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\section*{derived stack}

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

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

[[!include higher geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{central_motivation_derived_stacks_have_good_limits}{Central motivation: derived stacks have good limits}\dotfill \pageref*{central_motivation_derived_stacks_have_good_limits} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{derived_yoneda_embedding}{Derived Yoneda embedding}\dotfill \pageref*{derived_yoneda_embedding} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{derived stack} $X$ is an [[∞-stack]] -- an [[(∞,1)-sheaf]] -- on an [[(∞,1)-site]] $C$. For example, $C$ is most often the [[(∞,1)-category]] of [[simplicial commutative rings]] or [[E-∞ ring spectra]] (with the [[Zariski topology|Zariski]] or [[etale topology|etale]] topology)).

One says \emph{derived stack} in order to distinguish from the more restrictive notion of an [[∞-stack]] on a 1-categorical [[site]], such as for instance described at [[topological ∞-groupoid]].

For recall that a [[sheaf]] is a [[functor]] $F : C^{op} \to Set$ satisfying some [[descent]]-condition. So there are two steps in which the notion of [[sheaf]] may be [[vertical categorification|categorified]]:

\begin{enumerate}%
\item the codomain is categorified and the domain remains a 1-category
\item the codomain and the domain are categorified.

\end{enumerate}
The categorification of the codomain leads to the notion of [[stack]]s when [[Set]] replaced by [[Grpd]], and further to [[∞-stack]]s, when sets are replaced by [[∞-groupoid]]s.

But there is no natural reason why the domain should in general remain a 1-category if one passes to an [[infinity-category|∞-categorical]]-context. A \emph{derived stack} is a generalization of the notion of sheaf where both domain and codomain are taken to be $\infty$-categorical.

Following the general logic of [[models for ∞-stack (∞,1)-toposes]], derived stacks are typically [[model category|modeled]] by the [[model structure on SSet-enriched presheaves]] on an [[SSet-site]] or [[model site]] $C$. In such a model a derived stack is represented by an [[SSet]]-[[enriched functor]] $F : C^{op} \to SSet$ from an [[SSet]]-[[enriched category]] $C$ to [[SSet]] that satisfies a [[descent]] condition.

\hypertarget{central_motivation_derived_stacks_have_good_limits}{}\subsection*{{Central motivation: derived stacks have good limits}}\label{central_motivation_derived_stacks_have_good_limits}

One general idea for the use of higher and derived stacks is that

\begin{itemize}%
\item passing to a higher categorical codomain -- i.e. from Set-values [[sheaf|sheaves]] to higher [[groupoid]] valued sheaves -- is a means to obtain \emph{good colimits}, [[colimit]]s that do not lose information. For instance

\begin{itemize}%
\item in the category [[Diff]] of manifolds the quotient by a non-free action of a group may not exist


\item in [[sheaf|sheaves]] in $[Diff^{op},Set]$ it will exist, but will have the wrong properties in general with respect to some operations such as taking cohommology,


\item while finally in [[stack]]s $[Diff^{op}, Grpd]$ it exists as the corresponding smooth [[action groupoid]] or [[orbifold]] and in this form rembers in terms of the isomorphisms how the quotient was obtained. The cohomology of the stack is then indeed the equivariant cohomology of the original manifold.



\end{itemize}

\item similarly passing to higher categorical domain -- i.e. from presheaves on categories to presheaves on higher categories, is analogously a means to ensure that good [[limit]]s exist.



\end{itemize}
A detailed illustration and motivation of the need of these ``good limits'' that don't forget the way they were formed is

\begin{itemize}%
\item in the introductory section of

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

\end{itemize}

\item in the introductory section of

\begin{itemize}%
\item David Spivak, \emph{\href{http://math.berkeley.edu/~dspivak/thesis2.pdf}{Quasi-smooth derived manifolds}}

\end{itemize}


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

\begin{itemize}%
\item A derived refinement of the ordinary [[site]] [[CRing]]${}^{op}$ of formal duals to commutative tings is the [[SSet-site]] $SCRing^{op}$ of [[simplicial ring]]s. Derived stacks on this site are studied in [[derived algebraic geometry]].

An introduction to this is for instance in \href{http://www.crm.cat/HigherCategories/hc1.pdf#page=135}{chapter 5} of the lecture notes

\begin{itemize}%
\item [[Bertrand Toen]], \emph{Simplicial presheaves and derived algebraic geometry} course on \href{http://www.crm.cat/HigherCategories/}{Simplicial Methods in Higher Categories} (\href{http://www.crm.cat/HigherCategories/hc1.pdf#page=99}{pdf})

\end{itemize}
There are various slight variations of this. For instance using the [[monoidal Dold-Kan correspondence]], simplicial rings may be replaced with non-positively graded cochain [[dg-algebra]]s.

One application of derived stacks on $(dgAlg^-)^{op}$ is the [[BV-BRST formalism]] in [[physics]].


\item A derived refinement of the ordinary [[site]] $\mathbb{L}$ of [[smooth loci]] is the [[SSet-site]] $cs \mathbb{L}$ of [[cosimplicial object|cosimplicial]] [[smooth loci]]. Derived stacks on this are the objects in a theory that could be called [[derived synthetic differential geometry]].



\end{itemize}
\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\hypertarget{derived_yoneda_embedding}{}\subsubsection*{{Derived Yoneda embedding}}\label{derived_yoneda_embedding}

One obvious but notable phenomenon that occurs in derived stacks in general, but not in [[∞-stack]]s over a 1-categorical site, is that under the [[Yoneda lemma for (∞,1)-categories|Yoneda embedding for (∞,1)-categories]] a categorically discrete object, i,e. a [[n-truncated object in an (∞,1)-category|0-truncated object]] may be mapped to a higher categorical object.

Consider specifically an ordinary [[site]] $C$ and let $csC = [\Delta,C]$ be the corresponding [[SSet-site]] of [[cosimplicial object]]s. Write

\begin{displaymath}
\mathbf{Y} : C \hookrightarrow [\Delta,C] \stackrel{\mathbb{R}Y}{\to}
  [[Delta,C],SSet]
\end{displaymath}
for the composite that first regards objects of $C$ as constant cosimplicial objects and then applies the [[derived functor]] of the [[enriched category theory|enriched]] [[Yoneda embedding]], i.e. the functor

\begin{displaymath}
\mathbb{R}Y = Y(Q(-), P(-))
\end{displaymath}
for $Q$ a fibrant and $P$ a cofibrant replacement functor. Then of course the simplicial presheaf $\mathbf{Y}(U)$ for $U \in C$ may in general take values in [[simplicial set]]s with nontrivial higher [[simplicial homotopy group]]s, to the extent that the [[SSet]]-[[hom-object]]s $\mathbf{Y}(U) : V \mapsto [\Delta,C](V,U)$ in $[\Delta,C]$ are nontrivial.

This cannot happen for [[∞-stack]]s over 1-categorical site. For instance a [[topological space]] regarded as a [[topological ∞-groupoid]] is always an [[n-truncated object of an (∞,1)-category|0-truncated object]] as an [[∞-stack]].

A notable example of this is the case where $C =$ [[CRing]] and $U = Spec R$. While ordinarily this is 0-categorical, when regarded as a derived stack on formal duals of [[simplicial ring]]s this has in general a nontrivial [[free loop space object]], i.e. a nontrivial [[homotopy pullback]] of the form

\begin{displaymath}
\itexarray{
    \Spec \Omega^\bullet_K(R) &\to& Spec R
    \\
    \downarrow && \downarrow^{\mathrlap{(Id,Id)}}
    \\
    Spec R &\stackrel{(Id,Id)}{\to}& Spec R \times Spec R
  }
  \,,
\end{displaymath}
where, as indicated, the free loop space object is something like the [[de Rham space]] of $Spec R$. This means that regarded as a derived stack, the space $Spec R$ becomes an [[∞-groupoid]] whose morphisms are given by [[infinitesimal object|infinitesimal paths]] in the orighinal space.

By the $\infty$-erspective on [[Hochschild cohomology]] (as discussed there) this implies a bunch of nice relations. Details are in

\begin{itemize}%
\item [[Bertrand Toen]] [[Gabriele Vezzosi]], \emph{$S^1$-Equivariant simplicial algebras and de Rham theory} (\href{http://arxiv.org/abs/0904.3256}{arXiv:0904.3256})

\end{itemize}
The fact is also mentioned and used in passing every now and then (e.g. p. 9) in

\begin{itemize}%
\item [[David Ben-Zvi]], [[David Nadler]], [[John Francis]], \emph{[[geometric infinity-function theory|Integral transforms etc.]]}

\end{itemize}
But there must be a better reference, somewhere.

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

An overview is provided in

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

\end{itemize}
A set of lecture notes on the [[model structure on simplicial presheaves]] with an eye towrads algebraic sites and derived algebraic geometry is

\begin{itemize}%
\item [[Bertrand Toën]], \emph{Simplicial presheaves and derived algebraic geometry} , lecture at \href{http://www.crm.es/HigherCategories/}{Simplicial methofs in higher categories} (\href{http://www.crm.cat/HigherCategories/hc1.pdf}{pdf})

\end{itemize}
Details modeled on [[simplicially enriched category|simplicial categories]] have been developed in the series of articles by To\"e{}n and Vezossi.

This article generalizes the notion of [[site]] and [[model structure on simplicial presheaves]] from 1-categorical sites to [[simplicially enriched category|simplicial sites]] and hence to a [[model structure on SSet-enriched presheaves]]:

\begin{itemize}%
\item [[Bertrand Toën]], [[Gabriele Vezzosi]], \emph{Homotopical Algebraic Geometry I: Topos theory} (\href{http://arxiv.org/abs/math.AG/0207028}{arXiv})

\end{itemize}
Of central interest in derived algebraic geometry is the simplicial site of \emph{simplicial algebras}, which generalizes the familiar site of algebra used in algebraic geometry. This is introduced and studied in

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

\end{itemize}
Further developments in this direction are in

\begin{itemize}%
\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.AG/0210407}{arXiv})

\end{itemize}
The unifying picture, in particular independent of the choice of model for the [[(infinity,1)-category|(infinity,1)-categories]] is presented in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
Derived ($\infty$-)stacks are currently mostly, maybe exclusively, studied on algebraic sites $S$, where the category $S^{op} :=$ [[Alg]] is replaced with a category of ``$\infty$-algebras'' of sorts. The theory of these $\infty$-algebras is described in great detail in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[higher algebra|Noncommutative and Commutative algebra]]}

\end{itemize}
Concretely the need for the site of simplicial ring objects is discussed in the introduction of

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

\end{itemize}
and in the introduction of

\begin{itemize}%
\item [[David Spivak]], \emph{\href{http://math.berkeley.edu/~dspivak/thesis2.pdf}{Quasi-smooth derived manifolds}} .

\end{itemize}
The proof that [[simplicial object|simplicial]] [[algebra]]s are Quillen equivalent of [[differential graded algebra]]s -- so that derived stacks on simplicial algebras are the same as derived stacks on DGAs -- is in

\begin{itemize}%
\item [[Stefan Schwede]], [[Brooke Shipley]], \emph{Equivalences of monoidal model categories} , Algebr. Geom. Topol. 3 (2003), 287--334 (\href{http://arxiv.org/abs/math.AT/0209342}{arXiv}) .

\end{itemize}
[[!redirects derived stacks]] [[!redirects derived infinity-stack]] [[!redirects derived infinity-stacks]] [[!redirects derived ∞-stack]] [[!redirects derived ∞-stacks]]



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