\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*{derived microlocalization}

[[!redirects Derived microlocalization]] [[!redirects derived microlocalization]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{motivation_for_a_derived_notion_of_microlocalization}{Motivation for a derived notion of microlocalization}\dotfill \pageref*{motivation_for_a_derived_notion_of_microlocalization} \linebreak
\noindent\hyperlink{loop_space_and_deformation_to_the_normal_bundle_method_i}{Loop space and deformation to the normal bundle: method I}\dotfill \pageref*{loop_space_and_deformation_to_the_normal_bundle_method_i} \linebreak
\noindent\hyperlink{loop_space_and_deformation_to_the_normal_bundle_method_ii}{Loop space and deformation to the normal bundle: method II}\dotfill \pageref*{loop_space_and_deformation_to_the_normal_bundle_method_ii} \linebreak
\noindent\hyperlink{a_derived_analog_of_microlocalization}{A derived analog of microlocalization}\dotfill \pageref*{a_derived_analog_of_microlocalization} \linebreak
\noindent\hyperlink{a_derived_analytic_analog_of_microlocalization}{A derived analytic analog of microlocalization}\dotfill \pageref*{a_derived_analytic_analog_of_microlocalization} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

[[Microlocalization]] is a tool to study the propagation of singularities of solutions of partial differential systems, in order to study pullbacks of solutions of differential systems (which generalize products of distributions) and prove [[index theorems]]. Derived microlocalization is an adaptation of the theory of microlocalization to the setting of derived [[global analytic geometry]].

\hypertarget{motivation_for_a_derived_notion_of_microlocalization}{}\subsection*{{Motivation for a derived notion of microlocalization}}\label{motivation_for_a_derived_notion_of_microlocalization}

In non-smooth situations, the usual normal and conormal bundle used in classical microlocalization is not well behaved, and one needs to take derived versions of them. Moreover, to study [[global analytic index theory]], one needs a formulation of the theory of microlocalization in terms of derived loop stacks, in order to work out a cyclic and Hochshild version of index theorems (similar to the one developed by Kashiwara and Schapira in the [[microlocal formulation of index theory]]) that works on an arbitrary (e.g. integral) basis.

\hypertarget{loop_space_and_deformation_to_the_normal_bundle_method_i}{}\subsection*{{Loop space and deformation to the normal bundle: method I}}\label{loop_space_and_deformation_to_the_normal_bundle_method_i}

We would like to define a family over (say) the disc $D^1$ whose fiber at $0$ is $LX:=\Hom(S^1,X)$ and whose fiber at $1$ is $[LX/S^1]$. In the case of a usual scheme, this is done (by Vezzosi) by showing that $LX$ identifies with $\Hom(BG_a,X)$, and by making $G_a$ act multiplicatively on the action of $BG_a$ on $LX$, to get the trivial action at $0$ and the usual action at $1$. This gives the desired family. Recall that $BG_a$ is the affinization of $B\mathbb{Z}\cong S^1$.

\hypertarget{loop_space_and_deformation_to_the_normal_bundle_method_ii}{}\subsection*{{Loop space and deformation to the normal bundle: method II}}\label{loop_space_and_deformation_to_the_normal_bundle_method_ii}

[[Microlocalization]] in [[derived geometry]] involves the proper definition of the deformation to the [[normal bundle]] of a closed embedding $Y\subset X$ of global analytic spaces (or even stacks). One needs to consider a [[derived loop space]] approach to this construction because it allows to avoid the use of denominators in the definition of the [[Chern character]] (following Connes-Loday-Toen-Vezzosi) and in the development of more general [[global analytic index theory]] with integral coefficients.

The space of paths on $X$ based on $Y$ is the groupoid acting on $Y$ given by

\begin{displaymath}
P_Y X:=\{f\in \Hom(\Delta^1,X),\;f(0)\in Y,\;f(1)\in Y\}.
\end{displaymath}
More concretely, this is given by the homotopy pullback

\begin{displaymath}
\itexarray{
    P_Y X &\to& \Hom(\Delta^1,X)
    \\
    \downarrow && \downarrow^{\mathrlap{ev_0\times ev_1}}
    \\
    Y\times Y &\to& X\times X
}
\end{displaymath}
Notice that the natural projection $P_Y X\to Y\times Y$ makes $P_Y X$ a groupoid (of paths in $X$) acting on $Y$. In the case of the diagonal immersion $Y=M\hookrightarrow M\times M=X$, we get

\begin{displaymath}
P_Y X\cong LM:=\Hom(S^1,M).
\end{displaymath}
There is a natural projection $p:P_Y X\to \Hom(\Delta^1,X)\sim X$ and $P_Y X$ is equiped with the natural structure of a groupoid acting on $Y$ through the projection $P_Y X\to Y\times Y$. A more explicit description of the loop space (that is obtained by using the homotopy $\Delta^1\sim \Delta^0$) is given by the homotopy pullback

\begin{displaymath}
P_Y X=Y\times^h_X Y,
\end{displaymath}
which clearly has an interesting meaning only in the setting of derived geometry. See also at \emph{[[derived loop space]]} and at \emph{[[Hochschild cohomology]]}.

To make the above construction work for general Artin stacks, one needs to make it local for the smooth topology. This is done by replacing the loop space groupoid $P_Y X$ acting on $Y$ by its formal completion along the identity morphism. This gives a formal groupoid $\hat{P}_Y X$ that is equivalent modulo the [[HKR]] theorem (in characteristic $0$) to $T_Y[-1]X$ for a general Artin stack. Similarly, we will have $\hat{L}M:=\hat{P}_M(M\times M)\cong T[-1]M$ by [[HKR]].

The deformation to the normal bundle in strict derived global analytic geometry is then simply given by the (\textbf{false to be corrected}) formula (with $D^1=\mathbb{M}(R\{X\}^\dagger)$ for $R$ the base ind-Banach ring)

\begin{displaymath}
\widetilde{P_Y X}:=\{f\in \Hom_{D^1}(\Delta^1\times D^1,X\times D^1),\;f(0,0)\in Y,\; f(1,0)\in Y,\;f(-,1)\in (X\backslash Y)\Rightarrow f(x,t)\in (X\backslash Y)\forall t\neq 0\}.
\end{displaymath}
More concretely, this is given by the homotopy pullback (where $U(1)=\mathbb{M}(R\{X,Y\}^\dagger/(XY-1))$ and we use the homotopy equivalence $\Hom(\Delta^1,X)\sim X$)

\begin{displaymath}
\itexarray{
    \widetilde{P_Y X} &\to& \Hom_{D^1}(\Delta^1\times D^1,X\times D^1)
    \\
    \downarrow && \downarrow^{\mathrlap{ev_{(0,0)}\times ev_{(1,0)}\times \ev_{(-,U(1))}\times \ev_{(-,1)}}}
    \\
    Y\times Y\times \Hom(\Delta^1\times U(1),(X\backslash Y))\times (X\backslash Y) &\stackrel{i\times j\times k}{\to}& X\times X\times \Hom(\Delta^1\times U(1),X)
\times X}
\end{displaymath}
It has an evident natural projection $t:\widetilde{P_Y X}\to D^1$, and a natural projection $\widetilde{P_Y X}\to Y\times Y$ that makes it a family of groupoids parametrized by $D^1$ and acting on $Y$. There is also a natural projection $p:\widetilde{P_Y X}\to X$ given by $f\mapsto f(-,1)$. We will denote $s:P_Y X\to \widetilde{P_Y X}$ the fiber $t^{-1}(0)$.

One may complete $\widetilde{P_Y X}$ along the unit of its groupoid structures, to get a family of formal groupoids $\widehat{P_Y X}$ parametrized by $D^1$, whose fiber at $0$ will be the formal loop space $\hat{P}_Y X$ obtained by completing the path groupoid $P_Y X$ acting on $Y$ along its identity morphism, and whose fiber at $1$ will be the formal completion $\hat{X}_Y$ of $X$ along $Y$. There is a natural action of $S^1$ on $\widehat{P_Y X}$. We will still denote $t:\widehat{P_Y X}\to D^1$ the natural projection, $s:\widehat{P}_Y X\to \widehat{P_Y X}$ the fiber $t^{-1}(0)$, and $p:\widehat{P_Y X}\to X$ the natural projection (evaluation at $(-,1)$).

In the particular case of the diagonal embedding $Y=M\hookrightarrow M\times M=X$ over a field of characteristic $0$, the quotient of $X$ by this family of formal groupoids gives back (modulo a convenient [[HKR]] theorem) exactly Simpson's non-abelian Hodge structure, that gives a family of formal stacks over $D^1$ whose fiber at $0$ is the tangent space $TX$ (seen as the quotient of $X$ by the trivial action of $T[-1]X$) and whose fiber at $1$ is the so-called de Rham space $X_{dR}$ of $X$.

\hypertarget{a_derived_analog_of_microlocalization}{}\subsection*{{A derived analog of microlocalization}}\label{a_derived_analog_of_microlocalization}

We now work on an arbitrary derived Artin stack $M$, such that the diagonal $Y=M\to M\times M=X$ is an affine closed embedding.

The usual theory of [[microlocalization]] may be adapted to the derived global analytic setting by replacing the deformation to the normal bundle $\widetilde{T_Y X}\to \mathbb{A}^1$ by the formal deformation to the normal bundle $\widehat{P_Y X}\to D^1$.

The derived specialization functor is given on $F\in D^b(X)$ by

\begin{displaymath}
\nu_Y(F):=s^*p^*F\in D^b(\hat{P}_Y X).
\end{displaymath}
Remark that the loop space analog $L^*X$ of the odd cotangent bundle $T^*[1]X$ (that should be dual to the odd tangent bundle $T[-1]X$ (related to $LX$ by the [[HKR]] theorem) is given by $S^1\otimes X$ (external tensor product by the simplicial circle). We will have for every stack $Y$, the canonical equivalence

\begin{displaymath}
\Hom(S^1\otimes X,Y)\cong \Hom_{SSets}(S^1,\Hom_{dSt}(X,Y)).
\end{displaymath}
The derived Fourier-Sato transformation is given on $F\in D^b(\hat{P}_Y X)$ by

\begin{displaymath}
\Phi(F):=?.
\end{displaymath}
The derived microlocalization functor is given on $F\in D^b(M)$ by

\begin{displaymath}
\mu(F):=\Phi(\nu_Y(F))
\end{displaymath}
for $Y=M\hookrightarrow M\times M=X$.

\hypertarget{a_derived_analytic_analog_of_microlocalization}{}\subsection*{{A derived analytic analog of microlocalization}}\label{a_derived_analytic_analog_of_microlocalization}

One may replace the simplicial circle $S^1$, used in the definition of the derived loop space, by the unitary group $U(1)$ of [[overconvergent global analytic geometry]], to get a more analytic theory of microlocalization. In the complex situation, we will have $U(1)\cong S^1$ up to $D^1$-homotopy. Remark that the exponential map $exp(i-):\mathbb{R}\to S^1$ also has a meaning in overconvergent complex analytic geometry, if we see $\mathbb{R}\subset \mathbb{C}$ as a closed subset equipped with its germs of analytic functions.

In the global analytic setting (without imposing homotopy invariance), there is no reason to have $U(1)=D^1\times_{*\coprod *} *$. We thus prefer to use $P^1:=D^1\coprod_{U(1)}D^1$ as a natural global analytic parameter space for paths. We define

\begin{displaymath}
L^\dagger_Y X:=\{f\in Hom(P^1,X), f(0)\in Y,\; f(\infty)\in Y\}.
\end{displaymath}
Remark that up to $D^1_\mathbb{R}$-homotopy, we get

\begin{displaymath}
L^\dagger_Y X\sim L_Y X.
\end{displaymath}
We must now check that the natural morphism

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

\begin{itemize}%
\item [[David Ben-Zvi]], [[David Nadler]], \emph{Loops spaces and connections} (\href{http://arxiv.org/abs/1002.3636}{arXiv:1002.3636})


\item [[Damien Calaque]], [[Andrei Caldararu]] and [[Junwu Tu]]: \emph{PBW for an inclusion of Lie algebras} (\href{http://arxiv.org/abs/1010.0985}{arXiv:1010.0985})


\item [[Damien Calaque]], [[Andrei Caldararu]] and [[Junwu Tu]]: \emph{On the Lie algebroid of a derived self-intersection} (\href{http://arxiv.org/abs/1306.5260}{arXiv:1306.5260})



\end{itemize}


\end{document}