displaymath}
Hence

\begin{displaymath}
X_\ast^S \simeq R(S)
  \,.
\end{displaymath}
Now using the [[(∞,1)-Yoneda lemma]] we have for each $E \in \mathbf{H}[X_\ast]$ that

\begin{displaymath}
\begin{aligned}
    Hom( \Sigma \Omega R(K), E )
    &\simeq
    \Omega_{E(\ast)} Hom(\Omega R(K),E)
    \\
    & \simeq 
    \Omega_{E(\ast)} Hom( R(\Sigma K), E )
    \\
    & \simeq
    \Omega_{E(\ast)} E(\Sigma K)
  \end{aligned}
  \,.
\end{displaymath}
Hence for all $K \in \infty Grpd_{fin}^{\ast/}$

\begin{displaymath}
\begin{aligned}
    Hom(\Sigma \Omega R(K) \to R(K), E)
    & \simeq
    ( E(K) \longrightarrow  \Omega(\Sigma K) ) 
    \\
    & =
    (E \to T_1 E)(K)
  \end{aligned}
  \,,
\end{displaymath}
where in the last line we observe that the expression is that for the comparison map in def. \ref{P1}.

This means that the [[local objects]] are precisely those $E$ for which the morphism

\begin{displaymath}
E \longrightarrow T_1 E
\end{displaymath}
from def. \ref{P1} is an equivalence. With this the statement follows from theorem \ref{PnLocalizes}.

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

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


\item [[n-excisive (∞,1)-functor]]


\item [[model structure for excisive functors]]


\item [[spectrum object]]



\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}
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}
Review includes

\begin{itemize}%
\item [[Yonatan Harpaz]], section 5 of \emph{Introduction to stable $\infty$-categories}, October 2013 ([[HarpazStableInfinityCategory2013.pdf:file]])

\end{itemize}
A [[model structure for excisive functors]] was given in

\begin{itemize}%
\item Lydakis, \emph{Simplicial functors and stable homotopy theory} Preprint, available via Hopf archive, 1998 (\href{http://hopf.math.purdue.edu/Lydakis/s_functors.pdf}{pdf})

\end{itemize}
Discussion in terms of [[stable homotopy types]] is due to

\begin{itemize}%
\item [[Mathieu Anel]], [[Eric Finster]], [[André Joyal]], in preparation

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

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

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

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

[[!redirects excisive functor]] [[!redirects excisive functors]]

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

[[!redirects preexcisive functor]] [[!redirects preexcisive functors]]



\end{document}