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\begin{document}

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\section*{quasicoherent infinity-stack}

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

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,2)$-Topos theory}}\label{topos_theory}

[[!include (infinity,2)-topos theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{model_category_theoretic_presentation}{Model category theoretic presentation}\dotfill \pageref*{model_category_theoretic_presentation} \linebreak
\noindent\hyperlink{for_commutative_monoids}{For commutative monoids}\dotfill \pageref*{for_commutative_monoids} \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}

The notion of \emph{quasicoherent [[∞-stack]] of modules} is the anlog in [[(∞,1)-topos theory]] of the notion of [[quasicoherent sheaf]] in [[topos theory]].

A quasicoherent sheaf on a [[scheme]] $X$ is is equivalently a morphism of [[stack]]s $X \mapsto Mod$ into the canonical stack [[Mod]] $: Spec A \mapsto A Mod$ of [[module]]s, which corresponds to the [[bifibration]] $Mod \to CRing$ over the category of commutative rings/algebras: this is the [[tangent category]] of [[CRing]].

This general abstract description of quasi-coherent sheaves has a fairly direct generalization to [[(∞,1)-topos theory]] over arbitrary [[(∞,1)-site]]s:

for $C$ any [[(∞,1)-site]], the [[tangent (∞,1)-category]] $T (C^{op}) \to C^{op}$ is the [[Cartesian fibration|bifibration]] whose [[fiber]]s over an object $A \in C^{op}$ plays the role of the [[∞-groupoid]] of [[module]]s over $A$. Under the [[(∞,1)-Grothendieck construction]] this corresponds to an [[(∞,1)-presheaf]] $\infty Mod : C^{op} \to \infty Cat$

\begin{displaymath}
\infty Mod : Spec R \mapsto Stab( C^{op}/R )
\end{displaymath}
or directly in terms of test spaces $U$:

\begin{displaymath}
\infty Mod : U \mapsto Stab( U/C )
  \,.
\end{displaymath}
For the special case that $C = (sAlg^{op})^\circ$ [[opposite (∞,1)-category]] presented by the [[model structure on simplicial algebras]] over a characteristic 0 ground field we have (with example 8.6 of \emph{[[Stable ∞-Categories]]} ) this reproduces the notion of quasicoherent $\infty$-stack as considered in [[dg-geometry]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{udefn}
For $C$ an [[(∞,1)-site]] let the [[(∞,1)-functor]]

\begin{displaymath}
Mod_C : C^{op} \to (\infty,1)Cat
\end{displaymath}
be that corresponding under the [[(∞,1)-Grothendieck construction]] to the [[tangent (∞,1)-category]] $T C^{op} \to C^{op}$.

\end{udefn}
Let $\mathbf{H} := Sh_{(\infty,1)}(C)$ be the [[(∞,1)-category of (∞,1)-sheaves]] over $C$. Notice that this sits inside $[C^{op}, (\infty,1)Cat]$.

\begin{udefn}
For $X \in \mathbf{H}$, the \textbf{$(\infty,1)$-category of quasi-coherent $\infty$-stacks on $X$ is the [[(∞,1)-category of (∞,1)-functors]]}

\begin{displaymath}
QC(X) := [C^{op}, (\infty,1)Cat](X, Mod)
  \,.
\end{displaymath}
\end{udefn}
\hypertarget{model_category_theoretic_presentation}{}\subsection*{{Model category theoretic presentation}}\label{model_category_theoretic_presentation}

For [[derived geometry]] modeled on formal duals of [[algebras over an operad]], the [[model category]] presentation of quasi-coherent $\infty$-stacks is locally given by a [[model structure on modules over an algebra over an operad]].

The following considers the special case of the [[commutative operad]].

\hypertarget{for_commutative_monoids}{}\subsubsection*{{For commutative monoids}}\label{for_commutative_monoids}

Let $\mathcal{C}$ be a [[monoidal model category]]. Write $CMon(\mathcal{C})$ for the category of [[commutative monoid]]s in $\mathcal{C}$.

For instance

\begin{itemize}%
\item for $\mathcal{C} =$ [[Ab]] a monoid object in $\mathcal{C}$ is an ordinary [[ring]] and its formal dual an ordinary [[affine scheme]];


\item for $\mathcal{C} = sAb$, the category of abelian [[simplicial group]]s, a monoid in $\mathcal{C}$ is a [[simplicial ring]] and its formal dual is one notion of affine [[derived scheme]].



\end{itemize}
In (\href{To&#235;nVezzosiI}{To\"e{}nVezzosi, I}, \hyperlink{To&#235;nVezzosiII}{To\"e{}nVezzosi, II}) are discussed structures of a [[model site]]/[[sSet-site]] on $CMon(\mathcal{C})$.

For every commutative monoid $A \in \mathcal{C}$ There is naturally a [[model category]] structure on the category $A Mod$ of $A$-[[module]]s in $\mathcal{C}$.

(\ldots{})

Let $[C^{op},sSet]_{proj,loc}$ the local [[model structure on sSet-presheaves]] that [[presentable (∞,1)-category|presents]] the [[(∞,1)-category of (∞,1)-sheaves]]/[[∞-stack]]s on $C$

\begin{displaymath}
([C^{op}, sSet]_{proj,loc})^\circ \simeq \mathbf{H} := Sh_{(\infty,1)}(C)
  \,.
\end{displaymath}
as described as [[models for ∞-stack (∞,1)-toposes]].

So in this model an $\infty$-stack is a [[simplicial presheaf]] $C^{op} \to SSet$ on a simplicial site $C$ that takes values in [[Kan complex]]es and satisfies [[descent]] with respect to [[hypercover]]s in the [[homotopy category]] of $C$.

\begin{udefn}
The [[simplicial presheaf]] of \textbf{quasicoherent $\infty$-stacks} is

\begin{displaymath}
QC : C^{op} \to sSet
\end{displaymath}
is given by

\begin{displaymath}
QC : Spec A \mapsto N( A Mod_{cof, \sim} )
  \,,
\end{displaymath}
where on the right we have the [[nerve]] of the non-full [[subcategory]] of the model category $A Mod$ on cofibrant objects and weak equivalences between them.

\end{udefn}
This appears as (\hyperlink{ToenVezzosiII}{To\"e{}nVezzosiII, definition 1.3.7.1}.

\begin{uprop}
This $QC$ is indeed an [[∞-stack]] on the [[model site]] $C := Comm(\mathcal{C})^{op}$

\end{uprop}
This is (\hyperlink{ToenVezzsoi}{To\"e{}nVezzosiII, theorem 1.3.7.2}.

\begin{quote}%
notice: the $QC$ defined this way is not yet stabilized


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

\begin{itemize}%
\item [[quasicoherent sheaf]]


\item \textbf{quasicoherent $\infty$-stack}



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

The general discussion of the [[tangent (∞,1)-category]] is in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Deformation Theory]]} .

\end{itemize}
The model category theoretic presentation over model sites of commutative monoids is discussed in

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

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

\end{itemize}
A discussion specific to [[dg-geometry]] with an emphasis on the [[geometric ∞-function theory]] of quasicoherent $\infty$-stacks over [[perfect ∞-stack]]s is in

\begin{itemize}%
\item [[David Ben-Zvi|Ben-Zvi]], [[John Francis]], [[David Nadler]], \emph{Integral transforms and Drinfeld centers in derived algebraic geometry} (\href{http://arxiv.org/abs/0805.0157}{arXiv})

\end{itemize}
category: algebraic geometry [[!redirects quasicoherent ∞-stack]] [[!redirects quasicoherent ∞-stacks]] [[!redirects quasicoherent infinity-stacks]] [[!redirects quasicoherent (∞,1)-sheaf]] [[!redirects quasicoherent (∞,1)-sheaves]] [[!redirects quasicoherent (infinity,1)-sheaf]] [[!redirects quasicoherent (infinity,1)-sheaves]]



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