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\begin{document}

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\section*{Tannaka duality for geometric stacks}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\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{setup}{Setup}\dotfill \pageref*{setup} \linebreak
\noindent\hyperlink{ringed_toposes}{Ringed toposes}\dotfill \pageref*{ringed_toposes} \linebreak
\noindent\hyperlink{abelian_tensor_categories}{Abelian tensor categories}\dotfill \pageref*{abelian_tensor_categories} \linebreak
\noindent\hyperlink{GeometricStack}{Geometric stacks}\dotfill \pageref*{GeometricStack} \linebreak
\noindent\hyperlink{tannaka_duality_for_geometric_stacks}{Tannaka duality for geometric stacks}\dotfill \pageref*{tannaka_duality_for_geometric_stacks} \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}

Under mild conditions, a given [[site]] $C \subset T Alg^{op}$ of formal duals of [[algebra over a Lawvere theory|algebras over an algebraic theory]] admits [[Isbell duality]] exhibited by an [[adjunction]]

\begin{displaymath}
(\mathcal{O} \dashv Spec) : (T Alg^{\Delta})^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}}
  Sh_{(\infty,1)}(C)
\end{displaymath}
as described at [[function algebras on ∞-stacks]] Here $\mathcal{O}(X)$ is an $(\infty,1)$-algebra of functions on $X$.

This entry describes for certain [[algebraic stacks]] an analog of this situation where the 1-algebras are replaced by 2-algebras in the form of commutative algebra objects in the [[2-category]] of [[abelian categories]]: abelian [[symmetric monoidal categories]], and where the function algebras $\mathcal{O}(X)$ are replaced with category $QC(X)$ of [[quasicoherent sheaves]].

The replacement of the 1-algebra $\mathcal{O}(X)$ by the 2-algebra $QC(X)$ is the starting point for what is called [[derived noncommutative geometry]].

\hypertarget{setup}{}\subsection*{{Setup}}\label{setup}

\hypertarget{ringed_toposes}{}\subsubsection*{{Ringed toposes}}\label{ringed_toposes}

All [[topos]]es that we consider are [[Grothendieck topos]]es. A [[ringed topos]] $(S, \mathcal{O}_S)$ is a topos $S$ equipped with a ring object $\mathcal{O}_S$ -- a [[sheaf]] of rings -- called the [[structure sheaf]] -- on whatever [[site]] $S$ is the [[category of sheaves]] on. We write $\mathcal{O}_S Mod$ for the category of [[module]]s in $S$ (sheaves of modules) over $\mathcal{O}_S$.

We write $RngdTopos$ for the category of ringed toposes.

For $X$ a [[scheme]] or more generally an [[algebraic stack]], write $Sh(X_{et})$ for its [[little etale topos]].

\begin{udef}
A ringed topos $(S,\mathcal{O}_S)$ is a \textbf{[[locally ringed topos]]} with respect to the [[étale topology]] if for every object $U \in S$ and every family $\{Spec R_i \to Spec \mathcal{O}_S(U)\}$ of [[étale morphism]]s such that

\begin{displaymath}
\mathcal{O}_S(U) \to \prod_i R_i
\end{displaymath}
is [[faithfully flat]], there exists morphisms $E_i \to E$ in $S$ and factorizations $\mathcal{O}_S(U) \to R_i \to  \mathcal{O}_S(E_i)$ such that

\begin{displaymath}
\coprod_i E_i \to E
\end{displaymath}
is an [[epimorphism]].

\end{udef}
\begin{uprop}
If $S$ has [[point of a topos|enough points]] then $(S, \mathcal{O}_S)$ is local for the \'e{}tale topology precisely if the [[stalk]] $\mathcal{O}_S(x)$ at every [[point of a topos|point]] $x : Set \to S$ is a strictly [[Henselian ring|Henselian]] [[local ring]].

\end{uprop}
This is (\hyperlink{Lurie}{Lurie, remark 4.4}).

\begin{uprop}
\begin{itemize}%
\item The [[little étale topos]] $Sh(X_{et})$ of a [[Deligne-Mumford stack]] $X$ is locally ringed with respect to the \'e{}tale topology.

\end{itemize}
\end{uprop}
\hypertarget{abelian_tensor_categories}{}\subsubsection*{{Abelian tensor categories}}\label{abelian_tensor_categories}

\begin{udef}
An \textbf{abelian tensor category} (for the purposes of the present discusission) is a [[symmetric monoidal category]] $(C, \otimes)$ such that

\begin{itemize}%
\item $C$ is an [[abelian category]];


\item for every $x \in C$ the functor $(-) \otimes x : C\to C$ is additive and right-[[exact functor|exact]]: it commutes with finite [[colimit]]s.



\end{itemize}
A \textbf{complete abelian tensor category} is an abelian tensor category such that

\begin{itemize}%
\item it satisfies the axiom AB5 at [[additive and abelian categories]];


\item $(-) \otimes x$ commutes with \emph{all small} colimits.

(equivalently, we have a [[closed monoidal category]]).



\end{itemize}
An abelian tensor category is called \textbf{tame} if for any [[short exact sequence]]

\begin{displaymath}
0 \to M'\to M \to M''\to 0
\end{displaymath}
with $M''$ a \emph{flat object} (such that $x \mapsto x \otimes M''$ is an [[exact functor]]) and any $N \in C$ also the induced sequence

\begin{displaymath}
0 \to M'\otimes N \to M\otimes N \to M''\otimes N \to 0
\end{displaymath}
is exact.

\end{udef}
This appears as (\hyperlink{Lurie}{Lurie, def. 5.2}) together with the paragraph below remark 5.3.

\begin{udef}
For $C,D$ two \hyperlink{AbelianTensorCategory}{complete abelian tensor categories} write

\begin{displaymath}
Func_\otimes(C,D) \subset Func(C,D)
\end{displaymath}
for the [[core]] of the [[subcategory]] of the [[functor category]] on those [[functor]]s that

\begin{itemize}%
\item are [[symmetric monoidal functor]]s;


\item commute with all small [[colimit]]s (which implies they are [[additive functor|additive]] and [[exact functor|right exact]])


\item preserve flat objects and short exact sequences whose last object is flat.



\end{itemize}
Write

\begin{displaymath}
TCAbTens
\end{displaymath}
for the ([[strict 2-category|strict]]) [[(2,1)-category]] of \hyperlink{AbelianTensorCategory}{tame complete abelian tensor categories} with hom-[[groupoid]]s given by this $Func_\otimes$.

\end{udef}
This appears as (\hyperlink{Lurie}{Lurie, def 5.9}) together with the following remarks.

\begin{ulemma}
For $k$ a [[ring]], write $k Mod$ for its [[abelian category|abelian]] [[symmetric monoidal category]] of [[module]]s

Let $(S,\mathcal{O}_S)$ be a [[ringed topos]]. Then

\begin{displaymath}
\mathcal{O}_S Mod
\end{displaymath}
(the category of sheaves of $\mathcal{O}_S$-[[module]]s) is a tame \hyperlink{AbelianTensorCategory}{complete abelian tensor category}.

\end{ulemma}
This is (\hyperlink{Lurie}{Lurie, example 5.7}).

\begin{ulemma}
For $X$ an [[algebraic stack]], write

\begin{displaymath}
QC(X)
\end{displaymath}
for its category [[quasicoherent sheaves]].

This is a \hyperlink{AbelianTensorCategory}{complete abelian tensor category}

\end{ulemma}
\begin{ulemma}
If $X$ is a Noetherian geometric stack, then $QC(X)$ is the category of [[ind-object]]s of its full [[subcategory]] $Coh(X) \subset QC(X)$ of [[coherent sheaves]]

\begin{displaymath}
QC(X) \simeq Ind(Coh(X))
  \,.
\end{displaymath}
\end{ulemma}
This appears as (\hyperlink{Lurie}{Lurie, lemma 3.9}).

\hypertarget{GeometricStack}{}\subsubsection*{{Geometric stacks}}\label{GeometricStack}

\begin{udef}
A \textbf{[[geometric stack]]} is

\begin{itemize}%
\item an [[algebraic stack]] $X$ over $Spec \mathbb{Z}$


\item that is quasi-compact, in particular there is an [[epimorphism]] $Spec A \to X$;


\item with affine and [[representable morphism of stacks|representable]] diagonal $X \to X \times X$.



\end{itemize}
\end{udef}
\begin{uprop}
\begin{itemize}%
\item A quasicompact [[separated scheme]] is a geometric stack.


\item The classifying stack of a [[smooth scheme|smooth]] affine [[group object|group]] [[scheme]] is a geometric stack.



\end{itemize}
\end{uprop}
The geometricity condition on an algebraic stack implies that there are ``enough'' [[quasicoherent sheaves]] on it, as formalized by the following statement.

\begin{utheorem}
If $X$ is a \hyperlink{geometricStack}{geometric stack} then the [[bounded chain complex|bounded-below]] [[derived category]] of [[quasicoherent sheaves]] on $X$ is naturally [[equivalence of categories|equivalent]] to the full [[subcategory]] of the left-bounded derived category of smooth-[[etale site|etale]] $\mathcal{O}_X$-modules whose [[chain cohomology]] sheaves are quasicoherent.

\end{utheorem}
This is (\hyperlink{Lurie}{Lurie, theorem 3.8}).

\hypertarget{tannaka_duality_for_geometric_stacks}{}\subsection*{{Tannaka duality for geometric stacks}}\label{tannaka_duality_for_geometric_stacks}

\begin{utheorem}
Let $X$ be a \hyperlink{GeometricStack}{geometric stack}.

Then for every ring $A$ there is an [[equivalence of categories]]

\begin{displaymath}
RngdTopos(Sh((Spec A)_{et}),Sh(X_{et})) \simeq Hom_\otimes(QC(X), A Mod)
\end{displaymath}
hence (by the [[2-Yoneda lemma]])

\begin{displaymath}
X(Spec A) \simeq Hom_\otimes(QC(X), QC(Spec A))
  \,.
\end{displaymath}
More generally, for $(S, \mathcal{O}_S)$ any \hyperlink{localWRTEtaleTopology}{etale-locally} [[ringed topos]], we have

\begin{displaymath}
RngdTopos(S,Sh(X_{et})) \simeq Hom_\otimes(QC(X), \mathcal{O}_S Mod)
  \,.
\end{displaymath}
\end{utheorem}
This is (\hyperlink{Lurie}{Lurie, theorem 5.11}) in view of (\hyperlink{Lurie}{Lurie, remark 4.5}).

\begin{uremark}
It follows that forming [[quasicoherent sheaves]] constitutes a [[full and faithful (infinity,1)-functor|full and faithful (2,1)-functor]]

\begin{displaymath}
QC : GeomStacks \to TCAbTens^{op}
\end{displaymath}
from geometric stacks to \hyperlink{TCAbTens}{tame complete abelian tensor categories}.

This statement justifies thinking of $QC(X)$ as being the ``2-algebra'' of functions on $X$. This perspective is the basis for [[derived noncommutative geometry]].

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

\begin{itemize}%
\item [[analytification]]


\item [[2-algebraic geometry]]


\item [[spectrum of a tensor triangulated category]]


\item [[prime spectrum of a symmetric monoidal stable (∞,1)-category]]


\item [[Tannaka duality for Lie groupoids]]


\item [[Bondal-Orlov reconstruction theorem]]


\item [[2-ring]]



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

The above material is taken from

\begin{itemize}%
\item [[Jacob Lurie]], \emph{Tannaka duality for geometric stacks}, (\href{http://arxiv.org/abs/math/0412266}{arXiv:math.AG/0412266})

\end{itemize}
The generalization to geometric stacks in the context of [[Spectral Schemes]] is in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]}

\end{itemize}
Related discussion is in

\begin{itemize}%
\item [[Martin Brandenburg]], [[Alexandru Chirvasitu]], \emph{Tensor functors between categories of quasi-coherent sheaves} (\href{http://arxiv.org/abs/1202.5147}{arXiv:abs/1202.5147})

\end{itemize}


\end{document}