\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 geometry}

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

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

[[!include higher geometry - contents]]

\textbf{higher geometry} $\leftarrow$ [[Isbell duality]] $\to$ [[higher algebra]]


\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Formalizations}{Formalizations}\dotfill \pageref*{Formalizations} \linebreak
\noindent\hyperlink{GrosHigherToposes}{Gros (∞,1)-toposes}\dotfill \pageref*{GrosHigherToposes} \linebreak
\noindent\hyperlink{PetitHigherToposes}{Petit (∞,1)-toposes}\dotfill \pageref*{PetitHigherToposes} \linebreak
\noindent\hyperlink{relation_between_these_approaches}{Relation between these approaches}\dotfill \pageref*{relation_between_these_approaches} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{list_of_examples}{List of examples}\dotfill \pageref*{list_of_examples} \linebreak
\noindent\hyperlink{NoncommutativeAlgebraicGeometry}{Noncommutative algebraic geometry}\dotfill \pageref*{NoncommutativeAlgebraicGeometry} \linebreak
\noindent\hyperlink{ConnesStyleNoncommutativeGeometry}{Connes-style noncommutative geometry}\dotfill \pageref*{ConnesStyleNoncommutativeGeometry} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{Lurie}{References}\dotfill \pageref*{Lurie} \linebreak


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

\emph{Higher geometry} or \emph{homotopical geometry} is the study of concepts of [[space]] and [[geometry]] in the context of [[higher category theory]] and [[homotopy theory]].

\begin{uremark}
higher geometry = [[geometry]] + [[homotopy theory]]/[[higher category theory]]

\end{uremark}
Higher geometry subsumes notably the theory of [[orbifolds]] and [[geometric stacks]], as well as the theory of more general [[stacks]] such as [[moduli stacks]], and generalizes all this to [[∞-stacks]] and [[derived stacks]]. This way higher geometry includes what is called \emph{[[derived geometry]]} and it subsumes at least parts of (derived) [[noncommutative geometry]]. Many other phenomena are naturally part of higher geometry, see the list of \hyperlink{Examples}{Examples} below.

In any given instance of higher geometry, one starts with a notion of ``local models'' for the geometry. An \emph{affine space} will then be a formal dual of such a local model, and a general [[space]] will be formed by ``gluing'' these affine spaces in some appropriate way. There are two ways of formalizing this idea, coming from [[Alexander Grothendieck]]`s two definitions of [[scheme]] in [[algebraic geometry]] via [[locally ringed spaces]] and [[functors of points]]. Both are built on [[(∞,1)-topos theory]]: in one direction, a [[petit topos|petit]] [[(∞,1)-topos]] (with some additional [[structured (∞,1)-topos|structure]]) encodes a [[space]] itself; in another direction, a [[space]] is an object of a [[gros topos|gros]] [[(∞,1)-topos]] of [[∞-stacks]] on some [[(∞,1)-site]]. We discuss these axiomatizations below in \emph{\hyperlink{Formalizations}{Formalization}}.

These approaches do not apply to [[noncommutative algebraic geometry]], which requires a different approach to deal with a more complicated notion of gluing; we discuss this below in \emph{\hyperlink{NoncommutativeAlgebraicGeometry}{Noncommutative algebraic geometry}}.

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

We discuss two different (but closely related) formalizations of these ideas.

In

\begin{itemize}%
\item \emph{\hyperlink{GrosHigherToposes}{Gros (∞,1)-topos}}

\end{itemize}
we discuss the approach of considering one big [[(∞,1)-topos]] $\mathbf{H}$ (with ``big''/[[gros topos|gros]] being formalized for instance by [[cohesion]]) such that (some of) its objects are to be regarded as higher geometric spaces.

In

\begin{itemize}%
\item \emph{\hyperlink{PetitHigherToposes}{Petit (∞,1)-toposes}}

\end{itemize}
we discuss the approach of encoding a would-be higher geometric space $X$ by a [[structured (∞,1)-topos]] to be thought of as the [[petit topos|petit]] [[(∞,1)-topos]] of [[(∞,1)-sheaves]] $(Sh_\infty(X), \mathcal{O}_X)$ of $X$, canonically equipped with a [[structure sheaf]] $\mathcal{O}_X$.

\hypertarget{GrosHigherToposes}{}\subsubsection*{{Gros (∞,1)-toposes}}\label{GrosHigherToposes}

Let $\mathcal{G}$ be an [[(∞,1)-site]] whose objects are to be viewed as ``local models'' or ``test spaces'' for a geometry. Within the context of this geometry, we make the following definitions:

An \textbf{affine space} is a formal dual of an object of $\mathcal{G}$, so that the (∞,1)-category of affine spaces is the [[opposite (∞,1)-category|opposite]] of $\mathcal{G}$. A \textbf{stack} is an [[∞-stack]] on $\mathcal{G}$, so that the (∞,1)-category of stacks is the [[gros topos|gros]] [[(∞,1)-sheaf (∞,1)-topos]] on $\mathcal{G}$. Finally, a \textbf{space} is a [[stack]] $X$ that has a cover by a family of affine spaces $(f_i : U_i \to X)_i$, where each $f_i$ belongs to some nice class of morphisms (e.g. [[open immersions]], [[etale morphisms]] or [[smooth morphisms]]).

When the underlying (∞,1)-category of $\mathcal{G}$ is the (∞,1)-category of [[commutative algebras in a symmetric monoidal (∞,1)-category]], this is known as \emph{[[homotopical algebraic geometry]]}. When it is the (∞,1)-category of [[algebras over a Lawvere theory]], this is discussed at [[derived geometry]].

Following [[Bill Lawvere]], one may ask for a set of [[axioms]] on the [[(∞,1)-sheaf (∞,1)-topos]] $Sh_\infty(\mathcal{G})$ that ensure that it is appropriate to view [[(∞,1)-sheaves]] on $\mathcal{G}$ as generalized geometric spaces. One such set of axioms is \emph{[[cohesion]]}.

\hypertarget{PetitHigherToposes}{}\subsubsection*{{Petit (∞,1)-toposes}}\label{PetitHigherToposes}

As above, let $\mathcal{G}$ be an [[(∞,1)-category]] whose objects will be viewed as ``local models'' for the kind of geometry to be developed. Following [[Jacob Lurie]] (based on the theory of geometry via [[ringed toposes]] by [[Alexander Grothendieck]] and [[Monique Hakim]]), a $\mathcal{G}$-[[structured (∞,1)-topos]] is the data of an [[(∞,1)-topos]] together with a [[structure sheaf]] valued in $\mathcal{G}$. Given an appropriate choice of $\mathcal{G}$, one gets the following hierarchy of generalized spaces this way:

\begin{itemize}%
\item [[geometry (for structured (∞,1)-toposes)|test spaces]] $\hookrightarrow$ [[generalized scheme|spaces locally equivalent to test spaces]] $\hookrightarrow$ [[structured (∞,1)-topos|concrete spaces with structure sheaves taking values in test spaces]] $\hookrightarrow$ [[∞-stack|spaces probeable by test spaces]].

\end{itemize}
technically modeled by:

\begin{itemize}%
\item [[geometry (for structured (∞,1)-toposes)]] $\mathcal{G}$ $\hookrightarrow$

[[generalized scheme]]s $\hookrightarrow$ formal duals to $\mathcal{G}$-[[structured (∞,1)-topos]]es $\hookrightarrow$ [[(∞,1)-topos]] of [[∞-stack]]s on $\mathcal{G}$.



\end{itemize}
A plethora of proposals for formalizations of higher geometry find their home in this pattern, for instance most of the concepts listed at [[generalized smooth space]].

\hypertarget{relation_between_these_approaches}{}\subsection*{{Relation between these approaches}}\label{relation_between_these_approaches}

Given a gros [[cohesive (∞,1)-topos]] $\mathbf{H}$ and an object $X \in \mathbf{H}$, one may in turn assign to $X$ a petit [[structured (∞,1)-topos]] $Sh_{\mathbf{H}}(X)$ of [[internal sheaves]] over $X$. See at \emph{[[differential cohesion]]} for how this works. This connects the ``gros'' perspective back to the ``petit'' perspective.

Conversely, given a [[structured (∞,1)-topos]] one may consider its associated [[functor of points]], which will be an object in the [[gros topos|gros]] [[(∞,1)-topos]].

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

\hypertarget{list_of_examples}{}\subsubsection*{{List of examples}}\label{list_of_examples}

\begin{itemize}%
\item [[homotopical algebraic geometry]]

\begin{itemize}%
\item [[derived algebraic geometry]]


\item [[étale (∞,1)-site]], [[dg-geometry]], [[Hochschild cohomology]] of [[dg-algebra]]s


\item [[schematic homotopy type]]



\end{itemize}

\item [[derived noncommutative geometry]]

\begin{itemize}%
\item [[noncommutative geometry]]

\end{itemize}

\item [[higher differential geometry]]

\begin{itemize}%
\item [[motivation for higher differential geometry]]


\item [[differential geometry]], [[differential topology]]


\item [[derived smooth manifold]]


\item [[smooth ∞-groupoid]], [[∞-Lie algebroid]]



\end{itemize}

\item [[higher symplectic geometry]]


\item [[higher complex analytic geometry]]


\item [[higher prequantum geometry]]


\item [[higher Klein geometry]]


\item [[higher Cartan geometry]]



\end{itemize}
\hypertarget{NoncommutativeAlgebraicGeometry}{}\subsubsection*{{Noncommutative algebraic geometry}}\label{NoncommutativeAlgebraicGeometry}

The above frameworks for higher geometry are not suitable for describing [[noncommutative algebraic geometry]], because of the more complicated notions of [[localization]], gluing and [[descent]] in the latter setting. Indeed, noncommutative spaces are supposed to be obtained from affine ones (formal duals of [[associative algebras]] or [[dg-algebras]]) by gluing along \emph{[[bimodules]]}. A good setting for such gluing is that of [[pretriangulated dg-categories]] (or [[stable (∞,1)-categories]]). Thus in [[derived noncommutative algebraic geometry]], a noncommutative space is defined to be a [[stable (∞,1)-category]].

\hypertarget{ConnesStyleNoncommutativeGeometry}{}\subsubsection*{{Connes-style noncommutative geometry}}\label{ConnesStyleNoncommutativeGeometry}

The process of forming [[groupoid convolution algebras]] is a [[2-functor]] from suitable [[topological stack|topological]] and [[differentiable stacks]] to [[C\emph{-algebras]] with [[Hilbert bimodules]] between them. Much of [[Connes]]-style [[noncommutative geometry]] turns out to deal with objects in the image of this functor, and to the extent that it does, Connes-style noncommutative geometry may be regarded as being a way of speaking about higher geometry, specifically the [[higher differential geometry]] of [[differentiable stacks]].}

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

[[!include Isbell duality - table]]

For relation to [[physics]] see

\begin{itemize}%
\item [[higher category theory and physics]]


\item [[geometry of physics]]



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

In

\begin{itemize}%
\item [[Bill Lawvere]], \emph{Axiomatic cohesion} Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41--49. (\href{http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf}{pdf})

\end{itemize}
Both approaches to higher geometry are described, in the special case of [[derived algebraic geometry]], in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Derived Algebraic Geometry]]}, Ph.D. thesis.

\end{itemize}
The gros topos approach is described, in the case of [[homotopical algebraic geometry]], in

\begin{itemize}%
\item [[Bertrand Toën]], [[Gabriele Vezzosi]], \emph{Homotopical algebraic geometry II: geometric stacks and applications}, 2004, \href{http://arxiv.org/abs/math/0404373}{arXiv:math/0404373}.

\end{itemize}
A general exposition of the petit topos approach is proposed in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Structured Spaces]]} .

\end{itemize}
In

\begin{itemize}%
\item [[Bill Lawvere]], \emph{Axiomatic cohesion} Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41--49. (\href{http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf}{pdf})

\end{itemize}
an axiomatization of generalized geometry is proposed in terms of 1-[[category theory]]. The evident generalization of this to [[(∞,1)-category theory]] provides an axiomatization for higher geometry. This is discussed at

\begin{itemize}%
\item [[cohesive (∞,1)-topos]].

\end{itemize}
[[!redirects higher geometries]]



\end{document}