\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*{higher differential geometry}

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

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

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

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{formalizations}{Formalizations}\dotfill \pageref*{formalizations} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

\emph{Higher differential geometry} is the incarnation of [[differential geometry]] in [[higher geometry]]. Hence it is concerned with [[n-groupoid]]-versions of [[smooth spaces]] for \emph{higher} $n$, where the traditional theory is contained in the case $n = 0$. For $n = 1$ these higher structures are \emph{[[Lie groupoids]]}, \emph{[[differentiable stacks]]}, their infinitesimal approximation by [[Lie algebroids]] and the generalization to \emph{[[smooth stacks]]}. For higher $n$ this includes ([[deloopings]] of) [[Lie 2-groups]], [[Lie 3-groups]].

Fully generally, higher differential geometry hence replaces [[smooth manifolds]] (and possibly variants such as [[supermanifolds]], [[formal manifolds]], [[dg-manifolds]] etc.) by \emph{[[∞-stacks]]} ([[(∞,1)-sheaves]]) on the [[site]] of all such. Technically this means that higher differential geometry is the study of an \emph{[[(∞,1)-topos]]} into which standard [[differential geometry]] faithfully embeds. This then allows to speak of [[smooth ∞-groups]], [[Lie ∞-algebroids]].

If the ambient [[(∞,1)-topos]] is not [[n-localic (∞,1)-topos|1-localic]] (for instance over a genuine site of [[dg-manifolds]]) then one also speaks of \emph{[[derived differential geometry]]}.

See at \emph{[[motivation for higher differential geometry]]} for motivation.

The standard variants of [[differential geometry]] have their higher analogs, for instance [[symplectic geometry]] generalizes to [[higher symplectic geometry]] and [[prequantum geometry]] to [[higher prequantum geometry]].

\hypertarget{formalizations}{}\subsection*{{Formalizations}}\label{formalizations}

One axiomatization is [[cohesion]] and [[differential cohesion]].

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

\begin{itemize}%
\item [[Smooth∞Grpd]], [[SuperFormalSmooth∞Grpd]].


\item [[Lie 2-groupoid]]


\item [[double Lie algebroid]], [[Lie algebroid-groupoid]], [[double Lie groupoid]]



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

\begin{itemize}%
\item [[higher Lie theory]]


\item [[higher prequantum geometry]]


\item [[higher complex analytic geometry]]


\item [[L-infinity algebras in physics]]



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

Exposition is in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:Higher Structures|Higher Structures in Mathematics and Physics]]}, talk at \href{https://www.mfo.de/occasion/1651a/www_view}{Oberwolfach Workshop 1651a}, 2016 Dec. 18-23

\end{itemize}
Details are in

\begin{itemize}%
\item \emph{[[geometry of physics]]}, [[geometry of physics -- categories and toposes|categories and toposes]], \ldots{}

\end{itemize}
The most classical aspect of higher differential geometry is the theory of [[orbifolds]], [[Lie groupoids]] and [[Lie algebroids]] and their application in [[foliation theory]]. Original reference here include

\begin{itemize}%
\item [[Charles Ehresmann]], \emph{Cat\'e{}gories topologiques et cat\'e{}gories diff\'e{}rentiables} Colloque de G\'e{}ometrie Differentielle Globale (Bruxelles, 1958), 137--150, Centre Belge Rech. Math., Louvain, 1959;


\item [[Ieke Moerdijk]], [[Dorette Pronk]], \emph{Orbifolds, sheaves and groupoids}, K-theory 12 3-21 (1997) (\href{http://www.math.colostate.edu/~renzo/teaching/Orbifolds/pronk.pdf}{pdf}), \emph{Orbifolds as Groupoids: an Introduction} (\href{http://arxiv.org/abs/math.DG/0203100}{arXiv:math.DG/0203100})



\end{itemize}
and standard textbook accounts include

\begin{itemize}%
\item [[Ieke Moerdijk]], [[Janez Mrcun]] \emph{Introduction to Foliations and Lie Groupoids}, Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, Cambridge, (2003)


\item [[Kirill Mackenzie]], \emph{General Theory of Lie Groupoids and Lie Algebroids,} Cambridge University Press, 2005, xxxviii + 501 pages (\href{http://kchmackenzie.staff.shef.ac.uk/gt.html}{website})


\item [[Kirill Mackenzie]], \emph{Lie groupoids and Lie algebroids in differential geometry}, London Mathematical Society Lecture Note Series, 124. Cambridge University Press, Cambridge, 1987. xvi+327 pp (\href{http://www.ams.org/mathscinet-getitem?mr=896907}{MathSciNet})



\end{itemize}
For properly appreciating the [[homotopy theory]] of Lie groupoids and for passage to more general higher differential geometry it is crucial to understand Lie groupoids as [[smooth stacks]] which are [[geometric stack|geometric]]: \emph{[[differentiable stacks]]}. Each of the following references provides introduction to this point of view:

\begin{itemize}%
\item [[Jochen Heinloth]], \emph{Some notes on differentiable stacks} (\href{http://www.uni-due.de/~hm0002/stacks.pdf}{pdf})


\item [[Kai Behrend]], [[Ping Xu]], \emph{Differentiable Stacks and Gerbes} (\href{http://front.math.ucdavis.edu/0605.5694}{arXiv:0605.5694}).


\item Metzler, \emph{Topological and smooth stacks} (\href{http://arxiv.org/abs/math/0306176}{arXiv:math/0306176})



\end{itemize}
As a warmup for these considerations it may be useful to first look at [[smooth spaces]] given by just [[sheaves]] on the site of [[smooth manifolds]], see at

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[geometry of physics]] -- [[geometry of physics -- smooth spaces|smooth spaces]]}

\end{itemize}
Passing from here to more general [[smooth groupoids]], to [[smooth 2-groupoids]] and then eventually to [[smooth ∞-groupoids]] involves [[(∞,1)-topos theory]] proper, with some tools as discussed at \emph{[[model structure on simplicial presheaves]]} over the [[site]] of [[smooth manifolds]], or equivalently just over its [[dense subsite]] of [[Cartesian spaces]].

For motivation for this step see also

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[twisted smooth cohomology in string theory]]}, lectures at \emph{\href{http://maths-old.anu.edu.au/esi/2012/}{ESI program on quantum fields and K-theory}}, 2012

\end{itemize}
Introductory exposition includes the introductory sections of

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Cech Cocycles for Differential characteristic Classes]]}, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 (\href{http://arxiv.org/abs/1011.4735}{arXiv:1011.4735})


\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{A higher stacky perspective on Chern-Simons theory}, in Damien Calaque et al. (eds.) \emph{Mathematical Aspects of Quantum Field Theories} Mathematical Physics Studies, Springer 2014 (\href{http://arxiv.org/abs/1301.2580}{arXiv:1301.2580})



\end{itemize}
and sections 1.2.4 ([[geometry of physics -- smooth homotopy types]]) as well as section 1.2.5 ([[geometry of physics -- principal bundles]]) in the Introduction section of

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} (\href{http://arxiv.org/abs/1310.7930}{arXiv:1310.7930})

\end{itemize}
This goes on to discuss [[differential cohomology]] and the [[differential cohomology diagram]] formulated in [[stable homotopy type|stable]] objects in [[smooth ∞-groupoids]] (hence in [[sheaves of spectra]] on the [[site]] of [[smooth manifolds]]/[[Cartesian spaces]]) in higher differential geometry, see

\begin{itemize}%
\item [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], \emph{Differential cohomology theories as sheaves of spectra} (\href{http://arxiv.org/abs/1311.3188}{arXiv:1311.3188})

\end{itemize}


\end{document}