\chapter{Algebroid stacks}
\selectlanguage{english}

The aim of this appendix is to present some notions of the theory of algebroid stacks. They have been introduced by Kontsevich in \cite{Kos}. They are used in the deformation quantization of complex Poisson varieties. We assume that the reader has some familiarities with the language of stacks. Our presentation follows closely \cite[§2.1]{KS3} and \cite[chapter  19]{categories_and_sheaves}.

In this appendix $k$ is a commutative unital ring and $X$ is a topological space.


\section{Algebroid stacks}


Stacks are roughly speaking sheaves of categories. In some sense algebroid stacks are the counterpart in the language of stacks of the notion of sheaves of rings. 

\begin{defi}
A $k$-linear category is a category $\sC$ such that $\Hom_{\sC}(X,Y)$ is endowed with a $k$-module structure for any $X$, $Y \in \Ob(\sC)$ and such that the composition map $\Hom_{\sC}(X,Y) \times \Hom_{\sC}(Y,Z) \to \Hom_{\sC}(X,Z)$ is a $k$-bilinear map for any $X$, $Y$, $Z \in \sC$. 
\end{defi}

\begin{defi}
\begin{enumerate}[(i)]
\item A $k$-linear stack $\mathfrak{S}$ on $X$ is a stack such that for every open subset $U$ of $X$, $\mathfrak{S}(U)$ is a $k$-linear category.

\item Let $\mathcal{R}$ be a sheaf of commutative ring on $X$. An $\mathcal{R}$-linear stack, is a $\Z$-linear stack $\mathfrak{S}$ such that the sheaf $\mathcal{E}nd(\id_\mathfrak{S})$ of endomorphism of the identity functor $\id_\mathfrak{S}$ from $\mathfrak{S}$ to itself, is a sheaf of commutative $\mathcal{R}$-algebras.
\end{enumerate}
\end{defi}


\begin{defi}
\begin{enumerate}[(i)]
\item A $k$-algebroid stack $\cA$ on $X$ is a $k$-linear stack which is locally non-empty (i.e there exists a covering of X by open subsets $\lbrace U_i\rbrace_{i \in I}$ such that for every $i \in I$, the category $\cA(U_i)$ is non-empty) and such that for any open subset $U$ of $X$, two objects of $\cA(U)$ are locally isomorphic.

\item An $\mathcal{R}$-algebroid stack $\cA$ on $X$ is a $\Z$-algebroid stack endowed with a morphism of rings $\mathcal{R} \to \mathcal{E}nd(\id_\mathfrak{S})$.

\item An $\mathcal{R}$-algebroid $\cA$ is called an invertible $\mathcal{R}$-algebroid if $\mathcal{R}|_U \to \mathcal{E} nd_{\cA}(\sigma)$ is an isomorphism for any open subset $U$ of $X$ and any $\sigma \in \cA(U)$.
\end{enumerate}
\end{defi}

\begin{example}
If $A$ is a $k$-algebra, we denote by $A^+$ the $k$-linear category with one object and having $A$ as the endomorphism ring of this object. Let $\cA$ be a sheaf of algebras on $X$ and consider the prestack which associates to an open set $U$ the category $\cA(U)^+$. The associated stack to this prestack, denoted $\cA^{\dagger}$, is an algebroid stack.
\end{example}

For two $k$-linear category $\sC$ and $\sC^\prime$ one defines their tensor product $\sC \otimes \sC^\prime$ by setting $\Ob(\sC \otimes \sC^\prime)=\Ob(\sC) \times \Ob(\sC^\prime)$ and 
\begin{equation*}
\Hom_{\sC \otimes \sC^\prime}((M,M^\prime),(N,N^\prime))=\Hom_{\sC}(M,N) \otimes \Hom_{\sC^\prime}(M^\prime, N^\prime).
\end{equation*}


\begin{Rem}
The tensor product of two algebroid stacks is well-defined. If $\cA$ and $\cA^\prime$ are two algebroid stacks their tensor product is the $k$-linear stack associated with the prestack $U \mapsto \cA(U) \otimes \cA^\prime(U)$ where $U$ is an open subset of $X$.
\end{Rem}


An algebroid stacks can be described by what is called gluing datum. To define them, we need to introduce the notion of bi-invertible-module.

\begin{defi}
Let $\mathcal{R}$ and $\mathcal{R}^\prime$ be two sheaves of $k$-algebras. An $\mathcal{R} \otimes \mathcal{R}^\prime$-module $\mathcal{L}$ is called bi-invertible if there exists locally a section $w$ of $\mathcal{L}$ such that the morphisms $\mathcal{R} \ni r \mapsto (r \otimes 1) w \in \mathcal{L}$ and $\mathcal{R}^\prime \ni r^\prime \mapsto (1 \otimes r^\prime) w \in \mathcal{L}$ give isomorphisms of $\mathcal{R}$-modules and $\mathcal{R}^\prime$-modules, respectively.
\end{defi}

Let $\lbrace U_i\rbrace_{i \in I}$ be an open covering of $X$, we set $U_{ij}=U_i \cap U_j$, $U_{ijk}=U_i \cap U_j \cap U_k$, etc. 

\begin{defi}
An algebroid datum on $\mathcal{U}=\lbrace U_i \rbrace_{i \in I}$ is the data of
\begin{enumerate}[$\bullet$]
\item a $k$-algebroid $\cA$ on $X$,
\item for every $U_i \in \mathcal{U}$, $\sigma_i \in \cA(U_i)$,
\item for every $U_{ij}$, an isomorphism $\phi_{ij}: \sigma_j|_{U_{ij}} \stackrel{\sim}{\to} \sigma_{i}|_{U_{ij}}$.
\end{enumerate}
\end{defi}

\begin{defi}
A gluing datum on $\mathcal{U}= \lbrace U_i \rbrace_{i \in I}$ is the data of
\begin{enumerate}[$\bullet$]
\item for every $U_i \in \mathcal{U}$ a sheaf $\cA_i$ of $k$-algebras on $U_i$
\item for every $U_{ij}$ a bi-invertible $\cA_i \otimes \cA_j^{op}$-module $\mathcal{L}_{ij}$ on $U_{ij}$.
\item for every $U_{ijk}$ an isomorphism $a_{ijk}:\mathcal{L}_{ij} \otimes_{\cA_j} \mathcal{L}_{jk} \stackrel{\sim}{\to} \mathcal{L}_{ik}$ in $\Mod(\cA_i \otimes \cA_k^{op}|_{U_{ijk}})$.
\end{enumerate}
such that the diagram below in $\Mod(\cA_i \otimes \cA_k^{op}|_{U_{ijkl}})$ commutes:

\begin{equation*}
\xymatrix{\mathcal{L}_{ij} \otimes \mathcal{L}_{jk} \otimes \mathcal{L}_{kl} \ar[r]^-{a_{ijk}} \ar[d]^-{a_{jkl}} & \mathcal{L}_{ik} \otimes \mathcal{L}_{kl} \ar[d]^-{a_{ikl}}\\
\mathcal{L}_{ij} \otimes \mathcal{L}_{jl} \ar[r]^-{a_{ijl}} & \mathcal{L}_{il}
}
\end{equation*}
\end{defi}

\begin{prop}
Consider a gluing datum on $\mathcal{U}$. Then there exist an algebroid datum to which this gluing datum is associated. This algebroid datum is unique up to an equivalence of stacks, this equivalence being unique up to a unique isomorphism.
\end{prop}

\section{Module over an algebroid stack}

We denote by $k_X$ the constant sheaf with coefficient in $k$, by $\mathfrak{Mod}(k_X)$ the $k$-linear stack of sheaves of $k$-modules on $X$. If $\mathfrak{S}_1$ and $\mathfrak{S}_2$ are two $k$-linear stacks, we denote by $\Fct_k(\mathfrak{S}_1, \mathfrak{S}_2)$ the category of $k$-linear functors of stacks from $\mathfrak{S}_1$ to $\mathfrak{S}_2$. We set
%
\begin{equation*}
\Mod(\cA)=\Fct_k(\cA,\mathfrak{Mod}(k_X))
\end{equation*}
%
\begin{prop}
For a $k$-algebroid $\cA$, the $k$-linear prestack $U \mapsto \Mod(\cA|_U)$ is a stack and we denote it $\mathfrak{Mod}(\cA)$.
\end{prop}

If $\mathfrak{S}$ is a stack on $X$ we denote by $\mathfrak{S}^{op}$ the stack defined by $\mathfrak{S}^{op}(U):=\mathfrak{S}(U)^{op}$.

\begin{Rem}
An algebroid stack $\cA$ is not in general an object of $\Mod(\cA)$. But, it can be canonically identified to an object of $\Mod(\cA \otimes \cA^{op})$ via its morphisms sheaf $\fHom_{\cA}(\cdot,\cdot)$ as follows
\begin{equation*}
\cA \otimes \cA^{op} \ni (\sigma, \tau) \mapsto \fHom_{\cA}(\tau,\sigma) \in \mathfrak{Mod}(k_X).
\end{equation*}
\end{Rem}


\begin{defi}
An $\cA$-module is invertible if for any open subset $U$ of $X$ and any $\sigma \in \cA(U)$, the $\mathcal{E}nd(\sigma)$-module $\mathcal{L}(\sigma)$ is locally isomorphic to $\mathcal{E}nd(\sigma)$. We denote by $\mathrm{Inv}(\cA)$ the full subcategory of $\Mod(\cA)$ consisting of invertible $\cA$-module.
\end{defi}

\begin{prop}
If $\mathfrak{Inv}(\cA)$ denote the full substack of invertible modules of $\mathfrak{Mod}(\cA)$ then we have an equivalence of $k$-linear stacks
\begin{equation}
\cA \stackrel{\sim}{\to} \mathfrak{Inv}(\cA^{op}) \stackrel{\sim}{\to} \mathfrak{Inv}(\cA)^{op}.
\end{equation}
\end{prop}

We give a description of modules over an algebroid stack in term of local data.

Let $\cA$ be an algebroid stack on $X$. We can associate to $\cA$ a gluing datum as follows. Let $\mathcal{U}=\lbrace U_i \rbrace_{i \in I}$ be an open covering of $X$ such that for every $i \in I$ there is a $\sigma_i \in \cA(U_i)$ and choose one of these $\sigma_i$ for every $i \in I$.

To these data, we associate
\begin{enumerate}[$\bullet$]
\item $\cA_i:=\mathcal{E}nd(\sigma_i)$,
\item $\mathcal{L}_{ij}:= \fHom_{\cA_i|_{U_{ij}}}(\sigma_j|_{U_{ij}},\sigma_i|_{U_{ij}})$, (Then $\mathcal{L}_{ij}$ is a bi-invertible $\cA_i \otimes \cA_j^{op}$ module on $U_{ij}$),
\item the natural isomorphism
\begin{equation*}
a_{ijk}: \mathcal{L}_{ij} \otimes_{\cA_j} \mathcal{L}_{ij} \stackrel{\sim}{\to} \mathcal{L}_{ik} \; \mathrm{in} \; \Mod(\cA_i \otimes \cA_k^{op}|_{U_{ijk}}).
\end{equation*}
\end{enumerate} 

\begin{prop}
To specify a module over $\cA$, it is sufficient to give a family $\lbrace \cM_i,q_{ij} \rbrace_{i,j \in I}$ with $\cM_i \in \Mod(\cA_i)$ and the $q_{ij}$ are isomorphisms
\begin{equation*}
q_{ij}: \mathcal{L}_{ij} \otimes_{\cA_{ij}} \cM_j \stackrel{\sim}{\to} \cM_i
\end{equation*}
making the diagram commutative
\begin{equation*}
\xymatrix{\mathcal{L}_{ij} \otimes \mathcal{L}_{jk} \otimes \mathcal{M}_{k} \ar[r]^-{q_{jk}} \ar[d]^-{a_{ijk}} & \mathcal{L}_{ij} \otimes \mathcal{M}_{j} \ar[d]^-{q_{ij}}\\
\mathcal{L}_{ik} \otimes \mathcal{M}_{k} \ar[r]^-{q_{ik}} & \mathcal{M}_{i}.
}
\end{equation*}
\end{prop}

A similar description can be given for morphisms of $\cA$-modules.
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