\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 algebraic geometry}

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

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

[[!include higher geometry - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{terminology}{Terminology}\dotfill \pageref*{terminology} \linebreak
\noindent\hyperlink{motivation}{Motivation}\dotfill \pageref*{motivation} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{terminology_derived_versus_}{Terminology ``derived'' versus ``$\infty$-''}\dotfill \pageref*{terminology_derived_versus_} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{structured_toposes}{structured $(\infty,1)$-toposes}\dotfill \pageref*{structured_toposes} \linebreak
\noindent\hyperlink{RelationToDerivedNoncommutativeGeometry}{Relation to derived noncommutative geometry}\dotfill \pageref*{RelationToDerivedNoncommutativeGeometry} \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{Derived algebraic geometry} is the specialization of [[higher geometry]] and [[homotopical algebraic geometry]] to the [[(infinity,1)-category]] of [[simplicial commutative rings]] (or sometimes, coconnective commutative [[dg-algebras]]). Hence it is a generalization of ordinary [[algebraic geometry]] where instead of [[commutative rings]], [[derived schemes]] are locally modelled on [[simplicial commutative rings]].

Derived algebraic geometry is the correct setting for certain problems arising in [[algebraic geometry]] that involve [[intersection theory]] and [[deformation theory]] (see below).

\hypertarget{terminology}{}\subsection*{{Terminology}}\label{terminology}

Sometimes the term \emph{derived algebraic geometry} is also used for the related subject of [[spectral algebraic geometry]], where [[commutative ring spectra]] are used instead of [[simplicial commutative rings]]. Sometimes it may also refer to the subject of [[derived noncommutative algebraic geometry]].

\hypertarget{motivation}{}\subsection*{{Motivation}}\label{motivation}

There are several motivations for the study of derived algebraic geometry.

\begin{enumerate}%
\item The [[hidden smoothness principle]] of [[Maxim Kontsevich]], which conjectures that in classical [[algebraic geometry]], the non-[[smoothness]] of certain [[moduli spaces]] is a consequence of the fact that they are in fact truncations of [[derived moduli stacks]] (which are [[smooth]]).


\item [[intersection theory|Intersection theory]]: a [[geometric]] interpretation of the [[Serre intersection formula]] for non-flat [[intersections]].


\item [[deformation theory|Deformation theory]]: a [[geometric]] interpretation of the [[cotangent complex]]. (In derived algebraic geometry, the [[cotangent complex]] $\mathbf{L}_X$ of $X$ \emph{is} its [[cotangent space]].



\end{enumerate}
For more detail on the final two points, see \hyperlink{VezzosiWhatIs}{(Vezzosi, 2011)}.

\hypertarget{history}{}\subsection*{{History}}\label{history}

The original approach to derived algebraic geometry was via [[dg-schemes]], introduced by [[Maxim Kontsevich]]. Using [[dg-schemes]], [[Ionut Ciocan-Fontanine]] and [[Mikhail Kapranov]] constructed the first [[derived moduli spaces]] (derived [[Hilbert scheme]] and derived [[Quot scheme]]).

[[Bertrand Toen]] and [[Gabriele Vezzosi]] developed [[homotopical algebraic geometry]], which is [[algebraic geometry]] in any [[HAG context]], i.e. over any [[symmetric monoidal model category]] satisfying certain assumptions. As special cases they recover the [[algebraic geometry]] of [[Grothendieck]] and the [[higher stacks]] of [[Carlos Simpson]], and also develop new theories of [[derived algebraic geometry]], [[complicial algebraic geometry]], and [[brave new algebraic geometry]].

In his thesis [[Jacob Lurie]] also developed fundamentals of derived algebraic geometry, using the language of [[structured (infinity,1)-toposes]] where [[Bertrand Toen|Toen]]-[[Gabriele Vezzosi|Vezzosi]] used [[model toposes]]. He also developed a version of derived algebraic geometry which is locally modelled on [[E-∞ rings]], called [[spectral algebraic geometry]].

\hypertarget{terminology_derived_versus_}{}\subsection*{{Terminology ``derived'' versus ``$\infty$-''}}\label{terminology_derived_versus_}

The adjective ``derived'' means pretty much the same as the ``$\infty$-'' in [[∞-category]], so this is higher algebraic geometry in the sense being locally represented by higher algebras. The word stems from the use of ``derived'' as in [[derived functor]]. This came from the study of derived moduli problem. Namely to parametrize the moduli, one first looks at some space of ``cochains'' which are candidates for structures to parametrize. then one cuts those which indeed satisfy the axioms (``equations'') for the structures. Satisfying these equations is a limit-type construction, hence left exact and one is lead to right derived functors to improve; exactness on the right; this leads to use cochain complexes. The obtained moduli is too big as there are many isomorphic structures, so one needs to quotient by the automorphisms; this is a colimit type construction hence right exact. The improved quotient is the left derived functor, what is obtained by passing to stacks.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{structured_toposes}{}\subsubsection*{{structured $(\infty,1)$-toposes}}\label{structured_toposes}

In \hyperlink{LurieStructured}{Lurie, Structured spaces} a definition of [[derived algebraic scheme]] and [[derived Deligne-Mumford stack]] is given in the wider context of [[generalized scheme]]s realized as locally affine [[structured (∞,1)-topos]]es.

See these links for more details.

\begin{remark}
\label{}\hypertarget{}{}
This definition is based on the observation that it is a deficiency of the ordinary definition of [[scheme]] to demand that underlying a scheme is a [[topological space]] and that a better definition is obtained by demanding it to have an underlying [[locale]]. But a [[locale]] is a [[0-topos]]. This motivates then the definition of a [[generalized scheme]] as a (locally affine, [[structured (infinity,1)-topos|structured]]) [[(∞,1)-topos]].

\end{remark}
\hypertarget{RelationToDerivedNoncommutativeGeometry}{}\subsection*{{Relation to derived noncommutative geometry}}\label{RelationToDerivedNoncommutativeGeometry}

Under some conditions, [[derived schemes]] $X$ in the sense of ([[Structured Spaces|Lurie, Structured Spaces]]) are faithfully encoded by their [[stable (∞,1)-categories]] $QCoh(X)$ of [[quasicoherent sheaves]]. This is the content of [[Tannaka duality for geometric stacks]], ([[Quasi-Coherent Sheaves and Tannaka Duality Theorems|Lurie, Quasi-Coherent Sheaves and Tannaka Duality Theorems]]). Therefore one can turn this around and declare that a suitable [[stable (∞,1)-category]] $\mathcal{A}$ which is not of the form $QCoh(X)$ for an actual [[derived scheme]] $X$ represents a generalized, ``noncommutative'' derived scheme. This is much like a [[2-category theory]] or rather [[(∞,2)-category]] theory analog of how in [[algebraic geometry]] the opposite of the category of [[monoids]] (algebras) is regarded as a category of generalized spaces. This might be (and has been) called \emph{[[2-algebraic geometry]]}.

Accordingly, one can decide to regard the [[opposite (∞,1)-category]] of suitable (e.g. [[monoidal (∞,1)-category|monoidal]]) [[stable (∞,1)-categories]] as being a category of ``noncommutative derived schemes''. This is effectively the perspective on [[noncommutative algebraic geometry]] that [[Maxim Kontsevich]] has been promoting.

Often and traditionally, all this is expressed in terms of certain presentations for these [[stable (∞,1)-categories]] by [[triangulated category|triangulated]] [[derived categories]] or better, [[dg-enhancement|enhancements]] as [[dg-categories]].

In this fashion then in [[derived noncommutative algebraic geometry]], a [[space]] is by definition a [[dg-category]] that is smooth and proper in an appropriate sense. The relation between [[noncommutative algebraic geometry]] and [[derived algebraic geometry]] may then be summed up by the adjunction

\begin{displaymath}
Pf : \DSt(k)^{op} \rightleftarrows NCSp(k) : \mathcal{M}_-
\end{displaymath}
where $Pf(X)$ denotes the [[dg-category]] of [[perfect complexes]] on the [[derived stack]] $X$, and $\mathcal{M}_\mathcal{C}$ denotes the [[derived moduli stack of objects in a dg-category|derived moduli stack of objects]] in the [[dg-category]] $\mathcal{C}$. See [[derived moduli stack of objects in a dg-category]] for details.

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

\begin{itemize}%
\item [[homotopical algebraic geometry]]
\item [[E-∞ geometry]]
\item [[dg-geometry]]
\item [[spectral algebraic geometry]]
\item [[brave new algebraic geometry]]
\item [[higher stacks]]
\item [[higher differential geometry]]
\item [[derived category of coherent sheaves]]
\item [[derived noncommutative algebraic geometry]]

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

(For references on [[dg-schemes]], the historical precursor to [[derived schemes]], see there.)

The main references are

\begin{itemize}%
\item [[Jacob Lurie]], [[Derived Algebraic Geometry]]


\item [[Jacob Lurie]], \emph{[[structured (∞,1)-topos|Structured Spaces]]}



\end{itemize}
\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}.


\item [[Bertrand Toën]], [[Gabriele Vezzosi]], \emph{From HAG to DAG: derived moduli stacks}, in Axiomatic, enriched and motivic homotopy theory, 173--216, NATO Sci. Ser. II Math. Phys. Chem., 131, Kluwer Acad. Publ., Dordrecht, 2004, \href{http://arxiv.org/abs/math/0210407}{math.AG/0210407}.



\end{itemize}
The following notes deal with the theory modelled on coconnective [[commutative dg-algebras]].

\begin{itemize}%
\item [[Dennis Gaitsgory]], \emph{Notes on geometric Langlands: stacks}, \href{http://www.math.harvard.edu/~gaitsgde/GL/Stackstext.pdf}{pdf}.

\end{itemize}
The following notes deal with the theory modelled on [[E-infinity ring spectra]] ([[E-infinity geometry]]):

\begin{itemize}%
\item [[Clark Barwick]], \emph{Applications of derived algebraic geometry to homotopy theory}, lecture notes, mini-course in Salamanca, 2009, \href{https://dl.dropboxusercontent.com/u/1741495/papers/salamanca.pdf}{pdf}.

\end{itemize}
See also the surveys

\begin{itemize}%
\item [[Bertrand Toën]], \emph{Higher and derived stacks: a global overview}, lecture notes, 2006, \href{http://arxiv.org/abs/math/0604504}{arXiv:math/0604504}.


\item [[Gabriele Vezzosi]], \emph{What is a derived stack?}, 2011, \href{http://www.ams.org/notices/201107/rtx110700955p.pdf}{pdf}.



\end{itemize}
\begin{itemize}%
\item [[Bertrand Toën]], \emph{Derived algebraic geometry}, \href{http://arxiv.org/abs/1401.1044}{arxiv/1401.1044}

\end{itemize}
Discussion of [[derived noncommutative algebra]] over [[E-n algebras]] is in

\begin{itemize}%
\item [[John Francis]], \emph{Derived algebraic geometry over $\mathcal{E}_n$-Rings} (\href{http://math.northwestern.edu/~jnkf/writ/thezrev.pdf}{pdf})


\item [[John Francis]], \emph{The tangent complex and Hochschild cohomology of $\mathcal{E}_n$-rings} (\href{http://math.northwestern.edu/~jnkf/writ/cotangentcomplex.pdf}{pdf})



\end{itemize}


\end{document}