\section{Introduction}

\subsection{Motivation}
Topos theory studies sheaves of sets over (small) categories equipped with Grothendieck topologies, significantly formalizing and generalizing sheaves on topological spaces. It connects many theories in very different languages and hence provides a common foundation for various mathematical areas such as first order logic, model theory, algebraic geometry, topology, functional analysis, and algebra \cite{Jo}. For the interest of the authors, topos theory also provides a powerful tool for studying representations of groups and categories. An important contribution along this approach is established by Artin, who proved a fundamental result relating discrete $G$-sets of topological groups $G$ and sheaves of sets over particular categories equipped with atomic topology; see \cite{Ar}, \cite[II.1.9]{Mi} or \cite[III.9, Theorems 1 and 2]{MM}. In particular, if we take $G$ to be infinite symmetric group over countably many elements, then the associated category is precisely the opposite category of the category $\FI$ of finite sets and injections, and the category of sheaves of sets over the opposite category of $\FI$ with atomic topology is the famous Schanuel topos in categorical logic. Thus the result of Artin connects quite a few areas, including topos theory, representation stability theory introduced by Church, Ellenberg and Farb in \cite{CEF}, and continuous actions of topological groups on sets. This strategy has also been used to study representations of finite groups in \cite{WX, XX}. Explicitly, given a finite group $G$, one may construct a few small categories $\mathscr{C}$ (for instances, orbit categories) and equip them with various Grothendieck topologies. It is found that sheaves of modules over these categories are closely related to representations of $G$, and the categories of sheaves correspond to various abelian subcategories of the representation category of $G$. Consequently, one may investigate representation theory of $G$ via considering sheaves over $\mathscr{C}$.

Motivated by the above mentioned works, we want to build connections among sheaves of modules over categories equipped with particular Grothendieck topologies, representations of categories, and (continuous) representations of topological groups. Since it is direct to deduce from the fundamental result of Artin that the category of discrete representations of a topological group $G$ is equivalent to the category of sheaves of modules over a certain orbit category $\C$ of $G$ equipped with atomic topology, one only needs to relate sheaves over $\C$ to representations of $\C$. More generally, let $\C$ be a skeletally small category and $R$ a commutative ring. A presheaf of $R$-modules is a contravariant functor from $\C$ to the category $R \Mod$ of $R$-modules, or equivalently, a covariant functor from $\C^{\op}$ to the category $R \Mod$ (we sometimes call it a $\C^{\op}$-module, or a representation of $\C^{\op}$). When we impose a Grothendieck topology on $\C$, a presheaf of $R$-modules is a sheaf of $R$-modules with respect to this topology if it satisfies several equivalent axioms; see for instances \cite[III.4, Proposition 1]{MM}. In other words, sheaves of $R$-modules over $\C$ are representations of $\C^{\op}$ satisfying certain special requirements (which are, of course, dependent on the imposed topology), and the category of sheaves is a full subcategory of the abelian category of $\C^{\op}$-modules. However, these special requirements, though are explicit in a formal sense, are hard to check in practice. Therefore, for experts in other areas (especially representation theory) who hope to apply results in sheaf theory to their research, it is desirable to find an equivalent but more accessible approach; that is, to interpret concepts and results in sheaf theory in the language of representation theory of categories. In particular, there are several important questions to be answered: finding representation theoretic conditions such that a $\C^{\op}$-modules is a sheaf of $R$-modules over $\C$ if and only if it satisfies these conditions; reformulating sheafification functor in terms of natural functors in representation theory; finding equivalences between the category of sheaves and certain special module categories of $\C^{\op}$. This task is handled in the first part of this paper for a special Grothendieck topology: the atomic topology.

In the second part of this paper, we mainly consider discrete representations of certain topological groups $G$ such as the infinite symmetric group and the infinite general linear group over a finite field, which are of significant interest to representation theorists. We construct their associated orbit categories (which turn out to be well known combinatoric categories such as $\FI$ and $\VI$ studied in representation stability theory), and apply established results in representation stability theory to describe the structure of categories of discrete representations of $G$, and in particular to classify special objects in them, such as simple objects and indecomposable injective objects.

\subsection{From sheaf theory to representation theory}

Suppose that $\C$ is a skeletally small category and satisfies the right Ore condition. Then one may impose the atomic topology $J_{at}$ on $\C$ to get a site $\boldsymbol{\C} = (\C, J_{at})$. In this situation, the constant presheaf $\underline{R}$ assigning to each object in $\C$ the ring $R$ and to each morphism in $\C$ the identity map $\mathrm{id}_R$ is a sheaf of rings, and sheaves of $\RR$-modules over $\boldsymbol{\C}$  \cite{KS, Stack} are precisely presheaves of $R$-modules (defined as above) such that the underlying presheaves of sets are sheaves of sets.

On the other hand, one can consider a natural torsion theory in $\C^{\op} \Mod$. A $\C^{\op}$-module $V$ is \textit{torsion} if for every object $x$ and every element $v \in V(x)$, there is a morphism $\alpha: x \to y$ in $\C^{\op}$ such that $V(\alpha): V(x) \to V(y)$ sends $v$ to 0. For every $\C^{\op}$-module $V$ there is a natural short exact sequence $0 \to V_T \to V \to V_F \to 0$ such that $V_T$ is the maximal torsion submodule of $V$. Note that the assignment $V \mapsto V_T$ gives rise to a left exact functor $\tau$. We say that $V$ is \textit{torsion free} if $V_T = \tau V = 0$, and $V$ is \textit{saturated} if $\mathrm{R}^i \tau (V) = 0$ for $i=0, 1$, where $\mathrm{R}^i \tau$ is the $i$-th right derived functor of $\tau$. Since the category $\C^{\op} \Mod^{\tor}$ of torsion $\C^{\op}$-modules is an abelian subcategory of $\C^{\op} \Mod$, one may define the Serre quotient $\C^{\op} \Mod / \C^{\op} \Mod^{\tor}$, and obtains two functors:
\begin{align*}
\textsl{loc}: & \, \, \C^{\op} \Mod \longrightarrow \C^{\op} \Mod / \C^{\op} \Mod^{\tor};\\
\textsl{sec:} & \, \, \C^{\op} \Mod / \C^{\op} \Mod^{\tor} \longrightarrow \C^{\op} \Mod.\\
\end{align*}

It turns out that for certain atomic sites, one may translate notions on sheaves of $R$-modules in terms of the above notions in representation theory. As the first main result, for atomic topology, we have the following ``translation" from sheaf theory to representation theory:

\begin{center}
\begin{tabular}{c|c}
 Sheaf theory & Representation theory\\
  \hline
presheaves (of $R$-modules) over $\C$ & $\C^{\op}$-modules\\
separated presheaves over $\C$ & torsion free $\C^{\op}$-modules\\
sheaves (of $R$-modules) over $\BC=(\C,J_{at})$ & saturated $\C^{\op}$-modules\\
presheaf category $\PSh(\C,R)$ & $\C^{\op} \Mod$\\
sheaf category $\RR\Mod=\Sh(\BC, R)$ & Serre quotient $\C^{\op} \Mod / \C^{\op} \Mod^{\tor}$\\
sheafification $\sharp: \PSh(\C, R) \to \Sh(\BC, R)$ & $\textsl{sec} \circ \textsl{loc}: \C^{\op} \Mod \to \C^{\op} \Mod^{\sat}$\\
sheaf comohomology functors & right derived functors of the torsion functor\\
\end{tabular}
\end{center}

More precisely, we have:

\begin{theorem}
Let $\C$ be a skeletally small category (satisfying the right Ore condition) equipped with the atomic topology, and let $R$ be a commutative ring. Then:
\begin{enumerate}
\item A presheaf $V$ of $R$-modules over $\C$ is a sheaf if and only if $V$ is saturated as a $\C^{\op}$-module.
\item The category $\Sh(\BC, R)$ of sheaves of $R$-modules over $\BC$ is equivalent to the abelian category $\C^{\op} \Mod^{\sat}$ of saturated modules, and is equivalent to the Serre quotient category $\C^{\op} \Mod / \C^{\op} \Mod^{\tor}$.
\item The sheafification functor $\sharp: \PSh(\C, R) \to \Sh(\BC, R)$ corresponds to
\[
\textsl{sec} \circ \textsl{loc}: \C^{\op} \Mod \to \C^{\op} \Mod^{\sat}
\]
under the identification given in part (2).
\item Suppose that $\C$ has an initial object. Then for a sheaf $V \in \Sh(\BC, R)$, an object $x$ in $\C$, and $i \geqslant 1$, the sheaf cohomology group $\mathrm{R}^i \Gamma_x(V) \cong (\mathrm{R}^{i+1} \tau (V))_x$, the value of the $\C^{\op}$-module $\mathrm{R}^{i+1} \tau (V)$ on $x$.
\end{enumerate}
\end{theorem}

\begin{remark} \normalfont
Although the category $\C^{\op} \Mod^{\sat}$ is a full subcategory of $\C^{\op} \Mod$, its abelian structure is not inherited from that of $\C^{\op} \Mod$. This is an analogue of the following well known fact: the category of sheaves is a full subcategory of the category of presheaves, but they have different abelian structures.
\end{remark}

\begin{remark} \normalfont \label{sec}
Note that the images of $\textsl{sec}: \C^{\op} \Mod / \C^{\op} \Mod^{\tor} \longrightarrow \C^{\op} \Mod$ always lie in the subcategory $\C^{\op} \Mod^{\sat}$ of $\C^{\op} \Mod$; see \cite[Lemma III.2.2]{Gab}. Therefore, in part (3), $\textsl{sec}$ may be viewed as a functor from the Serre quotient to $\C \Mod^{\sat}$, and consequently, $\textsl{sec} \circ \textsl{loc}$ can be viewed as a functor to $\C \Mod^{\sat}$ as well.
\end{remark}


\subsection{A more transparent description via Nakayama functor}

Objects and the structure of the Serre quotient category $\C^{\op} \Mod /\C^{\op} \Mod^{\tor}$ are still mysterious for the purpose of application. One thus may ask for more explicit descriptions under certain special circumstances. We consider two types of categories equipped with suitable combinatorial structure. In particular, in the situation that $R$ is a field of characteristic 0, and that $\C^{\op}$ has the \textit{locally Noetherian} property (that is, the category $\C^{\op} \module$ of finitely generated $\C^{\op}$-modules over $R$ is abelian), one automatically obtains that the category $\mathrm{sh}(\BC, R)$ of ``finitely generated" objects in $\Sh(\BC, R)$ is equivalent to the Serre quotient $\C^{\op} \module / \C^{\op} \module^{\tor}$ formed by restricting to the categories of finitely generated objects. This quotient category already contains useful information which is of great interest to us; for example, all simple sheaves are contained in it. We then apply the strategy introduced in \cite{GLX} to establish the following result for quite a few infinite combinatorial categories $\C$ (including the category $\FI$ of finite sets and injections, the category $\VI$ of finite dimensional vector spaces over a finite field and linear injections, the category $\OI$ of finite linearly ordered sets and order-preserving injections, etc):

\begin{theorem}
Let $R$ be a field of characteristic 0, and let $\C^{\op}$ be a type II combinatorial category (see Subsection \ref{type ii cats}). Then the Nakayama functor induces the following equivalence:
\[
\mathrm{sh} (\BC, R) \cong \C^{\op} \module / \C^{\op} \module^{\tor} \cong \C^{\op} \fdmod
\]
where $\C^{\op} \fdmod$ is the category of finite dimensional $\C^{\op}$-modules.
\end{theorem}

Consequently, we know that objects in $\mathrm{sh} (\BC, R)$ have finite length, and have finite injective dimension. Furthermore, since it is much easier to classify simple objects in $\C^{\op} \fdmod$, via this equivalence, one can explicitly construct all simple sheaves in $\mathrm{sh} (\BC, R)$.

\subsection{Discrete representations of topological groups}

Let $G$ be a topological group and let $V$ be an $RG$-module. We say that $V$ is a \textit{discrete representation} of $G$ if $V$ is equipped with the discrete topology and the action of $G$ on $V$ is continuous. By the canonical result of Artin (\cite[III.9, Theorems 1 and 2]{MM}), one can construct an orbit category $\mathbf{S}_{\mathcal{U}}G$ with respect to a cofinal system $\mathcal{U}$ of open subgroups of $G$, and the category $RG \DMod$ of discrete representations of $G$ is equivalent to the category $\Sh(\mathbf{S}_{\mathcal{U}}G, {R})$. This result provides a novel approach to study representations of topological groups with nice combinatorial structure. The following table includes a few interesting examples:

\begin{center}
\begin{tabular}{c|ccc}
  $G$ & $\mathbf{S}_{\mathcal{U}}G$ & objects & morphisms\\
  \hline
  $\mathrm{Aut}(\mathbb{N})$ & $\FI^{\op}$ & finite sets & injections \\
  $\varinjlim_n \mathrm{Aut}([n])$ & $\FI^{\op}$ & finite sets & injections \\
  linear bijections over $\mathbb{F}_q^{\mathbb{N}}$ & $\VI^{\op}$ & finite diml. spaces over $\mathbb{F}_q$ & linear injections \\
  $\varinjlim_n \mathrm{GL}_n(\mathbb{F}_q)$ & $\VI^{\op}$ & finite diml. spaces over $\mathbb{F}_q$ & linear injections \\
  $\varinjlim_n \mathrm{SL}_n(\mathbb{F}_q)$ & $\VI^{\op}$ & finite diml. spaces over $\mathbb{F}_q$ & linear injections \\
  $\mathrm{Aut}(\mathbb{Q}, \leqslant)$ & $\OI^{\op}$ & finite linearly ordered sets & order-preserving injections \\
\end{tabular}
\end{center}

Sheaf theory over these categories has been widely applied in categorical logic theory; see for instance \cite[Example D3.4.1]{Jo}. More recently, their representations were also extensively investigated in a new and active area called representation stability theory, originated from the work \cite{CEF} of Church, Ellenberg and Farb. People have obtained a deep and comprehensive understanding of their representation theoretic and homological properties via a series of works; see for instances \cite{CE, CEF, CEFN, GL2, GLX, GS, LR, LY, Nag, Nag1, Nag2, PS, SS, SS2}. In particular, when $R$ is a field of characteristic 0 (or any field for $\mathbf{S}_{\mathcal{U}}G = \OI^{\op}$), one concludes that $\mathrm{sh}(\mathbf{S}_{\mathcal{U}}G, R) \cong (\mathbf{S}_{\mathcal{U}}G)^{\op} \fdmod$ via the Nakayama functor (see \cite{ GL2, GLX}). Consequently, through this equivalence we can explicitly classify all simple sheaves over $\mathbf{S}_{\mathcal{U}}G$, and in particular, we get the following parametrizations of simple discrete representations of topological groups listed in the above table:

\begin{theorem}
Let $R$ be a field of characteristic 0. Then:
\begin{enumerate}
\item simple discrete representations of $\mathrm{Aut}(\mathbb{N})$ are parameterized by the set $\sqcup_{n \in \mathbb{N}} \mathcal{P}_n$, where $\mathcal{P}_n$ is the set of partitions of $[n]$;
\item simple discrete representations of $\mathrm{GL}(\mathbb{F}_q^{\mathbb{N}})$ or $\varinjlim_n \mathrm{SL}_n(\mathbb{F}_q)$ are parameterized by the set $\sqcup_{n \in \mathbb{N}} \mathcal{I}_n$, where $\mathcal{I}_n$ is the set of isomorphism classes of simple representations of $\mathrm{GL}_n(\mathbb{F}_q)$;
\item simple discrete representations of $\mathrm{Aut}(\mathbb{Q}, \leqslant)$ are parameterized by $\mathbb{N}$.
\end{enumerate}
\end{theorem}

\begin{remark} \normalfont
The classification of simple discrete representations of $\mathrm{Aut}(\N)$ has been described in \cite{SS2},\footnote{They are called \textit{algebraic representations} rather than discrete representations. But it is not hard to see that these two notions coincide.}, though they consider the infinite symmetric group $\varinjlim_n \Aut([n])$ rather than the bigger group of all bijections. The second and third classifications, as far as the authors know, do not appear in literature. We also point out that isomorphism classes of irreducible representations of $\mathrm{GL}_n(\mathbb{F}_q)$ have been classified in \cite[Section 9]{Ze} when $R$ is the complex field.
\end{remark}

\begin{remark} \normalfont
In \cite{SS2} Sam and Snowden studied \textit{algebraic representations} of the infinite general linear group. Their work and the work described in this paper look quite similar, but are actually very different. To avoid possible confusion, let us briefly mention their setup. Let $V$ be a complex vector space with countable dimension, let $G$ be the infinite general linear group over $V$, and let $T_n$ be the tensor product of $n$ copies of $V$. Clearly, $V$ is a representation of $G$. A representation of $G$ is call \textit{algebraic} in \cite{SS2} if it is a subquotient of the direct sum of finitely many copies of $T_n$ ($n$ be different). In summary, what they considered are (algebraic) complex representations of the infinite general linear group over the complex field, while in this paper we consider (discrete) representations (over any field of characteristic 0) of the infinite general linear group over a finite field.
\end{remark}

\begin{remark} \normalfont
For $\mathrm{Aut}(\mathbb{Q}, \leqslant)$, the above parametrization of simple discrete representations holds for any field $R$.
\end{remark}

\subsection{Organization}

This paper is organized as follows. In Section 2 we give some background and preliminary knowledge on topos theory for the convenience of the reader. Sheaves of modules over categories equipped with atomic topology are studied in Section 3, where we setup the translation machinery between sheaf theory and representation theory of categories, and prove all results listed in Theorem 1.1. In Section 4 we consider three types of categories with suitable combinatorial structure, describe equivalences between $\mathrm{Sh} (\BC, R)$ and some other representation categories which are more transparent compared to the Serre quotient category, and relate sheaf cohomology to certain local cohomology. In particular, we introduce the Nakayama functor, and establish the equivalence between $\mathrm{sh} (\BC, R)$ and $\C^{\op} \fdmod$. This result is crucial for us to classify simple discrete representations of some interesting topological groups. Discrete representations of topological groups are studied in Section 5, where we describe the classical result of Artin, and give some examples for illustration. In Sections 6 and 7, we consider examples in the above table, classify simple sheaves for $\FI$, $\VI$ and $\OI$, and construct simple discrete representations of their associated topological groups.

\subsection{Notations}

Throughout this paper we stick to the following conventions unless otherwise stated: all functors are covariant functors, composition of maps or functors is from left to right, and actions of groups (resp., rings) on sets (resp., modules) are on the left side.

Some notations used throughout this paper are listed below:
\begin{itemize}
\item $\C$: a skeletally small category;
\item $\C^{\op}$: the opposite category of $\C$;
\item $J$: a Grothendieck topology on $\C$;
\item $J_{at}$: the atomic topology on $\C$, which satisfies the right Ore condition;
\item $\boldsymbol{\C} = (\C, J)$: a site (here $J$ is almost always $J_{at}$);
\item $\PSh(\C)$: the category of presheaves of sets over $\C$;
\item $\Sh(\boldsymbol{\C})$: the category of sheaves of sets over the site $\boldsymbol{\C}$;
\item $R$: a commutative ring;
\item $R\C$: the category algebra;
\item $\RR$: the constant presheaf of rings over $\C$ sending each object in $\C$ to $R$;
\item $P(x)$: the representable functor $R\C(x, -)$;
\item $\PSh(\C, R) = \C^{\op} \Mod$: the category of presheaves of $R$-modules over $\C$, or equivalently, the category of $\C^{\op}$-modules;
\item $\Sh(\BC, R)=\RR\Mod$: the category of sheaves of $R$-modules over the ringed site $(\BC, \RR)$ (equipped with the atomic topology), which are often called the $\RR$-modules;
\item $\mathrm{psh}(\C, R) = \C^{\op} \module$: the category of finitely generated $\C^{\op}$-modules;
\item $\mathrm{sh}(\BC, R)$: the category of finitely generated $\C^{\op}$-modules which are also sheaves of $R$-modules over the ringed site $(\BC, \RR)$ (equipped with the atomic topology);
\item $R \Mod$: the category of $R$-modules;
\item $\C \Mod^{\sat}$: the category of saturated $\C$-modules;
\item $\C \Mod^{\tor}$: the category of torsion $\C$-modules;
\item $\C \Mod^{\mathrm{tf}}$: the category of torsion free $\C$-modules;
\item $\C \module$: the category of finitely generated $\C$-modules;
\item $\C \module^{\sat}$: the category of finitely generated saturated $\C$-modules;
\item $\C \module^{\tor}$: the category of finitely generated torsion $\C$-modules;
\item $\C \fdmod$: the category of finite dimensional $\C$-modules;
\item $RG \DMod$: the category of discrete representations of a topological group $G$;
\item $\mathcal{P}_n$: the set of partitions of $[n]$;
\item $\mathcal{P}$: the disjoint union of $\mathcal{P}_n$ for all $n \in \N$.
\end{itemize}


\section{Preliminaries on topos theory}

For the convenience of the reader, in this section we introduce some basic concepts and results on topos theory. A good reference for this area is \cite{Jo} or \cite{MM}.


\subsection{Grothendieck topology and sites}

Throughout this paper we let $\C$ be a \textit{skeletally small} category; that is, the isomorphism classes of objects form a set, and for every pair of objects $x$ and $y$, the class $\C(x, y)$ of morphisms from $x$ to $y$ is also a set. For an object $x$, a \textit{sieve} on $x$ is a subfunctor of the representable functor $\C(-, x)$; or equivalently, a \textit{right ideal} $S$ of $\C$ consisting of morphisms ending at $x$ in the following sense: if $f: y \to x$ is a member in $S$, and $g: z \to y$ is a morphism, then $fg = f \circ g \in S$ as well.

We recall the following definition of Grothendieck topology \cite[III.2, Definition 1]{MM}.

\begin{definition}
A \textit{Grothendieck topology} on $\C$ is a rule $J$ assigning to each object $x$ a family $J(x)$ of sieves on $x$ such that the following axioms are satisfied:
\begin{enumerate}
\item for each object $x$, $\C(-, x) \in J(x)$;
\item if $S \in J(x)$, then for any morphism $f: y \to x$, $f^{\ast} (S) = \{ g: \bullet \to y \mid fg \in S \}$ is contained in $J(y)$;
\item if $S \in J(x)$ and $R$ is any sieve on $x$ such that $f^{\ast}(R) \in J(y)$ for any $f: y \to x$ contained in $S$, then $R \in J(y)$.
\end{enumerate}
The pair $\boldsymbol{\C}=(\C, J)$ is called a \textit{site}, and members in $J(x)$ are called \textit{covering sieves} of $x$.
\end{definition}

{We shall point out that in the literature there are different versions of definitions of Grothendieck topologies. In this paper we always follow the definitions and conventions of \cite{MM}, and do not require categories to have fibre products. We also note that the assumption that $\C$ is skeletally small is essential since the first condition in the above definition requires $\C(y, x)$ to be a set.

There is a special type of Grothendieck topology called the \textit{trivial topology}: for every object $x$ one has $J(x) = \{ \C(-, x)\}$. This is the coarsest topology we may impose on $\C$.

\subsection{Sheaves of sets}

Let $\Sets$ be the category of sets and maps. A \textit{presheaf} of sets over $\C$ is a contravariant functor $F: \C \to \Sets$, or equivalently, a covariant functor from $\C^{\op}$ to $\Sets$. Morphisms between two presheaves of sets are natural transformations. Note that the definition of presheaves of sets is independent of Grothendieck topologies on $\C$. We denote the category of presheaves of sets by $\PSh(\C)$.

Now we fix a Grothendieck topology $J$ on $\C$ and let $\boldsymbol{\C} = (\C, J)$ be the corresponding site. The following definitions are taken from \cite[III.4]{MM}.

\begin{definition}
Let $F$ be a presheaf of sets and $S$ a covering sieve of a certain object $x$ in $\C$. A \textit{matching family} for $S$ of elements of $F$ is a rule assigning to each morphism $f: y \to x$ in $S$ an element $v \in F(y)$ such that $v_f \cdot g = v_{fg}$ for any morphism $g: \bullet \to y$, where the action of $g$ on $v_f$ is determined by the contravariant functor $F: \C \to \Sets$. An amalgamation of this matching family is an element $v_x \in F(x)$ such that $v_x \cdot f = v_f$ for all $f \in S$ \footnote{To respect our previous convention specified in Introduction, for actions of morphisms on sets via contravariant functors, we write the action on the right side.}.
\end{definition}

\begin{definition}
A presheaf $F \in \PSh(\C)$ is \textit{separated} if every matching family for every object $x$ and every covering sieve $S \in J(x)$ has at most one amalgamation. It is a \textit{sheaf} on the site $\boldsymbol{\C} = (\C, J)$ if every matching family has exactly one amalgamation.
\end{definition}

In particular, when $J$ is the trivial topology, every presheaf is a sheaf. In general every sheaf is a separated presheaf, and morphisms between two sheaves are morphisms between them as presheaves. Consequently, the category of sheaves over $\BC$ is a full subcategory of $\PSh(\C)$, and we denote it by $\Sh(\BC)$ (or simply $\Sh(\C)$ when the Grothendieck topology $J$ is clear from the context). There is a natural inclusion functor $\iota: \Sh(\BC) \to \PSh(\C)$, and it has a left adjoint functor $\sharp: \PSh(\C) \to \Sh(\BC)$, called the \textit{sheafification} functor. It is well-known that the sheafification is a special type of localization; see \cite[Proposition 1.3]{GZ} or \cite{KS}. It means that $\Sh(\BC)$ is a localization of $\PSh(\C)$ by the multiplicative system of local isomorphisms \cite{KS} (or morphismes bicouvrants \cite{AGV}).

\subsection{Atomic topology}

In this paper we mostly consider a specific Grothendieck topology, called the atomic topology. To introduce its description, we need the following definition.

\begin{definition}
We say that $\C$ satisfies the \textit{left Ore condition} if for every pair of morphisms $f: x \to y$ and $f': x \to y'$, there exist an object $z$ and a pair of morphisms $g: y \to z$ and $g': y' \to z$ such that the following diagram commutes:
\[
\xymatrix{
x \ar[r]^-f \ar[d]_{f'} & y \ar@{-->}[d]^-g \\
y' \ar@{-->}[r]_-{g'} & z.
}
\]
The \textit{right Ore condition} is defined dually.
\end{definition}

For every object $x$ in $\C$, we let $J_{at}(x)$ be the collection of all nonempty sieves of $\C(-, x)$. The reader can check that the rule $J_{at}$ gives a Grothendieck topology if and only if $\C$ satisfies the right Ore condition; see \cite[III.2, Example (f)]{MM}. We call this topology the \textit{atomic topology}.

The following lemma gives a convenient criterion to determine whether a given presheaf over a category is a sheaf with respect to the atomic topology.

\begin{lemma}\cite[III.4, Lemma 2]{MM} \label{simple criterion}
Let $\C$ be a category with atomic topology and $F$ a presheaf of sets over $\C$ . Then $F$ is a sheaf of sets on $\BC$ if and only if for any morphism $f: y \to x$ and $v \in F(y)$ such that $v \cdot g = v \cdot h$ whenever $fg = fh$, then there exists a unique $u \in F(x)$ such that $u \cdot f = v$.
\end{lemma}

\begin{remark}\label{rmk 2.6} \normalfont
When replacing the words ``a unique" in the last sentence of this lemma by ``at most one", we obtain a criterion for a presheaf of sets to be separated.
\end{remark}

Recall that a presheaf $F$ is \textit{constant} if $F(x) = F(y)$ for any objects $x$ and $y$, and any morphism $f: y \to x$ acts (on the right side) as the identity map $F(y) \to F(x)$. It is easy to see that the constant presheaf is separated since every morphism acts as an injective map, and furthermore, by the previous lemma, it is actually a sheaf for the atomic topology.


\subsection{Sheaves of modules}

To consider applications of sheaf theory in representation theory, we need to study sheaves with extra algebraic structures, in particular, ringed sites and sheaves of modules. In this subsection we only give necessary definitions. For more details, please refer to \cite[Chapter 18]{Stack}.

\begin{definition}
Let $\boldsymbol{\C} = (\C, J)$ be a site. A \textit{presheaf of abelian groups} is a contravariant functor $\mathcal{O}: \C \to \mathrm{Ab}$, the category of abelian groups; it is called a \textit{sheaf of abelian groups} if the underlying presheaf of sets is a sheaf. Abelian presheaves (resp. sheaves) form abelian categories $\PSh(\C, \mathbb{Z})=\PSh(\C, \mathrm{Ab})$ (resp. $\Sh(\BC, \mathbb{Z})=\Sh(\BC, \mathrm{Ab})$).

More generally we have category of (pre)sheaves of $R$-modules, and they form abelian categories $\PSh(\C, R)=\PSh(\C, R\Mod)$ and $\Sh(\BC, R)=\Sh(\BC, R\Mod)$.

Similarly, one can define presheaves and sheaves of rings.
\end{definition}

\begin{definition}
Let $\mathcal{O}$ be a sheaf of rings. The pair $(\BC, \mathcal{O})$ is called a \textit{ringed site}, and $\mathcal{O}$ is called the \textit{structure sheaf} of the ringed site.
\end{definition}

\begin{definition}
Let $(\BC, \mathcal{O})$ be a ringed site. A \textit{presheaf of $\mathcal{O}$-modules} is a presheaf $F$ of abelian groups and a map of presheaf of sets
\[
\mathcal{O} \times F \to F
\]
such that for every object $x$, the map
\[
\mathcal{O}(x) \times F(x) \to F(x)
\]
defines an $\mathcal{O}(x)$-module structure of $F(x)$. The subcategory of $\PSh(\BC,\mathrm{Ab})$ that they form is $\PSh(\C, \mathcal{O})$.

We say that $F$ is a \textit{sheaf of $\mathcal{O}$-modules} if the underlying presheaf of sets is a sheaf. The subcategory of $\Sh(\BC,\mathrm{Ab})$ that they form is $\mathcal{O}\Mod$.
\end{definition}

Note that $\PSh(\C, \mathcal{O})$ is an abelian category, and kernels and cokernels are computed in the underlying category $\PSh(\C, \mathrm{Ab})$. The category $\mathcal{O} \Mod$ is also abelian: kernels are computed in $\PSh(\C, \mathcal{O})$; to compute cokernels, we first obtain the cokernels in $\PSh(\C, \mathcal{O})$, and then apply the sheafification functor $\sharp$. Furthermore, these two abelian categories are Grothendieck categories, and hence every object in them has an injective hull.

When $\mathcal{O}=\RR$ is the constant sheaf (of rings), we have $\RR\Mod=\Sh(\BC,R)$.


\section{Sheaves of categories with atomic topology}

From now on we suppose that $\C^{\op}$ \footnote{The reason we consider sheaf theory of $\C^{\op}$ rather than that of $\C$ is that presheaves of $\C^{\op}$ are exactly $\C$-modules, and in the main body of this paper we will study representation theory of $\C$ instead of that of $\C^{\op}$.} is a skeletally small category satisfying the right Ore condition, and impose the atomic topology $J_{at}$ on $\C^{\op}$; that is, for any object $x$, $J_{at}(x)$ is the collection of subfunctors $S$ of $\C^{\op}(-, x) = \C(x, -)$ such that $\C(x, y) \neq \emptyset$ for at least one object $y$. We also fix $\boldsymbol{\C}^{\op}$ to be the pair $(\C^{\op}, J_{at})$.

Let $R$ be a commutative ring and $\RR$ be the constant presheaf of rings such that $\RR(x) = R$ for every object $x$ in $\C^{\op}$. By Lemma \ref{simple criterion}, $\RR$ is a sheaf of rings. Let $(\BC^{\op}, \RR)$ be the ringed site. The main goal of this section is to characterize sheaves of $\RR$-modules in terms of representations of $\C$, and describe the structure of $\Sh(\BC^{\op}, R)$.

Since $\RR$ is a constant sheaf of rings, an object in $\PSh(\C^{\op}, R)$ is nothing but a \textit{$\C$-module}; that is, a covariant functor from $\C$ to the category $R \Mod$. Consequently, objects in $\Sh(\BC^{\op}, R)$ are $\C$-modules such that the underlying presheaves of sets are actually sheaves. Thus we may freely use the language of representation theory of $\C$. In particular, we also denote $\PSh(\C^{\op}, R)$ by $\C \Mod$.

\subsection{Characterizations of sheaves}

Now let $V$ be an object in $\PSh(\C^{\op}, R)$, or equivalently, a $\C$-module over the coefficient ring $R$. For an object $x$ in $\C$, we denote the value of $V$ on $x$ by $V_x$. An element $v \in V_x$ is \textit{torsion} if there is a morphism $f: x \to y$ in $\C$ such that $f \cdot v = 0$.

The following lemma tells us that we can define a torsion theory in the category $\C \Mod$.

\begin{lemma}
The torsion elements in $V$ form a submodule of $V$.
\end{lemma}

\begin{proof}
Let $v_x \in V_x$ and $v_y \in V_y$ be two torsion elements. We have to show that $v_x + v_y$ and $f \cdot v_x$ are torsion as well, where $f$ is a morphism in $\C$. Since $v_x$ and $v_y$ are torsion elements, by definition, there are morphisms $f_x: x \to u$ and $f_y: y \to w$ such that $f_x \cdot v_x = 0 = f_y \cdot v_y$. If $x \neq y$, then clearly $f_x \cdot v_y = 0$, and hence
\[
f_x \cdot (v_x + v_y) = f_x \cdot v_x + f_x \cdot v_y = 0
\]
If $x = y$, then by the left Ore condition, we can find morphisms $g$ and $g'$ such that $gf_x = g'f_y$. Consequently,
\[
gf_x \cdot (v_x + v_y) = gf_x \cdot v_x + gf_x \cdot v_y = 0 + g'f_y \cdot v_y = 0.
\]
In both cases, we have shown that $v_x + v_y$ is torsion.

We use the same strategy to prove that $f \cdot v_x$ is torsion, by considering the following commutative diagram:
\[
\xymatrix{
x \ar[r]^-{f_x} \ar[d]_f & z \ar[d]^-g \\
b \ar[r]_-{g'} & a.
}
\]
That is, $g' \cdot (f \cdot v_x) = gf_x \cdot v_x = 0$, so $f \cdot v_x$ is torsion.
\end{proof}

The left Ore condition is essential, as shown in the following example.

\begin{example} \normalfont
Let $Q$ be the Kronecker quiver with two objects $x$ and $y$, and two arrows $\alpha$ and $\beta$ from $x$ to $y$. Consider $V = \mathbb{R}Q / \langle \beta \rangle$. Then the image of $1_x$ (the identity morphism on $x$) in the quotient $V$ (still denote it by $1_x$) is torsion, since $\beta$ maps it to 0. However, $\alpha \cdot 1_x = \alpha$ is not torsion. Thus the torsion elements do not form a submodule of $V$.
\end{example}

We denote the submodule consisting of all torsion elements in $V$ by $V_T$, which is the maximal torsion submodule of $V$. We say that $V$ is \textit{torsion free} if $V_T = 0$, and is \textit{torsion} if $V_T = V$. It is easy to check that the category $\C \Mod^{\tor}$ of torsion $\C$-modules is an abelian subcategory of $\C \Mod$, and furthermore the assignment $V \to V_T$ give rises to a left exact functor $\tau$, called the \text{torsion functor}. Instead, the category $\C \Mod^{\mathrm{tf}}$ consisting of torsion free $\C$-modules is in general not abelian. However, these two subcategories form a \textit{torsion pair} in the following sense:

\begin{lemma} \label{torsion pair}
Notation as above. Then one has:
\begin{enumerate}
\item if $V$ is a torsion $\C$-module and $W$ is torsion free $\C$-module, then $\Hom_{\C \Mod} (V, W) = 0$;
\item $\Hom_{\C \Mod} (V, W) = 0$ for all $W \in \C \Mod^{\mathrm{tf}}$ if and only if $V \in \C \Mod^{\tor}$;
\item dually, $\Hom_{\C \Mod} (V, W) = 0$ for all $V \in \C \Mod^{\tor}$ if and only if $W \in \C \Mod^{\mathrm{tf}}$.
\end{enumerate}
\end{lemma}

\begin{proof}
Suppose that there is a nonzero homomorphism $\varphi: V \to W$. Then in particular there exist an object $x$ and a nonzero element $v \in V_x$ such that $\varphi_x(v) \neq 0$. But since $V$ is torsion, one can find a morphism $f: x \to y$ such that $f \cdot v = 0$. Consequently, $f \cdot \varphi_x(v) = \varphi_y (f \cdot v) = 0$. But $\varphi_x(v) \in W_x$, so $W$ is not torsion free, and hence (1) follows from this contradiction.

The ``if" directions of (2) and (3) are clear. For the other directions, take $W$ (resp., $V$) to be $V/V_T$ (resp., $W_T$). Then $\Hom_{\C \Mod} (V, V/V_T) = 0$ or $\Hom_{\C \Mod} (W_T, W) = 0$ implies that $V/V_T = 0$ or $W_T = 0$.
\end{proof}

The following lemma interprets separated presheaves as torsion free modules.

\begin{lemma} \label{separated presheaves}
Separated presheaves in $\PSh(\C^{\op}, R)$ are precisely torsion free $\C$-modules.
\end{lemma}

\begin{proof}
Suppose that $V$ is a torsion free module. Note that $V$ is torsion free if and only if for every morphism $f: x \to y$ in $\C$, the corresponding $R$-linear homomorphism $V_f: V_x \to V_y$ is injective. By Remark \ref{rmk 2.6} (note that we have to reverse all arrows since we are working with $\C^{\op}$ rather than $\C$), $V$ is a separated presheaf of $R$-modules. Conversely, if $V$ is not torsion free, then there exists a morphism $f: x \to y$ such that $V_f: V_x \to V_y$ is not injective. In particular, if we let $g = h = 1_y$ and let $v = 0$  in Lemma \ref{simple criterion}, then one can find two distinct elements (one of which is 0) in $V_x$ such that both are sent to $v = 0 \in V_y$ by $V_f$, and hence $V$ is not a separated presheaf.
\end{proof}

Our next task is to characterize sheaves. For this purpose, we introduce a few more notions.

The functor $\tau: \C \Mod \to \C \Mod^{\tor}$ defined by $V \mapsto V_T$ is left exact, and hence have right derived functors $\mathrm{R}^i \tau$, $i \in \N$. To compute $\mathrm{R}^i \tau (V)$ for a $\C$-module $V$, we only need to apply the functor $\tau$ to an injective resolution of $V$, and cohomology groups of this complex give $\mathrm{R}^i \tau(V)$ for $i \geqslant 0$. Clearly, $V$ is torsion free if and only if $\mathrm{R}^0 \tau (V) = \tau V = V_T = 0$.

The next definition is used in \cite[Subsection 4.1]{SS}, where a module is ``saturated" means that it is torsion free, and furthermore it has no nontrivial extension by torsion modules.

\begin{definition}
A $\C$-module $V$ is said to be \textit{saturated} if $\mathrm{R}^i \tau (V) = 0$ for $i = 0, 1$.
\end{definition}

We will show that objects in $\Sh(\BC^{\op}, R)$ are precisely saturated $\C$-modules. Before proving it, we need a few lemmas.

\begin{lemma}\label{embedding}
Every torsion free $\C$-module can be embedded into a torsion free injective $\C$-module.
\end{lemma}
\begin{proof}
Let $V$ be a torsion free $\C$-module. Then there is an exact sequence $0 \to V \to I \to C \to 0$ of $\C$-modules with $I$ injective. Consider the exact sequence $0 \to I_T \to I \to I_F \to 0$ of $\C$-modules, where $I_T = \tau I$ is torsion and $I_F = I/I_T$ is torsion free. Since $V$ is torsion free, the pull-back of $V \to I$ and $I_T \to I$ (which is isomorphic to $V \cap I_T$) is the zero module. This yields the following commutative diagram with exact rows and columns
\[
\xymatrix{
 & 0 \ar[d]\ar[r] & I_T \ar[d]\ar[r] & \ar[d]^{\alpha}\ar[r] I_T  & 0 \\
 0 \ar[r] & V \ar[r] & I \ar[r] & C \ar[r] & 0
}
\]
By the snake lemma there is an exact sequence $0 \to \Ker\alpha \to V \to I_F$. Note that $V$ is torsion free and $I_T$ is torsion. One gets that $\Ker\alpha=0$. That is, $V$ is isomorphic to a submodule of $I_F$. It suffices to show that $I_F$ is injective. But this is clear. Indeed, since the functor $\tau$ is the right adjoint of the exact embedding functor from $\C \Mod^{\tor}$ into $\C \Mod$, it preserves injective modules. Therefore, $I_T$ is injective, and hence the exact sequence $0 \to I_T \to I \to I_F \to 0$ splits. Consequently, $I_F$ is injective as well.
\end{proof}

\begin{remark} \normalfont
Since $\C \Mod$ is a Grothendieck category, one may strengthen the above result: every torsion free $\C$-module has a torsion free injective hull.
\end{remark}

Equivalent characterizations of saturated modules are collected in the following proposition.

\begin{proposition} \label{saturated modules}
Let $V$ be a $\C$-module. The following conditions are equivalent:
\begin{enumerate}
\item $V$ is saturated;
\item there is a short exact sequence $0 \to V \to I \to W \to 0$ such that $I$ is torsion free and injective, and $W$ is torsion free;
\item there is a morphism $\alpha: I^0 \to I^1$ between two torsion free and injective modules with $\ker \alpha \cong V$;
\item $\Ext_{\C \Mod}^1(T, V) = 0 = \Hom_{\C \Mod}(T, V)$ for any torsion module $T$.
\end{enumerate}
\end{proposition}

\begin{proof}

$(1)\Rightarrow(2)$: Since $V$ is saturated, in particular it is torsion free. By Lemma \ref{embedding}, one gets a short exact sequence $0 \to V \to I \to W \to 0$ such that $I$ is a torsion free injective $\C$-module, and it remains to show that $W$ is torsion free as well. Applying $\tau$ to the above exact sequence we get $\tau W = \mathrm{R}^0\tau (W) \cong \mathrm{R}^1 \tau (V) = 0$, and the conclusion follows.

$(2)\Rightarrow(1)$: Apply $\tau$ to the exact sequence.

$(2)\Rightarrow(3)$: Immediately follows from Lemma \ref{embedding}.

$(3)\Rightarrow(4)$: Apply $\Hom_{\C \Mod} (T, -)$ to the exact sequence $0 \to V \to I^0 \to I^1$ to get the conclusion.

$(4)\Rightarrow(2)$: Since $\Hom_{\C \Mod}(T, V) = 0$ for any torsion module $T$, $V$ is torsion free by Lemma \ref{torsion pair}, so we get a short exact sequence $0 \to V \to I \to W \to 0$ such that $I$ is a torsion free injective module. Now applying $\Hom_{\C \Mod} (T, -)$ again we deduce that $\Hom_{\C \Mod}(T, W) \cong \Ext_{\C}^1(T, V) = 0$, so $W$ is torsion free as well.
\end{proof}

Now we are ready to prove the main result of this subsection.

\begin{theorem} \label{main result 1}
A presheaf in $\PSh(\C^{\op}, R)$ is a sheaf of $R$-modules if and only if it is saturated as a $\C$-module.
\end{theorem}

\begin{proof}
Suppose that $V$ is an object in $\Sh(\BC^{\op}, R)$. In particular, it is a separated presheaf. By Lemma \ref{separated presheaves}, $V$ is torsion free. Then we get by Lemma \ref{embedding} a short exact sequence $0 \to V \to I \to W \to 0$ such that $I$ is a torsion free injective $\C$-module. Suppose that there exist an object $x$, an element $w \in W_x$, and a morphism $f: x \to y$ in $\C$ such that $f \cdot w = 0$. Let $\tilde{w} \in I_x$ be a preimage of $w$ and consider the left ideal $S = \{gf \mid g: y \to \bullet \}$ of $\C$ generated by $f$ (which is a covering sieve of $x$ in $\C^{\op}$). Since $gf \cdot w = 0$, one deduces that $gf \cdot \tilde{w} \in V_{\bullet}$. Thus $\{g f \cdot \tilde{w}: \mid g: y \to \bullet \}$ is a matching family for $S$ of elements of $V$. Since $V$ is a sheaf, one can find an amalgamation $\bar{w} \in V_x \subseteq I_x$ such that $ gf \cdot \bar{w} = gf \cdot \tilde{w}$ for every $g$ with domain $y$. In particular, by taking $g = 1_y$ one shall have $f \cdot \tilde{w} = f \cdot \bar{w}$. But $I$ is torsion free, so $\tilde{w} = \bar{w} \in V_x$, and hence $w$ as the image of $\tilde{w}$ is 0. Consequently, $W$ is torsion free, and $V$ is saturated by Proposition \ref{saturated modules}.

Conversely, suppose that $V$ is saturated as a $\C$-module, and in particular it is torsion free. Therefore, $V$ is a separated presheaf in $\PSh(\C^{\op}, R)$. By \cite[III.5, Lemma 2]{MM}, the canonical map $\eta: V \to V^+$ is injective, and by \cite[III.5, Lemma 5]{MM}, $V^+$ is a sheaf in $\Sh(\BC^{\op}, R)$. Therefore, one obtains the following exact sequence of $\C$-modules
\[
0 \to V \to V^+ \to W \to 0.
\]
Note that $V^+$ is a saturated $\C$-module by the result we just proved. We show that $V$ is a sheaf by contradiction.

If $V$ is a separated presheaf in $\PSh(\C^{\op}, R)$ but is not a sheaf in $\Sh(\BC^{\op}, R)$, then there exist an object $x$, a nonempty covering sieve $S \subseteq \C(x, -)$ of $x$ (note that $f$ is a morphism in $\C$, not in $\C^{\op}$), and a matching family $ \{v_f \in V_y \mid (f: x \to y) \in S \}$ for $S$ of elements of $V$ satisfying $g \cdot v_f = v_{gf}$ for any morphism $g: y \to \bullet$, such that there does not exist an element $v \in V_x$ satisfying $f \cdot v = v_f$ for all $f \in S$. Now identify $V$ with a subpresheaf of $V^+$ via the canonical map $\eta$. The above matching family of elements of $V$ becomes a matching family of elements of $V^+$. Since $V^+$ is a sheaf, we can find an element $\tilde{v} \in V_x^+$ serving as the unique amalgamation of this matching family of elements of $V^+$. Clearly, $\tilde{v}$ is not contained in $V_x$ since otherwise $\tilde{v}$ as an element in $V_x$ can serve as the unique amalgamation of the matching family of elements of $V$. Consequently, the image $\bar{v} \in W_x$ of $\tilde{v}$ is nonzero. However, for every $f: x \to \bullet$ in $S$, one has $f \cdot \tilde{v} = v_f \in V_{\bullet}$, and hence in the quotient module $W$ one must have $f \cdot \bar{v} = 0$. That is, $W$ not torsion free. Applying $\tau$ to the above short exact sequence we have:
\[
0 \to \tau V = 0 \to \tau V^+ = 0 \to \tau W \to \mathrm{R}^1 \tau (V) \to \mathrm{R}^1 \tau (V^+) = 0 \to \ldots.
\]
Consequently, $\mathrm{R}^1 \tau (V) \cong \tau W \neq 0$, and hence $V$ is not saturated. This contradiction tells us that $V$ must be a sheaf.
\end{proof}

As an immediate corollary, we can describe all injective objects in $\Sh(\BC^{\op}, R)$.

\begin{corollary} \label{injective sheaves}
Injective objects in $\Sh(\BC^{\op}, R)$ are precisely torsion free injective $\C$-modules.
\end{corollary}

\begin{proof}
Theorem \ref{main result 1} asserts that torsion free injective $\C$-modules $I$ are contained in $\Sh(\BC^{\op}, R)$. They are injective objects in this category. Indeed, given a momomorphism $0 \to I \to V$ in $\Sh(\BC^{\op}, R)$, it is also a momomorphism in $\PSh(\C^{\op}, R)$, and hence splits. Since the cokernel of this map is direct summand of the sheaf $V$, it is also a sheaf. Consequently, the momomorphism actually splits in $\Sh(\BC^{\op}, R)$. Thus $I$ is an injective sheaf. Furthermore, given any $V \in \Sh(\BC^{\op}, R)$, one can obtain a monomorphism $0 \to V \to I$ in $\PSh(\C^{\op}, R)$ such that $I$ is a torsion free injective $\C$-module, which is also a monomorphism in $\Sh(\BC^{\op}, R)$. Thus any sheaf can be embedded into a sheaf which is also a torsion free injective $\C$-module. Consequently, any injective sheaf must be a torsion free injective $\C$-module.
\end{proof}

\subsection{Categories of sheaves}

In this subsection we describe the structure of $\Sh(\BC^{\op}, R)$. As we already know, $\PSh(\C^{\op}, R)$ coincides with $\C \Mod$, and every sheaf in $\Sh(\BC^{\op}, R)$ is a saturated $\C$-module. Let $\C \Mod^{\sat}$ be the full subcategory of $\C \Mod$ consisting of saturated $\C$-modules.

Since $\C \Mod^{\tor}$ is a Serre subcategory of $\C \Mod$, one can form the Serre quotient category $\C \Mod / \C \Mod^{\tor}$, which is a Grothendieck category as well. Furthermore, by classical results of Gabriel on localization of abelian categories, one obtains the following diagram of functors:
\[
\xymatrix{
\C \Mod^{\sat} \ar[rr]^{\textsl{inc}} & & \C \Mod \ar[rr]<.5ex>^-{\textsl{loc}} & & \C \Mod / \C \Mod^{\tor} \ar[ll]<.5ex>^-{\textsl{sec}}.
}
\]
Here $\textsl{loc}$ is the \textit{localization} functor, $\textsl{inc}$ is the \textit{inclusion} functor, and $\textsl{sec}$ is the \textit{section} functor; for details, see \cite[III.2]{Gab}. We also have the following basic results:

\begin{proposition} \label{serre quotient}
Notation as above. Then one has:
\begin{enumerate}
\item $(\textsl{loc}, \textsl{sec})$ is an adjoint pair of functors;
\item $\textsl{loc} \circ \textsl{sec}$ is isomorphic to the identity functor on $\C \Mod / \C \Mod^{\tor}$;
\item $\textsl{sec}$ sends every object in $\C \Mod / \C \Mod^{\tor}$ into $\C \Mod^{\sat}$;
\item $\textsl{sec} \circ \textsl{loc} \circ \textsl{inc}$ is isomorphic to $\textsl{inc}$.
\end{enumerate}
\end{proposition}

\begin{proof}
These results can be deduced by applying results in \cite{Gab}. In particular:

(1) follows from \cite[Proposition III.2.4]{Gab};

(2) follows from \cite[Proposition III.2.3]{Gab}

(3) follows from \cite[Lemma III.2.2]{Gab};

(4) follows from the corollary of \cite[Proposition III.2.3]{Gab}.
\end{proof}

Now we can show:

\begin{theorem} \label{equivalences}
There exist equivalences of the following categories:
\[
\xymatrix{
\Sh(\BC^{\op}, R) \ar[r]^-{\sim} & \C \Mod^{\sat} \ar[r]^-{\sim} & \C \Mod / \C \Mod^{\tor}.
}
\]
\end{theorem}

\begin{proof}
By \cite[II.6, Lemma 4]{MM}, the functors $\textsl{loc}$ and $\textsl{sec}$ induces the second equivalence. An explicit equivalence is given by the functors
\[
\textsl{loc} \circ \textsl{inc}: \C \Mod^{\sat} \to \C \Mod /\C \Mod^{\tor}
\]
and
\[
\textsl{sec}:  \C \Mod /\C \Mod^{\tor} \to \C \Mod^{\sat},
\]
since by (3) of Lemma \ref{serre quotient} the image of $\textsl{sec}$ is contained in $\C \Mod^{\sat}$.

To show the first equivalence, we note that both categories have the same objects by Theorem \ref{main result 1}. Furthermore, for two objects $V$ and $W$ in $\Sh(\BC^{\op}, R)$, one has:
\[
\Hom_{\Sh(\BC^{\op},R)} (V, W) = \Hom_{\PSh(\C^{\op}, R)}(V, W) = \Hom_{\C \Mod} (V, W) = \Hom_{\C \Mod^{\sat}} (V, W).
\]
Furthermore, given a morphism $\alpha: V \to W$ in $\Sh(\BC^{\op}, R)$, its kernel is computed in the category $\C \Mod$; that is, the kernel of $\alpha$ in $\Sh(\BC^{\op}, R)$ is also the kernel of $\alpha$ in $\C \Mod$. But the kernel of $\alpha$ in $\C \Mod$ is also contained in $\C \Mod^{\sat}$. Therefore, kernels in $\Sh(\BC^{\op}, R)$ correspond to kernels in $\C \Mod^{\sat}$ under the identification given by Theorem \ref{main result 1}.

To compute the cokernel of $\alpha$ in $\Sh(\BC^{\op}, R)$, we first take the cokernel
\[
W/\alpha(V) \in \C \Mod,
\]
and then apply the sheafification functor to obtain the cokernel $(W/\alpha(V))^{\sharp} \in \Sh(\BC^{\op},R)$. When we view $\alpha: V \to W$ as a morphism in $\C \Mod^{\sat}$, its cokernel is given by $(\textsl{sec} \circ \textsl{loc}) (W/\alpha(V))$. To deduce the identification of the cokernel of $\alpha$ in $\Sh(\BC^{\op}, R)$ and the cokernel of $\alpha$ in $\C \Mod^{\sat}$, we only need to show that $\sharp$ corresponds to $\mathrm{\sec} \circ \textsl{loc}$ under the identification given by Theorem \ref{main result 1}, which is proved in the next proposition.
\end{proof}

\begin{proposition} \label{sheafification}
The sheafification functor $\sharp: \PSh(\C^{\op}, R) \to \Sh(\BC^{\op}, R)$ corresponds to $\textsl{sec} \circ \textsl{loc}: \C \Mod \to \C \Mod^{\sat}$ under the identification given in Theorem \ref{main result 1}.
\end{proposition}

As explained in Remark \ref{sec}, we regard $\textsl{sec}$ as a functor from $\C \Mod / \C \Mod^{\tor}$ to $\C \Mod^{\sat}$.

\begin{proof}
For $V \in \PSh(\C^{\op}, R)$ and $W \in \Sh(\BC^{\op}, R)$, one has:
\begin{align*}
& \Hom_{\Sh(\BC^{\op}, R)} (V^{\sharp}, W)\\
& \cong \Hom_{\PSh(\C^{\op}, R)} (V, \iota W) & \iota \text{ is the forgetful functor}\\
& \cong \Hom_{\C \Mod} (V, \textsl{inc} (W))\\
& \cong \Hom_{\C \Mod} (V, (\rm{\sec} \circ \textsl{loc} \circ \textsl{inc}) W) & \text{by (4) of Proposition \ref{serre quotient}}\\
& \cong \Hom_{\C \Mod / \C \Mod^{\tor}} (\textsl{loc} (V), (\textsl{loc} \circ \textsl{inc}) W) & \text{by (1) of Proposition \ref{serre quotient}}\\
& \cong \Hom_{\C \Mod / \C \Mod^{\tor}} ((\textsl{loc} \circ \textsl{sec} \circ \textsl{loc}) V, (\textsl{loc} \circ \textsl{inc}) W) & \text{by (2) of Proposition \ref{serre quotient}}\\
& \cong \Hom_{\C \Mod} ((\textsl{sec} \circ \textsl{loc}) V, (\textsl{sec} \circ \textsl{loc} \circ \textsl{inc}) W) & \text{by (1) of Proposition \ref{serre quotient}}\\
& \cong \Hom_{\C \Mod} ((\textsl{sec} \circ \textsl{loc}) V, \textsl{inc} (W)) & \text{by (4) of Proposition \ref{serre quotient}}\\
& \cong \Hom_{\C \Mod^{\sat}} ((\textsl{sec} \circ \textsl{loc}) V, W).
\end{align*}
This establishes the conclusion.
\end{proof}

\begin{remark} \normalfont
It is well known that $\Sh(\BC^{\op}, R)$ is a localization of $\PSh(\C^{\op}, R)$ by the Serre subcategory which consists of $\C$-modules $V$ such that $V^{\sharp}=0$. In the case of atomic topology, the above theorem and proposition give a very transparent description for the localization process.
\end{remark}

In general, given a presheaf $V \in \PSh(\C^{\op}, R)$, it is not an easy task to describe the associated sheaf $V^{\sharp}$, since the definition of $\sharp$ is complicated, see for instance \cite[III.5]{MM}. However, the description of $(\textsl{sec} \circ \textsl{loc}) V$ is much simpler: one takes the torsion free part $V_F$ of $V$, embeds it into an injective hull $I$ (which is saturated) to get a short exact sequence $0 \to V_F \to I \to C \to 0$ of $\C$-modules, and then constructs the pull back of $I \to C$ and $C_T \to C$ to obtain the following commutative diagram such that all rows and columns are exact:
\[
\xymatrix{
0 \ar[r] & V_F \ar[r] \ar@{=}[d] & \widetilde{V} \ar[r] \ar[d] & C_T \ar[d] \ar[r] & 0\\
0 \ar[r] & V_F \ar[r] & I \ar[d] \ar[r] & C \ar[r] \ar[d] & 0\\
 & & C_F \ar@{=}[r] & C_F.
}
\]
Then $V^{\sharp} = (\textsl{sec} \circ \textsl{loc}) V \cong \widetilde{V}$. Indeed, from the middle column we conclude that $\widetilde{V}$ is a saturated module, and
\[
(\textsl{sec} \circ \textsl{loc}) V \cong (\textsl{sec} \circ \textsl{loc}) V_F \cong (\textsl{sec} \circ \textsl{loc}) \widetilde{V} \cong \widetilde{V},
 \]
where the first isomorphism follows by applying the exact functor $\textsl{sec} \circ \textsl{loc}$ to the short exact sequence $0 \to V_T \to V \to V_F \to 0$ and noting that $\textsl{loc}$ sends the torsion module $V_T$ to 0, the second isomorphism follows by applying $\textsl{sec} \circ \textsl{loc}$ to the top row of the above diagram, and the third isomorphism holds since $\widetilde{V}$ is saturated.

In the rest of this subsection we describe another interpretation of $\Sh(\BC^{\op}, R)$ via the notion of morphism categories. Let $\mathfrak{I}$ be the additive category of torsion free injective $\C$-modules (which are precisely injective objects in $\Sh(\BC^{\op}, R)$). Following from Auslander, Reiten and Smal{\o} (see \cite[IV.1]{ARS}), we define the \textit{morphism category} $\Mor(\mathfrak{I})$ as follows:
\begin{enumerate}
\item[$\bullet$] objects are morphisms $\alpha: I^0 \to I^1$ in $\mathfrak{I}$;
\item[$\bullet$] morphisms from $\alpha: I^0 \to I^1$ to $\beta: J^0 \to J^1$ are pairs $(f^0,f^1)$ such that the following diagram commutes
\[
\xymatrix{
I^0 \ar[r]^-{\alpha} \ar[d]^-{f^0} & I^1 \ar[d]^-{f^1}\\
J^0 \ar[r]^-{\beta} & J^1.
}
\]
\end{enumerate}

We now define the functor $\ker: \Mor(\mathfrak{I}) \to \C \Mod^{\sat}$ as follows:
\begin{enumerate}
\item[$\bullet$] for each object $\alpha: I^0 \to I^1$ in $\Mor(\mathfrak{I})$, $\ker\alpha \in \C \Mod^{\sat}$ by Proposition \ref{saturated modules};
\item[$\bullet$] For each morphism $(f^0,f^1)$ in $\Mor(\mathfrak{I})$ from $\alpha: I^0 \to I^1$ to $\beta: J^0 \to J^1$, we let $\ker(f^0,f^1)$ be the unique morphism such that the diagram
    \[
\xymatrix{
0 \ar[r] & \ker\alpha \ar[d] \ar[r] & I^0 \ar[r]^\alpha \ar[d]^{f^0} & I^1 \ar[d]^{f^1}\\
 0 \ar[r] & \ker\beta \ar[r] & J^0  \ar[r]^\beta & J^1 \\
}
\]
is commutative.
\end{enumerate}
It is straightforward to see that $\ker$ is a full and dense functor, but not faithful. However, the reader can easily check that $\ker(f^0,f^1)=0$ if and only if there is a morphism $f: I^1 \to J^0$ such that $f\alpha=f^0$. Therefore, to obtain an equivalence between categories, we need to modulo some relations.

For two objects $\alpha: I^0 \to I^1$ and $\beta: J^0 \to J^1$ in $\Mor(\mathfrak{I})$, we let $\mathfrak{R}(\alpha,\beta)$ be the subgroup of $\Hom_{\Mor(\mathfrak{I})}(\alpha,\beta)$ consisting of those morphisms $(f^0,f^1)$ such that there exists a morphism $f: I^1 \to J^0$ with $f\alpha=f^0$. Then $\mathfrak{R}$ gives a relation on $\Mor(\mathfrak{I})$ in the sense of \cite{ARS}. As Auslander, Reiten and Smal{\o} proved in \cite[Proposition IV.1.2]{ARS}, we have the following result.

\begin{theorem}
The functor $\ker: \Mor(\mathfrak{I}) \to \C \Mod^{\sat}$ induces an equivalence between $\Mor(\mathfrak{I})/\mathfrak{R}$ and $\C \Mod^{\sat}$.
\end{theorem}

In summary, we obtain the following equivalences of categories:
\[
\xymatrix{
\Sh(\BC^{\op}, R) \ar[r]^-{\sim} & \C \Mod^{\sat} \ar[r]^-{\sim} & \C \Mod / \C \Mod^{\tor} \ar[r]^-{\sim} & \Mor(\mathfrak{I})/\mathfrak{R}.
}
\]


\subsection{Sheaf cohomology}

We already know that there is a close relation between the sheafification procedure and the torsion theory in $\C \Mod$. In this subsection we illustrate a relation between sheaf cohomology functors and derived functors $\mathrm{R}^{\bullet} \tau$ of the torsion functor $\tau$.

Given a presheaf $V \in \PSh(\C^{\op}, R)$, we can construct an injective resolution $0 \to V \to I^{\bullet}$ of presheaves. Now for every object $x \in \C^{\op}$, one defines a functor $\Gamma^p_x: \PSh(\C^{\op}, R) \to R \Mod$ by $V \mapsto V_x$, called an \textit{evaluation} functor, which is exact. Similarly, one can define an evaluation functor $\Gamma_x: \Sh(\BC^{\op}, R) \to R \Mod$, which is only left exact. When $\C^{\op}$ has a terminal object, or equivalently, $\C$ has an initial object, the $i$-th sheaf cohomology group at $x$, is defined to be $\mathrm{R}^i \Gamma_x(V)$, where $\mathrm{R}^i \Gamma_x$ is the $i$-th right derived functor of $\Gamma_x$.

To compute $\mathrm{R}^{\bullet} \Gamma_x(V)$, one constructs an injective resolution $0 \to V \to I^{\bullet}$ of sheaves and apply $\Gamma_x$ to obtain a complex $0 \to \Gamma_x V \to \Gamma_x I^{\bullet}$. Cohomology groups of this complex give $\mathrm{R}^{\bullet} \Gamma_x(V)$. For details, please refer to \cite[I.21.2]{Stack}.

The following result relate sheaf cohomology functors to right derived functors of $\tau$.

\begin{theorem} \label{sheaf cohomology}
Suppose that $\C$ has an initial object. For $V \in \Sh(\BC^{\op}, R)$ and object $x$ in $\C^{\op}$, one has $\mathrm{R}^i \Gamma_x(V) \cong (\mathrm{R}^{i+1} \tau (V))_x$ for $i \geqslant 1$, the value of the $\C$-module $\mathrm{R}^{i+1} \tau (V)$ on the object $x$.
\end{theorem}

Note that $\mathrm{R}^{i+1} \tau (V)$ is also a $\C$-module, and hence its value on $x$ is well defined.

\begin{proof}
Firstly we take a short exact sequence $0 \to V \to I^0 \to V^1 \to 0$ in $\C \Mod$ such that $I^0$ is a torsion free injective $\C$-module. Note that this is not an exact sequence in $\Sh(\BC^{\op}, R)$ since $V^1$ might not even be a sheaf, although it is a separated presheaf. Thus we apply the functor $\sharp$ to get the following short exact sequence in $\Sh(\BC^{\op}, R)$:
\[
0 \to V^{\sharp} = V \to (I^0)^{\sharp} = I^0 \to (V^1)^{\sharp} \to 0.
\]

 By our description of sheafification functor in subsection 3.3, we have the following commutative diagram of short exact sequences of $\C$-modules:
\[
\xymatrix{
0 \ar[r] & V^1 \ar[r] \ar@{=}[d] & (V^1)^{\sharp} \ar[r] \ar[d] & \tau V^2 \ar[d] \ar[r] & 0\\
0 \ar[r] & V^1 \ar[r] & I^1 \ar[r] & V^2 \ar[r] & 0.
}
\]

Now applying the evaluation functors $\Gamma_x^p$ and $\Gamma_x$, we obtain the following exact sequences
\begin{align*}
& 0 \to \Gamma^p_x V \to \Gamma^p_x I^0 \to \Gamma_x^p V^1 \to 0\\
& 0 \to \Gamma_x V \to \Gamma_x I^0 \to \Gamma_x ((V^1)^{\sharp}) \to \mathrm{R}^1 \Gamma_x(V) \to 0\\
& 0 \to \Gamma^p_x V^1 \to \Gamma^p_x ((V^1)^{\sharp}) \to \Gamma_x^p (\tau V^2) \to 0.
\end{align*}
Note that $\Gamma^p_x F = \Gamma_x F$ whenever $F \in \Sh(\BC^{\op}, R)$. Therefore, by comparing these sequences, one gets
\[
\mathrm{R}^1 \Gamma_x(V) \cong \Gamma_x^p (\tau V^2) \cong (\tau V^2)_x.
\]
However, it is straightforward to check that $\tau V^2 \cong \mathrm{R}^2 \tau (V)$, so the conclusion holds for $i = 1$. Now replacing $V$ by $(V^1)^{\sharp}$ we deduce the isomorphism for $i = 2$. The conclusion follows from an induction.
\end{proof}

\subsection{Finiteness conditions}

For future applications in the upcoming Section 4, we end this section with some remarks on finiteness conditions on presheaves and sheaves. We follow the terminologies from \cite[Section 3.5]{Po}.

\begin{proposition}
Let $R$ be a commutative ring. Then a presheaf $V \in \PSh(\C^{\op}, R)$ is of finite type if and only if the $\C$-module $V$ is finitely generated.
\end{proposition}

\begin{proof}
This follows from \cite[Chapter 3, Corollary 5.7]{Po} and the remark immediately after this corollary, where the small preaddtive category there can be taken as the $R$-linearization of the category $\C$ considered by us in this paper.
\end{proof}

We denote the category of presheaves of $R$-modules of finite type by $\mathrm{psh}(\C^{\op}, R)$, or $\C \module$. Clearly, when $R$ is Noethrian and the category algebra $R\C$ is locally Noetherian, $\C \module$ is an abelian subcategory of $\C \Mod$. We denote by $\mathrm{sh}(\BC^{\op},R) \subset \C\module$ the full subcategory of $\C \Mod$ consisting of sheaves which, considered as presheaves, are of finite type.

\begin{proposition}
Suppose that $R$ is a commutative Noetherian ring, and that $R\C$ is locally Noetherian. Then $\mathrm{sh}(\BC^{\op}, R)$ is an abelian subcategory of $\Sh(\BC^{\op}, R)$.
\end{proposition}

\begin{proof}
Let $\alpha: V \to W$ be a morphism in $\mathrm{sh}(\BC^{\op}, R)$. Clearly, the kernel of $\alpha$ (viewed as a morphism in $\Sh(\BC^{\op}, R))$ is a sheaf, and is of finite type as a presheaf by the locally Noetherian property of $R\C$. Therefore, the kernel of $\alpha$ actually lies in $\mathrm{sh}(\BC^{\op}, R)$. The cokernel of $\alpha$ is clearly a sheaf, and as a presheaf is of finite type.
\end{proof}

We must warn the reader that presheaves of finite type and sheaves/modules over a ringed site of finite type are quite different, see Remark 4.10.

\section{Some special cases}

In the previous section we have converted the sheaf theory of categories $\C^{\op}$ equipped with atomic topology to the representation theory of $\C$. In particular, the sheaf category $\Sh(\BC^{\op}, R)$ is equivalent to the Serre quotient $\C \Mod / \C \Mod^{\tor}$. Since this quotient category is still mysterious for practical purposes, in this section we consider the sheaf category over some special types of combinatorial categories $\C$, and obtain more transparent descriptions.

Throughout this section we assume that $\C$ satisfies the left Ore condition, so that $\C^{\op}$ is equipped with the atomic topology.

\subsection{Categories with enough maximal objects} \label{type i cats}

In this subsection we consider small skeletal categories $\C$ satisfying the following conditions (called \textit{Type I combinatorial categories}):
\begin{itemize}
\item $\C$ is \textit{directed}: the relation that $x \preccurlyeq y$ if and only if $\C(x, y) \neq \emptyset$ is a partial ordering on the set of objects in $\C$;
\item $\C$ \textit{has enough maximal objects}: for every object $x$ in $\C$, there is another object $y$ such that $x \preccurlyeq y$ and $y$ is maximal with respect to $\preccurlyeq$.
\item if $y$ is maximal with respect to $\preccurlyeq$, then $\C(y, y)$ is a group.
\end{itemize}

Examples of Type I combinatorial categories include: skeletal subcategories of the orbit category of a finite group $G$ consisting of cosets $G/H$ (where $H$ is a subgroup) and $G$-equivariant maps, skeletal subcategories of the category of finite sets and surjections (with the singleton set as the unique maximal element), the poset of open sets with respect to inclusions of a given topological space $X$ (with $X$ as the unique maximal element).

Now let $S$ be the set of maximal objects in $\C$. For $x \in S$, we denote $\C(x, x)$ by $G_x$. The main result of this section is:

\begin{theorem}
Suppose that $\C$ is a Type I combinatorial category. Then one has the following equivalence:
\[
\Sh(\BC^{\op}, R) \cong \prod_{x \in S} RG_x \Mod,
\]
where $\prod_{x \in S} RG_x \Mod$ is the product category of $RG_x \Mod$ indexed by $x \in S$. Explicitly, objects in $\prod_{x \in S} RG_x \Mod$ are families $(V_x)_{x \in S}$ such that $V_x$ is an $RG_x$-module.
\end{theorem}

\begin{proof}
Since we have shown that $\Sh(\BC^{\op}, R)$ is equivalent to $\C \Mod/ \C \Mod^{\tor}$, and it is also easy to see that $\prod_{x \in S} RG_x \Mod \cong \D \Mod$, where $\D$ is the full subcategory of $\C$ whose objects are precisely elements in $S$, one only needs to show that $\C \Mod/ \C \Mod^{\tor} \cong \D \Mod$. The proof of this conclusion is based on \cite[Proposition III.2.5]{Gab}.

The inclusion functor $\D \to \C$ induces a restriction functor $\textsl{res}: \C \Mod \to \D \Mod$, which is exact. It is well known that $\textsl{res}$ has a right adjoint functor $\textsl{coind}: \D \Mod \to \C \Mod$, called the \textit{coinduction} functor, which is a special case of the right Kan extension. Explicitly, given a $\D$-module $V$, $\textsl{coind}(V)$ is defined as follows: for every object $x$ in $\C$, one has
\[
(\textsl{coind}(V))_x = \Hom_{\D \Mod} (R\C(x, -), V).
\]
Furthermore, it is direct to check that $\textsl{res} \circ \textsl{coind}: \D \Mod \to \D \Mod$ is isomorphic to the identity functor on $\D \Mod$, using the fact that objects in $\D$ are maximal objects in $\C$ with respect to $\preccurlyeq$. Therefore, by \cite[Proposition III.2.5]{Gab}, we obtain the following equivalence:
\[
\C \Mod / \mathrm{ker} (\textsl{res}) \cong \D \Mod,
\]
where $\mathrm{ker} (\textsl{res})$ is the full subcategory of $\C \Mod$ consisting of $\C$-modules $V$ such that $\textsl{res} (V) = 0$. It remains to show that $\mathrm{ker} (\textsl{res}) = \C \Mod^{\tor}$, which is done in the next lemma.
\end{proof}

\begin{lemma}
Notation as above. Then $\mathrm{ker} (\textsl{res}) = \C \Mod^{\tor}$.
\end{lemma}

\begin{proof}
Clearly, $\mathrm{ker} (\textsl{res})$ consists of $V \in \C \Mod$ such that $V_x = 0$ for all objects $x$ in $\D$. But every such $\C$-module $V$ is torsion. Indeed, for an arbitrary object $y$ in $\C$ and any $v \in V_y$, by the combinatorial conditions imposed on $\C$, we can find a maximal object $x$ in $\C$ and a morphism $\alpha: y \to x$ such that $\alpha \cdot v \in V_x$. But $x$ is also an object in $\D$, so $V_x = 0$, and consequently $v$ is a torsion element in $V$. This shows that $V$ is torsion, and hence $\mathrm{ker}(\textsl{res}) \subseteq \C \Mod^{\tor}$.

Conversely, if $V$ is a torsion $\C$-module, then $V_x = 0$ for all objects $x$ in $\D$. Otherwise, assume that there exists an object $x$ in $\D$ with $V_x \neq 0$. But since $V$ is torsion, for an arbitrary $0 \neq v \in V_x$, we can find a morphism $\alpha: x \to y$ such that $\alpha \cdot v = 0$. We must have $x= y$ and $\alpha$ is an isomorphism as $x$ is maximal. Consequently, $\alpha \cdot v = 0$ implies that $v = 0$, contradicting the assumption that $v \neq 0$. This shows that $\C \Mod^{\tor} \subseteq \mathrm{ker} (\textsl{res})$.
\end{proof}

\begin{remark} \normalfont
The previous equivalence may be obtained as a special case of the Comparison Lemma, see \cite[C2.2.3]{Jo}. A statement of the same fashion, for all finite EI categories with dense topology, may be found in \cite{WX}.
\end{remark}

\begin{example}
Let $\C$ be the following category: objects are $[n] = \{1, 2, \ldots, n\}$, $n \geqslant 1$, and morphisms are surjections. It is easy to see that $\C$ is a skeletal category of the category of finite sets and surjections, and it has a terminal object $[1]$. Moreover, $\C$ satisfies the left Ore condition. Therefore, by the previous theorem, we deduce that $\Sh(\BC^{\op}, R) \cong R \Mod$.
\end{example}


\subsection{The Nakayama functor} \label{type ii cats}

Recall that a skeletally small category $\C$ is called an \textit{EI category} if every endomorphism in $\C$ is an isomorphism. Examples of EI categories include free categories of quivers without oriented cycles, posets, groups, and some important categories constructed from groups such as transporter categories, orbit categories, Frobenius categories; see for instance \cite{Luck, Webb}.

In this subsection we assume that $\C$ is a skeletally small category satisfying the following conditions (called \textit{Type II combinatorial categories}):
\begin{itemize}
\item $\C$ is a small skeletal EI category;
\item every morphism in $\C$ is a monomorphism;
\item $\C$ is \textit{locally finite}, that is, $|\C(x, y)| < \infty$ for all objects $x$ and $y$;
\item $\C$ is \textit{inwards finite}, that is, for every object $x$, the set $\{y \in \C \mid \C(y, x) \neq \emptyset \}$ is finite.
\end{itemize}

Now let $R$ be a field. We also impose the following conditions on $\C$-modules:
\begin{itemize}
\item (LN): $\C$ is \textit{locally Noetherian} over $R$, that is, every finitely generated $\C$-module is Noetherian;
\item (LS): $\C$ is \textit{locally self-injective}, that is, every projective $\C$-module is also injective;\footnote{This notion is introduced in \cite{GLX}. We remind the reader that we do not require injective $\C$-modules to be projective.}
\item (NV): for every finitely generated $\C$-module $V$ which is not torsion, there exists a representable $\C$-module $P(x) = R\C(x, -)$ such that $\Hom_{\C \Mod} (V, P(x)) \neq 0$.
\end{itemize}

\begin{remark} \normalfont
At a first glance, the conditions imposed on the category $\C$ and $\C$-modules seem very strong and artificial. However, it is very natural for the practice purpose. Indeed, skeletal categories of $\FI$, $\OI$, and $\VI$ and their modules over a field of characteristic 0 all satisfy these requirements; see \cite{GLX} and \cite{GL1}. We also mention that the third condition (NV) imposed on $\C$-modules is slightly different from \cite[Statement (2) of Proposition 3.10]{GLX}. This is because for skeletal categories of $\FI$ and $\VI$, finitely generated torsion modules are precisely finite dimensional modules, but for skeletal categories of $\OI$, there exist finitely generated torsion modules with infinite dimension.
\end{remark}

For a Type II combinatorial category $\C$, We define the \textit{Nakayama functor} as follows:
\[
\nu: \C \module \to \C \fdmod, \quad V \mapsto D (\Hom_{\C \module} (V, \bigoplus_{x \in \Ob(\C)} P(x)))
\]
where $D= \Hom_{R} (-, R)$ is the classical duality functor, $\C \module$ and $\C \fdmod$ are the categories of finitely generated $\C$-modules and finite dimensional $\C$-modules respectively. We also define the \textit{inverse Nakayama functor} to be
\[
\nu^{-1}: \C \fdmod \to \C \module, \quad W \mapsto \Hom_{\C^{\op} \module} (DW, R\C(-, x)).
\]

The following facts have been established in \cite{GLX}: the inwards finite condition guarantees that $\nu$ is well defined (that is, $\nu V$ is finite dimensional); the locally Noetherian condition (LN) guarantees that the category $\C \module$ of finitely generated $\C$-modules is an abelian category, and that $\nu^{-1}$ is well defined (that is, $\nu^{-1} W$ is finitely generated); the locally self-injective condition (LS) implies that $\nu$ is exact. Furthermore, $(\nu, \nu^{-1})$ is an adjoint pair, and $\nu \circ \nu^{-1}$ is isomorphic to the identity functor on $\C \fdmod$. Besides, for every $x \in \Ob(\C)$, $P(x)$ is torsion free since every morphism is a monomorphism, so $\ker (\nu)$ contains all finitely generated torsion modules. On the other hand, by the third condition (NV) on $\C$-modules, $\ker (\nu)$ only contains torsion modules. Therefore, we have the following result:

\begin{theorem} \cite[Theorem 3.6]{GLX}
Let $\C$ be a Type II combinatorial category, and suppose that $\C \Mod$ satisfies conditions (LN), (LS), and (LV). Then the Nakayama functor induces the following equivalence
\[
\xymatrix{
\C \module / \C \module^{\tor} \ar[r]^-{\sim} & \C \fdmod
}
\]
where $\C \module^{\tor}$ is the full subcategory of $\C \module$ of finitely generated torsion modules.
\end{theorem}

More explicitly, we have the following commutative diagram where $\bar{\nu}$ and $\bar{\nu}^{-1}$ give the equivalence $\C \module / \C \module^{\tor} \cong \C \fdmod$, and $\nu \circ \textsl{inc}$ and $\nu^{-1}$ give the equivalence between the category $\C \module^{\sat}$ of finitely generated saturated $\C$-modules and $\C \fdmod$:
\[
\xymatrix{
& & \C \module^{\tor} \ar[d]^{\textsl{inc}}\\
\C \module^{\sat} \ar[rr]^{\textsl{inc}} & & \C \module \ar[rr]<.5ex>^{\nu} \ar[dd]<.5ex>^{\textsl{loc}} & & \C \fdmod \ar[ll]<.5ex>^{\nu^{-1}} \ar@{-->}[lldd]<.5ex>^{\bar{\nu}^{-1}}\\
 \\
& & \C \module / \C \module^{\tor} \ar[uu]<.5ex>^{\textsl{sec}} \ar@{-->}[rruu]<.5ex>^{\bar{\nu}}
}
\]
For more details, please refer to \cite{GLX}.

\begin{remark}
If a Type II combinatorial category $\C$ is subcanonical with respect to the atomic topology, then each representable functor $P(x)=R\C(x,-)$ gives a sheaf of modules on $\C^{\op}$. Thus $\{P(x)\}_{x\in\Ob\C}$ is a set of generators of $\Sh(\BC^{\op},R)$. According to \cite[Page 92]{Po}, it makes sense to talk about ``$\{P(x)\}_x$-finitely generated'' objects in $\Sh(\BC^{\op},R)$. For obvious reasons, we shall simply call them ``\textit{finitely generated sheaves}''. Since $\Sh(\BC^{\op},R)$ is a full subcategory of $\C\Mod$, a sheaf is finitely generated if and only if it is as a presheaf.
\end{remark}

For a Type II combinatorial category $\C$, we denote by $\mathrm{sh}(\BC^{\op}, R)$ the (abelian) full subcategory of $\Sh(\BC^{\op},R)$ consisting of all finitely generated sheaves. It follows from \cite[Section 3.5, Proposition 5.8]{Po} that finitely generated sheaves are of finite type in $\Sh(\BC^{\op},R)$.

\begin{corollary} \label{finitely generated sheaves}
Suppose that $\C$ is a Type II combinatorial category, $\C \Mod$ satisfies conditions (LN), (LS), and (LV), and let $R$ be a field. Then:
\begin{enumerate}
\item the following categories are equivalent:
\[
\xymatrix{
\mathrm{sh}(\BC^{\op}, R) \ar[r]^-{\sim} & \C \module^{\sat} \ar[r]^-{\sim} & \C \module / \C \module^{\tor} \ar[r]^-{\sim} & \C \fdmod;
}
\]
\item every object in $\mathrm{sh}(\BC^{\op}, R)$ is of finite length, and has finite injective dimension;
\item the sheafification $\sharp: \C \module \to \mathrm{sh}(\BC^{\op}, R)$ corresponds to $\nu^{-1} \circ \nu$ under the equivalences given in (1);
\item every indecomposable injective object in $\mathrm{sh}(\BC^{\op}, R)$ is isomorphic to an indecomposable direct summand of $P(x)$ for a certain $x \in \Ob(\C)$;
\item the functor $\nu$ induces a bijection between the set of isomorphism classes of simple objects in $\C \module^{\sat}$ and the set of isomorphism classes of simple $R\C(x, x)$-modules, where $x$ ranges over all objects in $\C$.
\end{enumerate}
\end{corollary}

\begin{proof}
Statement (1) is follows from the finitely generated version of Theorem \ref{equivalences} restricted to $\C \module$ (which is abelian by the local Noetherianity of $\C$ over $R$) as well as the previous theorem.

Statement (2) is also clear since objects in $\C \fdmod$ satisfy these properties.

For Statement (3), we note that the functor $\sharp$ corresponds to $\textsl{sec} \circ \textsl{loc}: \C \module \to \C \module^{\sat}$, but by the above commutative diagram we also have
\[
\textsl{sec} \circ \textsl{loc} \cong (\nu^{-1} \circ \bar{\nu}) \circ (\bar{\nu}^{-1} \circ \nu) = \nu^{-1} \circ (\bar{\nu} \circ \bar{\nu}^{-1}) \circ \nu \cong \nu^{-1} \circ \nu.
\]

For Statement (4), we observe the following facts: $P(x)$ is a torsion free injective $\C$-module, and hence is contained in $\C \module^{\sat}$; $\nu P(x)$ is injective in $\C \fdmod$; and every indecomposable injective object in $\C \fdmod$ is isomorphic to a direct summand of $\nu P(x)$ for a certain object $x$ in $\C$. Now let $I$ be an indecomposable object in $\C \module^{\sat}$ (identified with $\mathrm{sh}(\BC^{\op}, R)$), then $\nu I$ is isomorphic to a direct summand of $\nu P(x)$ for a certain object $x$. Now applying the functor $\nu^{-1}$ we deduce that $I \cong (\nu^{-1} \circ) I$ is isomorphic to a direct summand of $(\nu^{-1} \circ \nu)P(x) \cong P(x)$.

For Statement (5), we note that every simple $R\C(x, x)$-module can be viewed as a simple $\C$-module, and every simple $\C$-module is of this form. For details, please refer to \cite[Corollary (4.2)]{Webb} or \cite{Luck}.
\end{proof}

\begin{remark} \normalfont
The first conclusion in this corollary can be used to construct simple objects in $\mathrm{sh}(\BC^{\op}, R)$. Indeed, since for many examples it is easy to classify all simple objects in $\C \fdmod$, by applying the inverse Nakayama functor one obtains all simple objects in $\mathrm{sh}(\BC^{\op}, R)$.
\end{remark}

\begin{remark}
Given a ringed site $(\BC^{\op},\mathcal{O})$, the characterization of $\mathcal{O}$-modules with finiteness conditions seems to be more subtle, which heavily relies on the structure sheaves on $\BC^{\op}$. For instance if we take $\mathcal{O}$ to be the constant sheaf of rings $\RR$, then $\RR\Mod=\Sh(\BC^{\op},R)$. In order to recognize local types of modules, see \cite[Sec 18.1]{KS} or \cite[Sec 18.23]{Stack}, one may rely on the definition and use various localization morphisms of sites $\BC^{\op}/[i]\to\BC^{\op}$, $i\in\mathbb{N}$, where $\C^{\op}/[i]$ is the category over $[i]$ with induced topology. It is easy to see that $\C^{\op}/[i]\cong([i]\backslash\C)^{\op}$.

Let $\C$ be the skeletal subcategory of $\FI$ consisting of objects $[n]$, $n \in \N$. Then by direct calculation we have an isomorphism of categories $[i]\backslash \C \cong \C$, $\forall i$. The forgetful functor is a natural embedding $j_{[i]} : [i]\backslash \C \to \C$ sending objects $(f,[j])$ to $[j]$ for all $j\ge i$. Thus, the restriction along $j_{[i]}$, ${j_{[i]}}_*: \PSh(\C^{\op},R) \to \PSh(\C^{\op}/[i], R)$ is the \textit{shift functor} $\Sigma^i$ introduced in \cite{CEF}, which is exact, satisfying $\Sigma P([n]) \cong P([n]) \oplus P([n-1])^n$. The restriction has two adjoint functors, the left and right Kan extensions along $j_{[i]}$.

Taking into account the atomic topology on $\C^{\op}$, the functor $j_{[i]}$ induces atompic topology on $\BC^{\op}/[i]$ and thus becomes both continuous and cocontinuous, giving rise to a morphism of ringed sites $(\BC^{\op}/[i], \RR_i) \to (\BC^{\op},\RR)$, where $\RR_i(\cong\RR)$ is the constant sheaf on $\BC^{\op}/[i](\cong\BC^{\op})$. We obtain a functor $\RR\Mod=\Sh(\BC^{\op}, R) \to \Sh(\BC^{\op}/[i], R)=:\RR_i\Mod$, again denoted by ${j_{[i]}}_*$. It has a right adjoint ${j_{[i]}}_*$, as well as a left adjoint ${j_{[i]}}_!$. They come from the above-mentioned Kan extensions.

Although each sheaves $P([n])\in\RR\Mod=\Sh(\BC^{\op}, R)$ is finitely generated in the sense of Remark 4.7, for $n>0$, it is not a module of finite type over $\RR$ in the sense of \cite[Section 18.23]{Stack}, as the dimension of $P([n])_m= R \C([n],[m])$ strictly increases when $m$ does. When $n=0$, $P([0])= \RR$ and it is certainly of finite type.
\end{remark}

Now we give a few examples such that the conclusion of Corollary \ref{finitely generated sheaves} can apply.

\begin{example}
When $R$ is a field of characteristic 0, the skeletal subcategory of the category $\FI$ (resp., $\VI$) consisting of objects $[n]$ (resp., $\mathbb{F}_q^n$), $n \in \mathbb{N}$, satisfies all conditions specified in Corollary \ref{finitely generated sheaves}; see \cite{GLX}.
\end{example}

\begin{example}
When $R$ is a field, the skeletal subcategory of the category $\OI$ consisting of objects $[n]$, $n \in \mathbb{N}$, satisfies all conditions specified in Corollary \ref{finitely generated sheaves}; see \cite{GL1}.
\end{example}

\begin{example}
When $R$ is a field of characteristic 0, the skeletal subcategory of $\FI^m$ (the product category of $m$ copies of $\FI$) consisting of objects $[n_1] \times \ldots \times [n_m]$, $n_i \in \mathbb{N}$, satisfies all conditions specified in Corollary \ref{finitely generated sheaves}; see \cite{Zen}.
\end{example}

A few other interesting examples of Type II combinatorial categories are introduced by Pol and Strickland in \cite{Pol}. Let $\mathcal{G}$ be the category of finite groups and conjugacy classes of surjective group homomorphisms. More explicitly, for two surjective homomorphism $\alpha: G \to H$ and $\beta: G \to H$, we say that $\alpha$ and $\beta$ lie in the same conjugacy class if there exists an element $h \in H$ such that $\alpha (g) = h\beta(g)h^{-1}$ for $g \in G$. A full subcategory $\mathcal{U}$ of $\mathcal{G}$ is called a \textit{multiplicative global family} if it is closed under taking finite products, quotient groups, and subgroups.

\begin{example}
Let $p$ be a prime number and $n$ a natural number. Then each of the following full subcategories of $\mathcal{G}$ is a multiplicative global family:
\begin{itemize}
\item $\mathcal{Z}$: the category of finite abelian groups;
\item $\mathcal{Z}[p^{\infty}]$: the category of finite abelian $p$-groups;
\item $\mathcal{G}[p^n]$: the category of finite groups of exponent dividing $p^n$;
\item $\mathcal{Z}[p^n]$: the category of finite abelian groups of exponent dividing $p^n$;
\item $\mathcal{Z}[p]$: the category of elementary abelian $p$-groups.
\end{itemize}
\end{example}

It is easy to see that the opposite categories of the categories listed in the above example are Type II combinatorial categories. Furthermore, Pol and Strickland proved the following result:

\begin{proposition} \cite[Propositions 12.16 and 15.1]{Pol}
Let $\mathcal{U}$ be a multiplicative global family, and let $R$ be a field of characteristic 0. Then torsion free injective objects in $\mathcal{U}^{\op} \Mod$ coincide with projective objects in it, and every torsion free object can be embedded into a projective one.
\end{proposition}

From this result we may deduce:

\begin{corollary}
Let $\mathcal{U}$ be a multiplicative global family, and let $R$ be a field of characteristic 0. Then the conditions (LS) and (NV) hold. Moreover, $\mathcal{U}$ satisfies the right Ore condition, so one can impose the atomic topology on it.
\end{corollary}

\begin{proof}
The first statement immediately follows from the previous proposition. Now we check that $\mathcal{U}$ satisfies the right Ore condition. Let $[f]: G \to H$ and $[f']: K \to H$ be two morphisms, which are represented by two surjective group homomorphisms $f$ and $f'$ respectively. Then the product $G \times H$ is contained in $\mathcal{U}$ since it is closed under products, so is the fiber product $G \times_K H$ as $\mathcal{U}$ is closed under taking subgroups. One can check that $G \times_K H$ is also the pull-back of $[f]$ and $[f']$ in $\mathcal{U}$. Thus $\mathcal{U}$ has pull-backs, and hence satisfies the right Ore condition.
\end{proof}

Unfortunately, it is still unknown whether $\mathcal{U}^{\op}$ has the locally Noetherian property (LN) over a field $R$ of characteristic 0. Pol and Strickland show that if $\C$ is the opposite category of $\mathcal{Z}[p^{\infty}]$ or $\mathcal{Z}[p^n]$, then $\C$ has this property; see \cite[Theorem 13.15]{Pol}.

In a summary, one has:

\begin{proposition} \label{equivalence of Z}
Suppose that $R$ is a field of characteristic 0, and let $\C$ be a skeletal category of the following categories:
\[
\FI, \, \VI, \, \OI, \, \FI^m, \, \mathcal{Z}[p^{\infty}]^{\op}, \, \mathcal{Z}[p^n]^{\op}.
\]
Then one has
\[
\mathrm{sh}(\BC^{\op}, R) \cong \C \fdmod
\]
\end{proposition}

\begin{proof}
Apply Corollary \ref{finitely generated sheaves}.
\end{proof}

\begin{remark} \normalfont
The requirement that the field $R$ has characteristic 0 can be dropped for $\OI$. For $\mathcal{Z}[p^{\infty}]^{\op}$, $\mathcal{Z}[p^n]^{\op}$, one only requires that the characteristic of $R$ is distinct from $p$.
\end{remark}


\subsection{Local cohomology}

Theorem \ref{sheaf cohomology} relates sheaf cohomology functors to derived functors of the torsion functor $\tau$. In this subsection we give another interpretation of sheaf cohomology for some special categories in terms of local cohomology. For this purpose, we introduce the following definitions.

Let $\C$ be a small skeletal EI category. The \textit{category algebra} $R\C$ is a free $R$-module with basis elements all morphisms in $\C$, and multiplication is given by the following rule: for two morphisms $\alpha$ and $\beta$, we define $\alpha \cdot \beta = \alpha \circ \beta$ if they can be composed, and 0 otherwise. Note that $R\C$ is an associative algebra, and it has the identity if and only if $\C$ has only finitely many objects. Furthermore, there is a natural inclusion functor from $\C \Mod$ to $R\C \Mod$, and thus we may regard a $\C$-module as an $R\C$-module. For details, please refer to \cite{Luck, Mit, Webb}. We shall also point out that the above definitions and results hold for an arbitrary small category $\C$, though in this subsection we only consider small skeletal EI categories.

Now let $\mathfrak{m}$ be the free $R$-module spanned by all non-invertible morphisms in $\C$. It is easy to check that $\mathfrak{m}$ is actually a two-sided ideal of $R\C$. We use this two-sided ideal to define the local cohomology of $\C$-modules, as the second author and Ramos did for the category $\FI$ in \cite{LR}.

Recall that $P(x) = R\C(x, -)$ for $x \in \Ob(\C)$. For $n \geqslant 1$, we define the functor
\[
\fHom_{\C \Mod} (R\C/\mathfrak{m}^n, \bullet): \C \Mod \to \C \Mod, \quad V \mapsto \bigoplus_{x \in \Ob(\C)} \Hom_{\C \Mod} (P(x)/\mathfrak{m}^n P(x), V).
\]
Note that this is well defined since $\bigoplus_{x \in \C} (P(x)/\mathfrak{m}^n P(x), V)$ carries the structure of $\C$-module, and in general
\[
\Hom_{\C \Mod}(R\C / \mathfrak{m}^n, V) \ncong \fHom_{\C \Mod} (R\C/\mathfrak{m}^n, V)
\]
as the former one might not be a $\C$-module (although it is an $R\C$-module).

The functor $\fHom_{\C \Mod} (R\C/\mathfrak{m}^n, \bullet)$ is left exact, and hence has right derived functors. Since taking filtered colimits is exact, the right derived functors can be explicitly described as follows:
\[
\fExt_{\C \Mod}^i (R\C/\mathfrak{m}^n, \bullet) = \bigoplus_{x \in \C} \Ext_{\C \Mod}^i (P(x)/\mathfrak{m}^n P(x), \bullet).
\]
Finally, we define the local cohomology functors as
\[
\HH^i_{\mathfrak{m}} (\bullet) = \varinjlim_n \fExt_{\C \Mod}^i (R\C/\mathfrak{m}^n, \bullet).
\]

Now we impose the following combinatorial conditions on $\C$ (called \textit{Type III combinatorial categories}):
\begin{itemize}
\item $\C$ is a small skeletal EI category;
\item objects in $\C$ are parameterized by $\N$ (that is, $\Ob(\C) = \{x_n \mid n \in \N \}$);
\item $\C(x_m, x_n) \neq \emptyset$ if and only if $m \leqslant n$;
\item the group $G_n = \C(x_n, x_n)$ acts transitively on the morphism set $\C(x_m, x_n)$ form the left hand side.
\end{itemize}

Examples of Type III combinatorial categories include skeletal subcategories of $\FI$ and $\VI$, see \cite{GL1, GLX}. For these categories, we have:

\begin{theorem} \label{local cohomology}
Suppose that $\C$ is a Type III combinatorial category and has an initial object, and let $V \in \Sh(\BC^{\op}, R)$. Then for every object $x$ in $\C$ and $i \geqslant 1$, one has
\[
\mathrm{R}^i \Gamma_x(V) \cong (\HH_{\mathfrak{m}}^{i+1}(V))_x,
\]
the value of the $\C$-module $\HH_{\mathfrak{m}}^{i+1}(V)$ on the object $x$.
\end{theorem}

\begin{proof}
By Theorem \ref{sheaf cohomology}, it suffices to show that
\[
\mathrm{R}^i \tau (V) \cong \HH_{\mathfrak{m}}^i(V)
\]
for $i \geqslant 1$. Since $\mathrm{R}^i \tau (\bullet)$ and $\HH_{\mathfrak{m}}^i (\bullet)$ are the $i$-th derived functors of $\tau$ and $\HH_{\mathfrak{m}}^0(\bullet)$ respectively, we only need to show
\[
\HH_{\mathfrak{m}}^0 (\bullet) = \varinjlim_n \fHom_{\C \Mod} (R\C/\mathfrak{m}^n, \bullet) \cong \tau(\bullet).
\]
That is, for every $W \in \C \Mod$, $\HH_{\mathfrak{m}}^0 (W) = \tau W = W_T$ (note that both $\tau$ and $\HH^0_{\mathfrak{m}}$ are defined in $\C \Mod = \PSh(\C^{\op}, R)$ rather than $\Sh(\BC^{\op}, R)$), or equivalently, $(\HH_{\mathfrak{m}}^0(W))_x = (W_T)_x$ for every object $x$ in $\C$.

It is easy to see that $\Hom_{\C \Mod} (P(x)/\mathfrak{m}^n P(x), W)$ consists of elements $w$ (the images of the module homomorphism determined by $1_x \mapsto w$) in $W_x$ such that $f \cdot w = 0$ for all $f \in \mathfrak{m}^n$. Therefore,
\[
\Hom_{\C \Mod} (P(x)/\mathfrak{m}^n P(x), W) \subseteq (W_T)_x,\]
and hence
\[
(\HH_{\mathfrak{m}}^0(W))_x = \varinjlim_n \Hom_{\C \Mod} (P(x)/\mathfrak{m}^n P(x), W) = \bigcup_{n \geqslant 1} \Hom_{\C \Mod} (P(x)/\mathfrak{m}^n P(x), W) \subseteq (W_T)_x.
\]
On the other hand, if $w \in W_x$ is a torsion element, then there exists a morphism $f: x \to y$ such that $f \cdot w = 0$. Clearly, $y \neq x$ since otherwise $f$ is an isomorphism in $\C$. Thus $f \in \mathfrak{m}$. Since objects in $\C$ are parameterized by $\N$, we can find $m, n \in \N$ such that $x = x_m$ and $y = x_n$ with $m < n$. Let $s = n -m$. We claim that $\mathfrak{m}^s \cdot v = 0$. If this holds, then
\[
w \in \Hom_{\C \Mod} (P(x)/\mathfrak{m}^s P(x), W) \subseteq \varinjlim_n \Hom_{\C \Mod} (P(x)/\mathfrak{m}^n P(x), W) = (\HH_{\mathfrak{m}}^0(W))_x,
\]
and the conclusion of this theorem is established.

Now we prove the claim. Let $g: x_a \to x_b$ be a morphism contained in $\mathfrak{m}^s$. If $a \neq m$, then $g \cdot w = 0$ trivially holds, so we can assume that $a = m$. Since $g \in \mathfrak{m}^s$, it can be written as a composite of $s$ non-invertible morphisms. Consequently, $b \geqslant a + s = n$, so we can find a morphism $h: y = x_n \to x_b$. Note that $hf \in \C(x_m, x_b)$. Since $\C(x_b, x_b)$ acts transitively on $\C(x_m, x_b)$, we can also find an automorphism $\delta: x_b \to x_b$ satisfying $\delta hf = g$. Therefore, $g \cdot w = (\delta h f) \cdot w = 0$ as well. This finishes the proof of the claim.
\end{proof}


\section{Discrete representations of topological groups}


In this section we extend the classical result of Artin from sheaves of sets to sheaves of modules, and hence build a bridge connecting sheaves of modules and representations of topological groups. Throughout this section let $G$ be a topological group, and let $X$ be a discrete set (that is, a set equipped with the discrete topology). Note that $\Aut(X)$ is also a topological group with a topology inherited from the product topology on $X^X$. Let $\rho: G \to \Aut(X)$ be a group homomorphism. We say that the action of $G$ on $X$ induced by $\rho$ is \textit{discrete} if $\rho$ is continuous with respect to the topologies of $G$ and $\Aut(X)$, and call $X$ a \textit{discrete $G$-set}.\footnote{In \cite[III.9]{MM} the action is called continuous, and $X$ is called a continuous $G$-set.} It is easy to see that $\rho$ is continuous if and if only for every $x \in X$, the stablizer subgroup $\Stab_G(x)$ is an open subgroup of $G$.


\subsection{An equivalence between sheaves and discrete representations}

Following \cite[III.9]{MM}, let $\mathbf{B}G$ be the category of discrete $G$-sets, and morphisms between two $G$-sets are $G$-equivariant maps (which are automatically continuous since $G$-sets are equipped with discrete topology). Let $\mathbf{S}G$ be the full subcategory of $\mathbf{B}G$ consisting of objects of the form $G/H$ where $H$ is an open subgroup (thus the quotient topology on $G/H$ is discrete). For a cofinal system of open subgroups $\mathcal{U}$ (that is, every open subgroup contains a member in $\mathcal{U}$), we let $\mathbf{S}_{\mathcal{U}}G$ be the full subcategory of $\mathbf{S}G$ consisting of objects of the form $G/H$ with $H \in \mathcal{U}$. Morphisms in $\mathbf{S}_{\mathcal{U}}G$ are induced by elements in $G$. Explicitly, every morphism from $G/H$ to $G/K$ is of the form $\alpha_g: xH \mapsto xg^{-1}K$ such that $gHg^{-1} \leqslant K$. In literature $\mathbf{S}G$ and $\mathbf{S}_{\mathcal{U}}G$ are often called \textit{orbit categories} associated to the set of open subgroups and the set $\mathcal{U}$ respectively.

The following classical result establishes connections between sheaves of sets over orbit categories and discrete $G$-sets.

\begin{theorem} \cite[Theorems 1 and 2]{MM} \label{mm}
Let $G$ be a topological group and $\mathcal{U}$ be a cofinal system of open subgroups of $G$. Then one has the following equivalences of categories
\[
\Sh(\mathbf{S}G) \cong \mathbf{B}G \cong \Sh(\mathbf{S}_{\mathcal{U}}G),
\]
where all sheaves are taken with respect to the atomic topologies.
\end{theorem}

The first equivalence is given by two functors
\begin{align*}
\phi: \mathbf{B}G \to \Sh(\mathbf{S}G),\\
\psi: \Sh(\mathbf{S}G) \to \mathbf{B}G.
\end{align*}
Explicitly, given $X \in \mathbf{B}G$ and an open subgroup $H$, one lets
\[
\phi(X)(H) = X^H = \{ x \in X \mid h \cdot x = x, \, \forall h \in H \}.
\]
Conversely, given $F \in \Sh(\mathbf{S}G)$, one sets
\[
\psi(F) = \varinjlim_{H} F(H)
\]
where the colimit is taken over the set of all open subgroups of $G$ ordered by inclusion. The second equivalence can be obtained in a similar way.

Now let $R$ be a commutative ring. Since a sheaf of $R$-modules is considered as a contravariant functor into $R \Mod$ meeting the existence and uniqueness condition of amalgamations, the preceding equivalences in Theorem \ref{mm} automatically induce the following equivalences:
\[
RG \DMod \cong \Sh(\mathbf{S}G, R) \cong \Sh(\mathbf{S}_{\mathcal{U}}G, R)
\]
where all sheaves are taken over atomic topologies, and $RG \DMod$ is the category of all \textit{discrete representations} $V$ of $G$ (or discrete $RG$-modules); that is, $V$ is an $RG$-module equipped with discrete topology such that the action of $G$ on $V$ is continuous with respect to the topologies of $G$ and $V$. Clearly, an $RG$-module $V$ is discrete if and only if $V$ is a discrete $G$-set, or equivalently, for every $v \in V$, the stabilizer subgroup $\Stab_G(v)$ is open. By this criterion, submodules and quotient modules of discrete modules are also discrete. Furthermore, the $R$-linearization of a discrete $G$-set is a discrete representation.

\begin{lemma} \label{discrete sets and modules}
Let $X$ be a discrete $G$-set and let $R$ be a commutative ring. Then the free $R$-module $\underline{X}$ (with discrete topology) spanned by $X$ is a discrete representation of $G$.
\end{lemma}

\begin{proof}
The induced action is defined as follows: for $v = r_1 x_1 + \ldots + r_n x_n \in \underline{X}$ with $r_i \in R$ and $x_i \in x$, and $g \in G$, one sets $g \cdot v = r_1 (g \cdot x_1) + \ldots r_n (g \cdot x_n)$. Since $\underline{X}$ is equipped with the discrete topology, $\underline{X}$ is a discrete representation if and only if $\Stab_G(v)$ is an open subgroup of $G$. However, one knows that
\[
\Stab_G(x_i) \cap \ldots \cap \Stab_G(x_n) \leqslant \Stab_G(x),
\]
and each $\Stab_G(x_i)$ is open since $X$ is a discrete $G$-set. Therefore, $\Stab_G(x)$ is also open since it contains an open subgroup.
\end{proof}

\begin{remark} \normalfont
It is easy to see that $\Stab_G(v)$ in the above proof acts on the finite set $\{x_i \mid i \in [n] \}$, and $\Stab_G(x_i) \cap \ldots \cap \Stab_G(x_n)$ is a normal subgroup of $\Stab_G(v)$. Moreover, the quotient group acts on this finite set faithfully. Therefore, the quotient group is isomorphic to a subgroup of $\Aut([n])$. Consequently, $\Stab_G(x)$ is a finite disjoint union of open sets, each of which is homeomorphic to $\Stab_G(x_i) \cap \ldots \cap \Stab_G(x_n)$.
\end{remark}

\begin{remark} \normalfont
The regular $RG$-module $RG$ is discrete if and only if $G$ is equipped with discrete topology. Indeed, consider $v = 1_R1_G \in RG$. Then $\Stab_G(v) = 1_G$. Thus the regular module is discrete only if the trivial subgroup is open, if and only if $G$ is equipped with the discrete topology.
\end{remark}


We illustrate applications of the above theorem by a few examples.

\begin{example} \normalfont
Let $G = \Aut(\N)$ be the set of all permutations on $\N$ with the topology inherited from the product topology of $\N^{\N}$ (which is nothing but the pointwise convergence topology), and let $\mathcal{U}$ be the set of subgroups $U_S = \mathrm{Stab}_G(S)$ where $S$ is a finite subset of $\N$. Then $G$ is a topological group and $\mathcal{U}$ is a cofinal system of open subgroups of $G$. Furthermore, $\mathbf{S}_{\mathcal{U}}G$ is equivalent to the full subcategory of $\FI^{\op}$ consisting of finite subsets of $\N$, which is equivalent to $\FI^{\op}$. Therefore, by the above theorems, one has
\[
\mathbf{B}\Aut(\N) \cong \Sh(\FI^{\op})
\]
which is called the \textit{Schanule topos}, and
\[
R\Aut(\N) \DMod \cong \Sh(\FI^{\op}, R).
\]
For details, see \cite[III.9]{MM} as well as \cite[Examples A2.1.11(h) and D3.4.10]{Jo}.
\end{example}

\begin{example} \normalfont
Let $G = S_{\infty}$ be the subgroup of $\Aut(\N)$ consisting of permutations on $\N$ fixing all but finitely many elements in $\N$; that is, $G = \varinjlim_n \Aut([n])$, called the \textit{infinite symmetric group} over countably many elements. Equip $G$ with the topology inherited from $\Aut(\N)$. Then $\mathcal{U} = \{ \Stab_G(S) \mid S \subset \N, \, |S| < \infty\}$ is a fundamental system of open subgroups of the identity in $G$. As in \cite[III.9]{MM}, one can show that $\mathbf{S}_{\mathcal{U}}G$ is also equivalent to the full subcategory of $\FI^{\op}$ consisting of finite subsets of $\N$. Thus we have
\[
\mathbf{B} S_{\infty} \cong \Sh(\FI^{\op})
\]
and
\[
RS_{\infty} \DMod \cong \Sh(\FI^{\op}, R).
\]
In particular, $RS_{\infty} \DMod \cong R\Aut(\N) \DMod$. This is not surprising as $S_{\infty}$ is a dense subgroup of $\Aut(\N)$, a fact which can be checked by using only the definition, so their continuous representations with respect to an arbitrary topology (not only the discrete topology) can be identified.
\end{example}

\begin{example} \normalfont \label{fim}
Let $m$ be a positive integer and let $\Aut(\N)^m$ be the product group of $m$ copies of $\Aut(\N)$. Equip $G$ with the product topology. Then one can show
\[
R\Aut(\N)^m \DMod \cong \Sh((\FI^m)^{\op}, R)
\]
where $\FI^m$ is the product category of $m$ copies of $\FI$.
\end{example}

\begin{example} \normalfont \label{orbit category of VI}
Let $\mathbb{F}_q$ be a finite field, and let $V = \mathbb{F}_q^{\N}$ be the vector space with basis elements indexed by $\N$; or equivalently,
\[
V = \{ f: \N \to \mathbb{F}_q \mid f(n) = 0 \text{ for } n \gg 0 \}.
\]
Equip $V$ with the discrete topology. Then $V$ is a topological vector space. Define $\mathrm{End}(V)$ to be the set of all maps from $V$ to itself. This is a topological semigroup equipped with the product topology. In particular, for $f \in \mathrm{End}(V)$, a fundamental system of open neighborhoods of $f$ is given by
\[
U_S = \{f' \in \mathrm{End}(V) \mid f'|_S = f|_S \}
\]
where $S$ ranges over all finite subsets of $V$.

Let $\Aut(V)$ be the topological group consisting of invertible elements in $\mathrm{End}(V)$. This is a topological group. In particular, for $\mathrm{Id} \in \Aut(V)$, the family of following open subgroups is a fundamental system of open neighborhoods:
\[
\Stab_{\Aut(V)}(S) = \{f \in \Aut(V) \mid f(s) = s, \, \forall s \in S \}
\]
where $S$ ranges over all finite subsets of $V$.

The automorphism group $\Aut(V)$ has the following subgroup respecting the vector space structure of $V$:
\[
G = \mathrm{GL}(V) = \{f \in \Aut(V) \mid f \text{ is a linear map} \}.
\]
A fundamental system of open neighborhoods of the identity is given by the following open subgroups
\[
\Stab_G(S) = \{f \in G \mid f(s) = s, \, \forall s \in S \} = \Stab_G (V(S)),
\]
where $V(S)$ is the $\mathbb{F}_q$-space spanned by a finite subset $S$ of $V$, and $\Stab_G (V(S))$ consists of elements in $G$ fixing every element in $V(S)$.

Let $\mathcal{U} = \{\Stab_G(S) \mid S \subset V, \, |S| < \infty \}$. Then $\mathcal{U}$ is a cofinal system of open subgroups of $G$.
\end{example}

We have:

\begin{proposition}\label{VI}
Notation as above. Then $\mathbf{S}_{\mathcal{U}}G$ is equivalent to the category $\VI^{\op}$, and hence
\[
R\mathrm{GL}(\mathbb{F}_q^{\N}) \DMod \cong \Sh(\VI^{\op}, R).
\]
\end{proposition}

\begin{proof}
To show this, we define a contravariant functor $\rho: \mathbf{S}_{\mathcal{U}} G \to \VI$ as follows. It sends an object $G/\Stab_G(S)$ to $V(S)$ in $\VI$. Clearly, $\rho$ is essentially surjective. Let
$$\alpha \in \mathbf{S}_{\mathcal{U}}G (G/\Stab_G(S), G/\Stab_G(T))$$ be a morphism. Then $\alpha$ can be represented by an element $g \in G$ such that $g \Stab_G(S) g^{-1} \subseteq \Stab_G(T)$. This happens if and only if $g^{-1} (T) \subseteq S$; that is, the restriction of $g^{-1} |_{V(T)}$ is contained in $\VI(V(T), V(S))$. We thus set $\rho(\alpha) = g^{-1} |_{V(T)}$.

We complete the proof by showing that this functor is well defined, faithful, and full. Suppose that
$$\alpha \in \mathbf{S}_{\mathcal{U}}G (G/\Stab_G(S), G/\Stab_G(T))$$
are represented by two different elements $g, h \in G$. This happens if and only if $g^{-1} \Stab_G(T) = h^{-1} \Stab_G(T)$, if and only if $hg^{-1} \in \Stab_G(T)$. But this is true if and only if $h^{-1}(t) = g^{-1}(t)$ for all $t \in T$; that is, $h^{-1} |_{V(T)} = g^{-1} |_{V(T)}$ as $g$ and $h$ are linear maps. Thus $\rho$ is well defined. This argument also shows that $\rho$ is faithful. Since every morphism in  $\VI(V(T), V(S))$ can be extended to an element in $G$, the functor $\rho$ is also full. The conclusion then follows.
\end{proof}

\begin{example} \normalfont
Let $G = \mathrm{GL}_{\infty} (\mathbb{F}_q) = \varinjlim_n \mathrm{GL}_n(\mathbb{F}_q)$. Using the same argument as the previous example, we can show
\[
\mathbf{B} \mathrm{GL}_{\infty} (\mathbb{F}_q) \cong \Sh(\VI^{\op})
\]
and
\[
R\mathrm{GL}_{\infty} (\mathbb{F}_q) \DMod \cong \Sh(\VI^{\op}, R)
\]
as well. In particular, $R\mathrm{GL}(\mathbb{F}_q^{\N}) \DMod \cong R\mathrm{GL}_{\infty} (\mathbb{F}_q) \DMod$. This is not surprising as well since $\mathrm{GL}_{\infty} (\mathbb{F}_q)$ is a dense subgroup of $\mathrm{GL}(\mathbb{F}_q^{\N})$.
\end{example}

\begin{example} \normalfont \label{SL}
Let $G = \mathrm{SL}_{\infty} (\mathbb{F}_q) = \varinjlim_n \mathrm{SL}_n(\mathbb{F}_q)$, which is a subgroup of $\mathrm{GL}(\mathbb{F}_q^{\N})$. Then one can show $\mathbf{B}G \cong \Sh(\VI^{\op})$ by mimicking the construction in Example \ref{orbit category of VI} and noting the following facts: a linear injection $f: V(S) \to V(T)$ can extend to an element $\tilde{f} \in \mathrm{SL}_n(V(T))$ via viewing $V(S)$ as a subspace of $V(T)$; and $\tilde{f}$ can be regarded as an element in $G$ in a natural way. Consequently, one has
\[
R \mathrm{SL}_{\infty} (\mathbb{F}_q) \DMod \cong \Sh(\VI^{\op}, R).
\]
\end{example}

\begin{example} \normalfont
Let $G = \Aut(\mathbb{Q}, \leqslant)$ be the group of all order-preserving permutations on the poset $(\mathbb{Q}, \leqslant)$, the set of rational numbers with its natural linear order. It can be topologized by letting $\mathcal{U} = \{\Stab_G (S) \mid S \subset \mathbb{Q}, \, |S| < \infty \}$ be a fundamental system of open neighborhoods of the identity element. Then $\mathbf{S}_{\mathcal{U}}G \cong \OI^{\op}$, and hence
\[
\mathbf{B}\Aut(\mathbb{Q}, \leqslant) \cong \Sh(\OI^{\op})
\]
and
\[
 R\Aut(\mathbb{Q}, \leqslant) \DMod \cong \Sh(\OI^{\op}, R).
\]
For details, see \cite[Example D3.4.1]{Jo}.

\end{example}

Using the technique illustrated in the above examples, the reader may consider more cases such as orthogonal groups, symplectic groups, and unipotent groups over $\mathbb{F}_q$.


\section{Discrete representations of $\Aut(\mathbb{N})$}

In this section we consider discrete representations of $G = \Aut(\N)$. Let $\C$ be the skeletal category of $\FI$ consisting of objects $[n]$, $n \in \N$. Then from previous sections we deduce the following equivalences of abelian categories:
\[
RG \DMod \cong \Sh(\BC^{\op}, R) \cong \C \Mod^{\sat} \cong \C \Mod /\C \Mod^{\tor}.
\]
Furthermore, when $R$ is a field of characteristic 0, we have:
\[
RG \dmodule \cong \mathrm{sh}(\BC^{\op}, R) \cong \C \module^{\sat} \cong \C \module /\C \module^{\tor} \cong \C \fdmod
\]
where $RG \dmodule$ is the category of finitely generated discrete representations of $G$, which will be defined later. We will use these equivalences to explicitly construct all simple discrete representations of $G$.


\subsection{Simple sheaves of $\FI$}

In this subsection we let $R$ be a field of characteristic 0, and classify simple objects in $\Sh(\BC^{\op}, R)$. The reader is suggested to refer to \cite{CEF} for details on representation theory of $\FI$ over a field of characteristic 0. By Corollary \ref{finitely generated sheaves}, isomorphism classes of simple objects in $\Sh(\BC^{\op}, R)$ are isomorphism classes of $\nu^{-1} S$ where $S$ is a simple $\C$-module. Thus we get the following result:

\begin{proposition}
Isomorphism classes of simple objects in $\Sh(\BC^{\op}, R)$ are parameterized by all possible partitions of $[n]$ for all $n \in \N$.
\end{proposition}

\begin{proof}
By Corollary \ref{finitely generated sheaves}, isomorphic classes of simple $\C$-modules are parameterized by isomorphism classes of simple $R\Aut([n])$-modules with $n \in \N$, while the latter ones are indexed by partitions of $[n]$ by a classical result of representation theory of symmetric groups.
\end{proof}

Note that in $\C \fdmod$ every simple object is the socle of an indecomposable injective object. This of course holds for $\mathrm{sh}(\BC^{\op}, R)$. However, by Statement (4) of \ref{finitely generated sheaves}, an indecomposable injective object in $\mathrm{sh}(\BC^{\op}, R)$ is precisely an indecomposable projective $\C$-module $P(\lambda)$, which by \cite{CEF}, is induced by a simple $R\Aut([n])$-module corresponding to a partition $\lambda$ of $[n]$ for a certain $n \in \N$. Therefore, simple objects in $\mathrm{sh}(\BC^{\op}, R)$ are just simple socle of $P(\lambda)$, where $\lambda$ ranges over all partitions of $[n]$ for all $n \in \N$.

Explicitly, let $L_{\lambda}$ be a simple $R\Aut([n])$-module corresponding to a partition $\lambda$ of $[m]$. Then for the corresponding indecomposable projective $\C$-module $P(\lambda)$, one has
\[
P(\lambda) \cong R\C \otimes_{R\Aut([m])} L_{\lambda} \cong P(m) \otimes_{R\Aut([m])} L_{\lambda}
\]
where $R\Aut([m]) = R\C([m], [m])$ is viewed as a subalgebra of $R\C$ and $P(m) = R\C([m], -)$. Furthermore, one has $(P(\lambda))_n = 0$ for $n < m$, and for $n \geqslant m$, one has
\[
(P(\lambda))_n = \bigoplus_{\mu} L_{\mu}
\]
where $\mu$ is the partition of $[n]$ whose corresponding Young diagram $Y_{\mu}$ is obtained by adding one box to $n - m$ distinct columns of $Y_{\lambda}$. As a consequence, we see that simple summands in $(P(\lambda))_n$ are pairwise non-isomorphic for $n \geqslant m$.

The following lemma gives a criterion to determine whether a submodule of $P(\lambda)$ is a simple object in $\C \module^{\sat}$; or equivalently, a simple object in $\mathrm{sh}(\BC^{\op}, R)$.

\begin{lemma}
Let $\lambda$ be a partition of $[m]$. A saturated submodule $V$ of $P(\lambda)$ is a simple object in $\C \module^{\sat}$ if and only if $V_n$ is a nonzero simple $R\Aut([n])$-module for all $n \geqslant m$.
\end{lemma}

\begin{proof}
Suppose that all $V_n$ are nonzero simple $R\Aut([n])$-modules for $n \geqslant m$. Let $U$ be a nonzero subobject of $V$ in $\C \module^{\sat}$. Then there exists a certain $N \in \N$ such that $U_N \neq 0$, and hence $U_n \neq 0$ for $n \geqslant N$ as $U$ is a torsion free module. Thus the quotient module $V/U$ must satisfy $(V/U)_n = 0$ for $n \geqslant N$, so $V/U$ is a finite dimensional module. Applying the sheafification functor to $0 \to U \to V \to V/U \to 0$ of $\C$-modules we obtain a short exact sequence of sheaves $0 \to U^{\sharp} \to V^{\sharp} \to 0$ since $(V/U)^{\sharp} = 0$. Thus $U = U^{\sharp} = V^{\sharp} = V$ (since both $U$ and $V$ are saturated modules, we have $U = U^{\sharp}$ and $V = V^{\sharp}$). Consequently, $V$ is a simple object in $\C \module^{\sat}$.

Conversely, if there is a certain $n \geqslant m$ such that $V_n$ is not simple, then we can write
\[
V_n = L_{\mu_1} \oplus \ldots \oplus L_{\mu_r}
\]
where $\mu_i$'s are partitions of $[n]$ and $r \geqslant 2$. We can also assume that $\mu_1$ is the maximal partition with respect to the lexicographic order. Let $U$ be the $\C$-submodule of $V$ generated by $L_{\mu_2}, \, \ldots, \, L_{\mu_r}$. We claim that $V/U$ is not torsion, and hence $V$ is not a simple object in $\C \module^{\sat}$. Indeed, applying the Nakayama functor we obtain an exact sequence $0 \to \nu U \to \nu V \to \nu (V/U) \to 0$. Since $U$ and $V/U$ are not torsion, the first and last terms in the sequence are nonzero. Therefore, $\nu V$ is not simple in $\C \fdmod$, so $V \in \C \module^{\sat}$ is not simple as well.

Now we turn to the proof of the above claim. Since $V$ is a submodule of $P(\lambda)$, and by the structure of this projective module, we know that for $l \geqslant n$, $U_l$ contains a simple summand corresponding to a partition of $[l]$ whose Young diagram is obtained by adding $l -n$ boxes to the first row of the Young diagram of $\mu_1$. However, since $\mu_1$ is the maximal partition among $\{ \mu_i \mid 1 \leqslant i \leqslant r\}$, the $\C$-submodule $U$ generated by $\mu_i$ for $i \geqslant 2$ cannot contain such a summand in $U_l$. Consequently, for $l \geqslant n$, $(V/U)_l \neq 0$. Thus $V/U$ is not finite dimensional, and hence not torsion. This finishes the proof.
\end{proof}

Given $m \in \N$ and a partition
\[
\lambda: \lambda_1 \geqslant \lambda_2 \geqslant \ldots \geqslant \lambda_r
\]
of $[m]$, we define a \textit{uniform partition} $\boldsymbol{\lambda}$ for all $n \geqslant m + \lambda_1$. Explicitly, $\boldsymbol{\lambda}(n)$ is the following partition
\[
n - \lambda_1 \geqslant \lambda_1 \geqslant \lambda_2 \geqslant \ldots \geqslant \lambda_r,
\]
of $[n]$. Denote the simple $R\Aut([m])$-module corresponding to this partition by $L_{\boldsymbol{\lambda}(m)}$. When $n < m + \lambda_1$, we let $L_{\boldsymbol{\lambda}(m)}$ be the zero module by convention.

In \cite[Proposition 3.4.1]{CEF}, Church, Ellenberg, and Farb described a $\C$-module $L_{\boldsymbol{\lambda}}$ of the following structure:
\[
\xymatrix{
L_{\boldsymbol{\lambda}(0)} \ar[r] & L_{\boldsymbol{\lambda}(1)} \ar[r] & L_{\boldsymbol{\lambda}(2)} \ar[r] & \ldots
}
\]
The main result of this subsection asserts that these modules exhaust all simple objects in $\C \module^{\sat}$ up to isomorphism.

\begin{theorem} \label{simple FI-sheaves}
For every partition $\lambda$ of $[m]$, $m \in \N$, the above $\C$-module $L_{\boldsymbol{\lambda}}$ is a simple saturated module, and the correspondence $\lambda \mapsto L_{\boldsymbol{\lambda}}$ is a bijection between partitions of $[m]$, $m \in \N$ and isomorphism classes of simple objects in $\C \module^{\sat}$.
\end{theorem}

\begin{proof}
We firstly show that $L_{\boldsymbol{\lambda}}$ is contained in $\C \module^{\sat}$ and is a simple object in it. The proof of this fact heavily relies on the explicit structure of $P(\lambda)$ and Pieri's formula. The proof of \cite[Proposition 3.4.1]{CEF} shows that $L_{\boldsymbol{\lambda}}$ is a submodule of $P(\lambda)$. Therefore, to show that $L_{\boldsymbol{\lambda}}$ is saturated, one needs to check that the quotient module $P(\lambda) / L_{\boldsymbol{\lambda}}$ is torsion free. This condition is equivalent to saying that for every simple summand $L_{\mu}$ of the $R\Aut([n])$-module $P(\lambda)_n$ with $\mu \neq \boldsymbol{\lambda}(n)$, $n \geqslant m + \lambda_1 - 1$, one has $\alpha(L_{\mu}) \nsubseteq L_{\boldsymbol{\lambda}(n+1)}$ for every morphism $\alpha \in \C([n], [n+1])$; or equivalently, $R\C([n], [n+1]) L_{\mu} \nsubseteq L_{\boldsymbol{\lambda}(n+1)}$ since $\C([n], [n+1])$ as a $\C([n+1], [n+1])$-set has only one orbit. However, since $\mu \neq \boldsymbol{\lambda}(n)$, by \cite[Lemma 3.2.3]{CEF}, the $R\Aut([n+1])$-module $R\C([n], [n+1]) L_{\mu}$ contains a simple summand corresponding to the partition $\mu_1 + 1 \geqslant \mu_2 \geqslant \ldots \geqslant \mu_r$, which is different from the partition $\boldsymbol{\lambda}(n+1)$. Thus $R\C([n], [n+1]) L_{\mu}$ cannot be contained in $L_{\boldsymbol{\lambda}(n+1)}$. In this way we prove that $L_{\boldsymbol{\lambda}}$ is a saturated module. Furthermore, by the previous lemma, it is a simple saturated module since its value on each object is either 0 or a simple module.

Now we show that every simple object $V \in \C \module^{\sat}$ is isomorphic to $L_{\boldsymbol{\lambda}}$ for a partition $\lambda$ of a certain $[m]$. Since every simple object in $\C \module^{\sat}$ is the simple socle of an indecomposable projective $\C$-module, there exists a partition $\lambda$ of a certain $[m]$ such that $V$ is isomorphic to the simple socle of $P(\lambda) \in \C \module^{\sat}$. We show that $V_i \cong L_{\boldsymbol{\lambda}(i)}$ as $R\Aut([i])$-modules for every $i \in \N$. Firstly, $V_i$ cannot contain a simple summand $L_{\mu}$ such that $\mu$ is a partition of $i$ not equal to $\boldsymbol{\lambda}(i)$, since otherwise $R\C(i, i+1) \cdot L_{\mu} \subseteq V_{i+1}$ will contain at least two simple summands (see \cite[Proposition 3.4.1, Lemma 3.2.3]{CEF}), contradicting the fact that $V_{i+1}$ is simple (by the previous lemma). Consequently, $V_i$ is either 0 or isomorphic to $L_{\boldsymbol{\lambda}(i)}$, and hence $V$ is a submodule of $L_{\boldsymbol{\lambda}}$ up to isomorphism. Therefore, we have an exact sequence $0 \to V \to L_{\boldsymbol{\lambda}} \to (L_{\boldsymbol{\lambda}}/V)^{\sharp} \to 0$ of saturated modules. Since the middle term is simple and the first term is nonzero, we get $V \cong L_{\boldsymbol{\lambda}}$ as desired.
\end{proof}

\begin{remark} \normalfont
As mentioned before, $L_{\boldsymbol{\lambda}}$ is the simple socle of $P(\lambda)$ in $\C \module^{\sat}$. For example, when $\lambda$ is the unique partition of $0$, $L_{\boldsymbol{\lambda}}$ coincides with the trivial $\C$-module $V$ such that $V_n = R$ for $n \in \N$ and $V(\alpha)$ is the identity map for every morphism $\alpha$ in $\C$. When $\lambda$ is the unique partition of $1$, $L_{\boldsymbol{\lambda}}$ is the simple socle of $P(\lambda)$ with the following structure
\[
\xymatrix{
L_{\boldsymbol{\lambda}(0)} \ar[r] & L_{\boldsymbol{\lambda}(1)} \ar[r] & L_{\boldsymbol{\lambda}(2)} \ar[r] & \ldots
}
\]
where $L_{\boldsymbol{\lambda}(0)} = L_{\boldsymbol{\lambda}(1)} = 0$, and $L_{\boldsymbol{\lambda}(n)}$ is the standard representation of $R\Aut([n])$ for $n \geqslant 2$.
\end{remark}

\begin{remark} \normalfont
We shall point out that these simple saturated modules were observed and studied by people fifty year ago; see \cite[Theorem 2.7]{Tek}.
\end{remark}

\begin{remark} \normalfont
One can obtain a new proof for the following result in \cite{CEF}: every finitely generated $\C$-module has representation stability. Indeed, as a special case, the simple saturated modules classified in the above theorem certainly satisfy the representation stability. Since objects in $\C \module^{\sat}$ have finite length, we deduce that every object in it has representation stability as well. For an arbitrary $V \in \C \module$, by the procedure described after Proposition \ref{sheafification}, $V$ and its associated saturated $\C$-module $V^{\sharp}$ coincide on all but finitely many objects in $\C$; that is, $V_n = (V^{\sharp})_n$ for $n \gg 0$. Therefore, $V$ has representation stability as well.
\end{remark}

\subsection{Simple discrete representations of $\Aut(\N)$}

In this subsection we let $R$ be a field of characteristic 0, and classify all indecomposable injective discrete representations and simple discrete representations of $G = \Aut(\N)$.

Firstly we construct the discrete $RG$-modules corresponding to $\C$-modules $P(n) = R\C([n], -)$, $n \in \N$. Let $\mathrm{Inj}([n], \N) = \{f: [n] \to \N \mid f \text{ is injective} \}$, and let $G$ acts on $\mathrm{Inj}([n], \N)$ via composition of functions. Then $\mathrm{Inj}([n], \N)$ is a discrete $G$-set since for $f \in \mathrm{Inj}([n], \N)$, one has $\Stab_G(f) = \Stab_G(f([n]))$, which is clearly an open subgroup of $G$. Furthermore, two elements $g_1$ and $g_2$ in $G$ lie in the same left coset if and only if $g_1(i) = g_2(i)$ for $i \in [n]$. Consequently, the map
\[
G / \Stab_G([n]) \to \mathrm{Inj}([n], \N), \quad g \mapsto g \mid _{[n]}
\]
is an isomorphism of discrete $G$-sets, where $g \mid_{[n]}$ denotes the restriction of $g$ (viewed as a function) on $[n]$. Thus
\[
\mathrm{Inj}([n], \N) \cong G/\Stab_G([n]),
\]
and correspondingly, for the free $R$-module $F(n)$ spanned by $\mathrm{Inj}([n], \N)$, one has
\[
F(n) \cong RG \otimes_{R\Aut(\N \setminus [n])} R,
\]
the representation of $\Aut(\N)$ induced from the trivial $R\Aut(\N \setminus [n])$-module.

Let $\phi$ be the functor mentioned after Theorem \ref{mm}. Then one has:

\begin{proposition} \label{free Aut(N)-modules}
The $RG$-module $F(n)$ corresponds to the free $\C$-module $P(n)$ under $\phi$; that is, $\phi(F(n)) \cong P(n)$.
\end{proposition}

\begin{proof}
Let $\mathcal{U}$ be the cofinal system of open subgroups of $G$ consisting of $\Stab_G(S)$ where $S \subset \N$ is a finite set, and let $U_i = \Stab_G([i])$, which is isomorphic to $\Aut(\N \setminus [i])$. By the definition of $\phi$, one has to show that $(F(n))^{U_i} = R\C([n], [i])$ for $i, n \in \N$, where we identify the set $\C([n], [i])$ and the set $\{f \in \mathrm{Inj}([n], \N) \mid f([n]) \subseteq [i] \}$.

Clearly, if $f([n]) \subseteq [i]$, then $f \in (F(n))^{U_i}$, so $R\C([n], [i]) \subseteq (F(n))^{U_i}$ under the identification. To show the inclusion of the other direction, we take $x \in (F(n))^{U_i}$ and write $x = r_1 f_1 + \ldots + r_s f_s$, where $r_j \in R$ and $f_j \in \mathrm{Inj}([n], \N)$. Suppose that there exists a certain $f_j$ such that $f_j([n])$ is not contained in $[i]$, and define
\[
N = \max (f([n]) \cup \ldots \cup f_s([n])).
\]
By our assumption, $N > i$. Now choose an element $g \in G$ such that
\begin{equation*}
g(j) = \left \{\begin{array} {rl}
j, & \text{ if } j \in [i]; \\
j+1, &  \text{ if } i+1 \leqslant j \leqslant N;\\
i+ 1, & \text{ if } j = N+1;\\
j, & \text{ if } j > N.
\end{array} \right.
\end{equation*}
Then $g \in U_i$. Furthermore, one has
\[
\max ((gf_i)([n]) \cup \ldots \cup (gf_s)([n]) = N + 1
\]
and hence $gx \neq x$, contradicting the fact that $x \in (F(n))^{U_i}$. Consequently, every $f_j$ in the linear combination of $x$ must satisfy $f_j([n]) \subseteq [i]$, so $x \in R\C([n], [i])$ under the identification.
\end{proof}

We say that a discrete $RG$-module $V$ is \textit{finitely generated} if $V$ is isomorphic to a quotient of $\bigoplus_{n \in \N} (F(n))^{c_n}$ with $\sum_{n \in \N} c_n < \infty$. Note that the kernel of the quotient map is also discrete and finitely generated (by the equivalence $RG \DMod \cong \C \Mod^{\sat}$ and the locally Noetherian property of $R\C$). Thus the full subcategory consisting of finitely generated discrete $RG$-modules is an abelian subcategory of $RG \DMod$, denoted by $RG \dmodule$. We shall warn the reader that here the meaning of finitely generated is different from its usual sense (the algebraic sense) in representation theory: for instance, the regular $RG$-module is finitely generated in the algebraic sense, but is not a discrete module, and hence is not a finitely generated discrete module in the sense defined above. With this definition, one has the following equivalences
\[
RG \dmodule \cong \C \module / \C \module^{\mathrm{tor}} \cong \C \module^{\sat} \cong \C \fdmod.
\]

\begin{remark} \normalfont
Representation theory of infinite symmetric group has been extensively studied by a lot of authors. For instances, unitary representations of $\Aut(\N)$ were considered in \cite{Lieb, Tek}. In particular,  Liebermann proved in \cite{Lieb} that the category of unitary representations of $\Aut(\N$) is semisimple, and classified all simple unitary representations. However, the category $RG \module^{\mathrm{dis}}$ considered in this paper is not semisimple. We shall also mention that the equivalence $RG \dmodule \cong \C \fdmod$, where $G = \varinjlim_n \Aut([n])$, was established by Sam and Snowden in \cite[Theorem 6.2.4 and Corollary 6.2.5]{SS} using the language of twisted commutative algebras.
\end{remark}

Given a finite set $[n]$, we identify $\Aut([n])$ with the subgroup of $\Aut(\N)$ consisting of permutations $g: \N \to \N$ such that $g(i) = i$ for $i > n$. Let $\lambda: \lambda_1 \geqslant \lambda_2 \geqslant \ldots \geqslant \lambda_r$ be a partition of $n$, and let $L_{\lambda}$ be the simple $R\Aut([n])$-module corresponding to $\lambda$. Then there is a primitive idempotent $e_{\lambda} \in R\Aut([n]) \subset R\Aut(\N)$ such that $R\Aut([n]) e_{\lambda} \cong L_{\lambda}$. Now we define $F(\lambda) = F(n)e_{\lambda}$, which is a submodule of $F(n)$. The next proposition asserts that all indecomposable injective objects in $RG \dmodule$ are of this form.

\begin{proposition}
Each $F(\lambda)$ is an indecomposable injective object in $RG \dmodule$, and every indecomposable injective object in $RG \dmodule$ is of this form up to isomorphism.
\end{proposition}

\begin{proof}
By the equivalence $\C \module^{\sat} \cong RG \dmodule$, one only needs to describe discrete $RG$-modules corresponding to $P(\lambda) = R\C \otimes_{R\Aut([n])} L_{\lambda}$ for partitions $\lambda$ of $[n]$ when $n$ ranges over all elements in $\N$ since these $P(\lambda)$ exhaust all indecomposable injective objects in $\C \module^{\sat}$ up to isomorphism. Now using the same technique as in Proposition \ref{free Aut(N)-modules}, one can check that they are precisely $F(\lambda)$ up to isomorphism; that is, $\phi(F(\lambda)) \cong P(\lambda)$.
\end{proof}

\begin{remark} \normalfont
An equivalent construction of $F(\lambda)$ is as follows. Regard $L_{\lambda}$ as a simple representation of the subgroup $H = \Aut([n]) \times \Aut(\N \setminus [n])$, where $\Aut(\N \setminus [n])$ acts on it trivially. Then $F(\lambda) \cong RG \otimes_{RH} L_{\lambda}$ is the induced representation of $\Aut(\N)$. This construction has been described in the literature; see \cite{Lieb, Tek}, though what they considered are unitary representations over Hilbert spaces. In particular, they showed that every simple unitary representation of $\Aut(\N)$ is of the form $F(\lambda)$ up to isomorphism. However, these $F(\lambda)$ are not simple in our situation.
\end{remark}

The partition $\lambda$ of $n$ induces another partition $\tilde{\lambda}: \lambda_1 \geqslant \lambda_1 \geqslant \lambda_2 \geqslant \lambda_3 \geqslant \ldots \geqslant \lambda_r$ of $n+ \lambda_1$. Note that $R\C([n], [n+ \lambda_1]) e_{\lambda}$ as an $R\Aut([n+ \lambda_1])$-module contains a direct summand isomorphic to the simple $R\Aut([n+ \lambda_1])$-module $L_{\tilde{\lambda}}$ (by the branching rule). Now choose a nonzero element $\epsilon_{\tilde{\lambda}} \in L_{\tilde{\lambda}}$, which is a finite linear combination of injections from $[n]$ to $[n+\lambda_1]$. Define $S(\lambda) = F(n+\lambda_1) \epsilon_{\tilde{\lambda}}$, an $RG$-submodule of $F(n)$. The next proposition asserts that every simple discrete representation of $\Aut(\N)$ is of this form up to isomorphism.

\begin{proposition}
Notation as above. Then $S(\lambda)$ is the simple socle of $F(\lambda)$, and every simple discrete $R\Aut(\N)$-module is of the form $S(\lambda)$ up to isomorphism.
\end{proposition}

\begin{proof}
Firstly, $S(\lambda)$ is a submodule of $F(\lambda)$ since
\begin{align*}
S(\lambda) & = F(n+\lambda_1) \epsilon_{\tilde{\lambda}}\\
& = F(n+\lambda_1) (R\Aut([n+\lambda_1]) \epsilon_{\tilde{\lambda}}) & \text{ as } F(n+\lambda_1) = F(n+\lambda_1) (R\Aut([n+\lambda_1])\\
& = F(n+\lambda_1) L_{\tilde{\lambda}} & R\Aut([n+\lambda_1]) \epsilon_{\tilde{\lambda}} = L_{\tilde{\lambda}} \text{ as } L_{\tilde{\lambda}} \text{ is simple}\\
& \subseteq (F(n+\lambda_1) \cdot \C([n], [n+\lambda_1])) e_{\lambda} & \text{ as } L_{\tilde{\lambda}} \subseteq R\C([n], [n+ \lambda_1]) e_{\lambda}\\
& = F(n) e_{\lambda} & \text{ as } F(n+\lambda_1) \cdot \C([n], [n+\lambda_1]) = F(n)\\
& = F(\lambda).
\end{align*}

Since $F(\lambda)$ corresponds to the $\C$-module $P(\lambda)$ under $\phi$, the indecomposable injective object in $\C \module^{\sat}$, it suffices to show that $S(\lambda)$ corresponds to the simple saturated $\C$-module $L_{\boldsymbol{\lambda}}$ defined in Subsection 6.1. Explicitly, we want to show
\[
S(\lambda)^{U_i} = L_{\boldsymbol{\lambda}(i)}
\]
where $\boldsymbol{\lambda}(i): i- \lambda_1 \geqslant \lambda_1 \geqslant \ldots \geqslant \lambda_r$ is a partition of $[i]$.

By the structure of $P(\lambda)$, we know that
\[
L_{\boldsymbol{\lambda}(i)} = R\C(n+\lambda_1, i) \cdot L_{\tilde{\lambda}} = R\C(n+\lambda_1, i) \epsilon_{\tilde{\lambda}}.
\]
Clearly, we have
\[
S(\lambda)^{U_i} = (F(n+\lambda_1)\epsilon_{\tilde{\lambda}})^{U_i} \supseteq R\C(n+\lambda_1, i)\epsilon_{\tilde{\lambda}} = L_{\boldsymbol{\lambda}(i)}
\]
by identifying injections $f: [n+\lambda_1] \to \N$ such that $f([n+\lambda_1]) \subseteq [i]$ with elements in $\C(n+\lambda_1, i)$.

On the other hand, if $x \in (F(n+\lambda_1)\epsilon_{\tilde{\lambda}})^{U_i}$, we can write
\[
x = (c_1 f_1 + \ldots + c_r f_r) (d_1 g_1 + \ldots + d_s g_s)
\]
where
\[
\epsilon_{\tilde{\lambda}} = d_1 g_1 + \ldots + d_s g_s, \, c_i, d_j \in R, \, f_i \in F(n+\lambda_1), \, g_j \in \C(n, n+ \lambda_1).
 \]
 By the proof of Proposition \ref{free Aut(N)-modules}, we know that $(f_ig_j)([n]) \subseteq [i]$ since $x$ is fixed by $U_i$. If $f_i ([n+\lambda_1])$ is not contained in $[i]$, one can replace $f_i$ by $f_i' \in \C(n+\lambda_1, i)$ such that $f_ig_j = f_i'g_j$. Via this process, we can assume that each $f_i$ satisfies $f_i ([n+\lambda_1]) \subseteq [i]$. That is, $x \in R\C(n+\lambda_1, i)\epsilon_{\tilde{\lambda}}$ by identifying injections $f: [n+\lambda_1] \to \N$ such that $f([n+\lambda_1]) \subseteq [i]$ with elements in $\C(n+\lambda_1, i)$. This proves the inclusion of the other direction, and hence the equality.
\end{proof}

\begin{example} \normalfont \label{an example}
Consider the unique partition $\lambda$ of $[1]$. Then the indecomposable injective discrete $R\Aut(\N)$-module $F(\lambda)$ is precisely $F(1)$, and $S(\lambda)$ is spanned by elements of the form $f_i - f_j$ where $f_i$ and $f_j$ are functions $[1] \to \N$ which map 1 to $i$ and $j$ respectively.
\end{example}

\begin{remark} \normalfont
From this proposition we know that when $R$ is a field of characteristic 0, the Grothendieck group
\[
K_0(R\Aut(\N) \DMod) \cong \bigoplus_{n \in \N} K_0(R\Aut([n] \Mod).
\]
Furthermore, by the work of Zelevinsky described in \cite{Ze}, $\mathscr{H} = \bigoplus_{n \in \N} K_0(R\Aut([n] \Mod)$ has a special Hopf algebra structure, called a \textit{positive self-adjoint Hopf algebra}, whose multiplication and comultiplication are induced by the induction and restriction of representations of symmetric groups respectively. By this isomorphism, one can also impose a positive self-adjoint Hopf algebra structure on $K_0(R\Aut(\N) \DMod)$. For details about positive self-adjoint Hopf algebras, please refer to \cite{Ze}.
\end{remark}

\subsection{Representation stability of $\Aut(\N)$}

In this subsection we consider the shift functor and representation stability of $\Aut(\N)$, translated from those of $\FI$-modules (see \cite{CEF}). The shift functor in $R\Aut(\N) \DMod$ is induced by the following self-embedding function
\[
\iota: \N \to \N, \quad i \mapsto i+1.
\]
This is not a bijection, but it induces a self-embedding of groups
\[
\iota^{\ast}: \Aut(\N) \to \Aut(\N)
\]
by sending $f \in \Aut(\N)$ to $f^{\ast} \in \Aut(\N)$ such that $f^{\ast} (1) = 1$ and $f^{\ast}(i) = f(i-1) + 1$ for $i \geqslant 2$. Clearly, $\iota^{\ast}$ is a continuous and open group homomorphism, and hence induces a functor $\Sigma$ from $R\Aut(\N) \DMod$ to itself. We call it the \textit{shift functor} of degree 1. This shift functor corresponds to the shift functor for $\FI$-modules introduced in \cite{CEF, CEFN} (note that the shift functor of $\FI$-modules preserves saturated modules). Therefore, we can deduce results about the action of $\Sigma$ on $R\Aut(\N) \DMod$ from corresponding results in representation theory of $\FI$. In particular, combining \cite[Theorem A]{Nag} and \cite[Theorem 1.5]{GL2}, we obtain the following result:

\begin{theorem}
Let $V$ be a finitely generated discrete $R\Aut(\N)$-module over a field $R$ of characteristic 0. Then for $n \gg 0$, $\Sigma^n V$ is an injective object in $R\Aut(\N) \dmodule$.
\end{theorem}

It is a fundamental result that $\FI$-modules have certain stability patterns; see for instance \cite{CEF, CEFN}. Now we describe the precise meaning of representation stability for $R\Aut(\N)$-modules. Let $V$ be an arbitrary discrete $R\Aut(\N)$-module. We obtain an $\FI$-module $\phi V: \FI \to R \Mod$ by setting
\[
(\phi V)(S) = V^{U_S}
\]
for any finite set $S \subset \N$, where $U_S = \Stab_G(S)$ as before. If the $\FI$-module $V$ satisfies certain moderate conditions (see \cite{GL}), then there exists an integer $N_V \in \N$ such that for any finite set $S$, one has (see \cite[Theorem C]{CEFN} or \cite[Theorem 3.2]{GL})
\[
(\phi V)_S \cong \varinjlim_{\substack{T \subseteq S \\ |T| \leqslant N_V }} (\phi V)_T.
\]
But we also have
\[
V \cong \varinjlim_{\substack{S \subseteq \N \\ |S| < \infty }} (\phi V)_S
\]
by Theorem \ref{mm}. Combining these two isomorphisms, we obtain
\[
V \cong \varinjlim_{\substack{T \subseteq \N \\ |T| \leqslant N_V }} (\phi V)_T = \varinjlim_{\substack{T \subseteq \N \\ |T| \leqslant N_V }} V^{U_T}.
\]

The minimal number $N_V$ such that the above isomorphism holds is called the \textit{stable range} (see \cite{CEFN}), which can be computed as follows. Firstly, view $\phi V$ as an object in the category $\FI \Mod$, which has enough projective modules. Thus we have a projective presentation
\[
\bigoplus_{n \in \N} P(n)^{c_n} \to \bigoplus_{n \in \N} P(n)^{d_n} \to \phi V \to 0.
\]
For each such presentation, we obtain an element $\sup \{n \mid c_n \neq 0 \text{ or } d_n \neq 0\} \in \N \cup \{ \infty \}$, and set $N_V$ to be the minimum of these elements obtained by exhausting all possible projective presentations of $V$. In other words, the stable range $N_V$ of $V$ is precisely the \textit{presentation degree} of $\phi V$; see \cite[Theorem 3.2]{GL}. Furthermore, since $\phi V$ is a saturated module, in particular it is torsion free. Then by \cite[Theorem 1.9]{LY}, one has $N_V \leqslant 2t_0(V)$, where $t_0(V)$ is the minimal element of the set consisting of $\sup \{n \mid d_n \neq 0\}$ obtained by exhausting all possible projective presentations of $\phi V$.

\begin{example}
Let $\lambda$ be the unique partition of $[0] = \emptyset$. Then the stable range of $S(\lambda)$ is 0, so
\[
S(\lambda) \cong \varinjlim_{\substack{T \subseteq \N \\ |T| \leqslant 0 }} (S(\lambda))^{U_T} = S(\lambda)^{\Aut(\N)}.
\]
Thus the action of $\Aut(\N)$ on $S(\lambda)$ is trivial, and actually, $S(\lambda)$ is the trivial $R\Aut(\N)$-module. Similarly, if we let $\lambda$ be the unique partition of $[1]$, and let $S(\lambda)$ be the associated simple discrete $R\Aut(\N)$-module, which is spanned by elements of the form $f_i - f_j$ where $f_i$ and $f_j$ are functions $[1]$ to $\N$ sending 1 to $i$ and $j$ respectively; see Example \ref{an example}. Then the $\FI$-module (more precisely, the $\C$-module) $\phi S(\lambda)$ has the following structure:
\[
\xymatrix{
0 \ar[r] & 0 \ar[r] & L_{1, 1} \ar[r] & L_{2, 1} \ar[r] & L_{3, 1} \ar[r] & \ldots
}
\]
A direct computation shows that the stable range of $S(\lambda)$ is 3. Then one has
\[
S(\mu) \cong \varinjlim_{\substack{T \subseteq \N \\ |T| \leqslant 3 }} (S(\mu))^{U_T}.
\]
\end{example}


\section{Other examples}

In this section we briefly describe sheaves of modules over some other combinatorial categories, and discrete representations of their associated topological groups.

\subsection{The category $\OI$ and the topological group $\Aut(\mathbb{Q}, \leqslant)$}

In this subsection let $R$ be a field, and let $\C$ the category whose objects are $[n]$ equipped with the canonical ordering, $n \in \N$, and whose morphisms are order preserving injections (or equivalently, strictly increasing maps). By Corollary \ref{finitely generated sheaves}, simple objects in $\mathrm{sh}(\BC^{\op}, R)$ are parameterized by $\N$ since $\C([n], [n])$ is the trivial group for $n \in \N$. They have been classified in \cite{GL1}, so let us briefly summarize the result.

For $n \in \N$ and $i \in [n+1]$, we denote by $\alpha_{n, i}$ the morphism from $[n]$ to $[n+1]$ such that $i \in [n+1]$ is not contained in the image of $\alpha_{n, i}$. For instance, $\alpha_{n, n+1}$ is the canonical inclusion $[n] \to [n+1]$. Now for $i \in [n]$, let $K_{n, i} = \langle \alpha_{n,i} - \alpha_{n, i+1} \rangle$ be the two-sided ideal of the category algebra $R\C$ generated by $\alpha_{n, i} - \alpha_{n, i+1}$, and define $K_n = \cap_{i \in [n]} K_{n,i}$. For $n = 0$, by convention we let $K_0 = P(0)$ spanned by all morphisms starting from $[0]$.

Clearly, $K_n$ is a submodule of $P(n) = R\C([n], -)$, and we have:

\begin{theorem}\cite[Theorem 4.9]{GL1} \label{simple OI-sheaves}
For $n \in \N$, $K_n$ is a simple object in $\C \module^{\sat}$, and every simple object in $\C \module^{\sat}$ is isomorphic to $K_n$ for a certain $n \in \N$.
\end{theorem}

Let $G = \Aut(\mathbb{Q}, \leqslant)$ be the group of order-preserving permutations over $\mathbb{Q}$. Since
\[
RG \DMod \cong \C \Mod^{\sat},
\]
as an immediate corollary, we have:

\begin{proposition}
Isomorphism classes of simple discrete $RG$-modules are parameterized by $\N$.
\end{proposition}

\begin{example} \normalfont
The simple discrete $RG$-module corresponding to $n = 0$ is the trivial module. For $n = 1$, the corresponding simple discrete module is the vector space over $R$ spanned by elements $f_x - f_y$, where $f_x$ (resp., $f_y$) means the functions $[1] \to \mathbb{Q}$ sending $1$ to $x \in \mathbb{Q}$ (resp., $y \in \mathbb{Q}$).
\end{example}

Given $n \in \N$, let $F(n)$ be the free $R$-module spanned by the set of all order-preserving maps from $[n]$ to $\mathbb{Q}$. It is easy to see that $F(n)$ is a discrete $RG$-module, and furthermore, $F(n)$ corresponds to the free $\C$-module $P(n)$; that is, $\phi(F(n)) = P(n)$. We say that a discrete $RG$-module $V$ is \textit{finitely generated} if there is a surjective homomorphism
\[
\bigoplus_{n \in \N} (F(n))^{c_n} \to V
\]
with $\sum_{n \in \N} c_n < \infty$. By the locally Noetherian property of $R\C$, the full subcategory consisting of finitely generated discrete $RG$-modules is an abelian subcategory of $RG \DMod$, denoted by $RG \dmodule$. We then have (by Proposition \ref{equivalence of Z}):
\[
RG \dmodule \cong \C \module / \C \module^{\mathrm{tor}} \cong \C \module^{\sat} \cong \C \fdmod.
\]

\begin{proposition}
Each $F(n)$ is an indecomposable injective object in $RG \dmodule$, and every indecomposable injective object in $RG \module$ is of this form up to isomorphism.
\end{proposition}

\begin{proof}
The conclusion follows from the equivalence $\phi: \C \module^{\sat} \cong RG \dmodule$ and the classification of indecomposable injective $\C$-modules in \cite[Theorem 14.2]{GS}: every projective $\C$-module is injective, and conversely, every torsion free indecomposable injective $\C$-module (which are precisely indecomposable injective objects in $\C \module^{\sat}$ by Corollary \ref{injective sheaves}) is isomorphic to $P(n)$ for a certain $n \in \N$. Therefore, every indecomposable injective object in $RG \dmodule$ corresponds to $P(n)$ for a certain $n \in N$ under the equivalence given by $\phi$. But these objects are exactly $F(n)$ for $n \in \N$.
\end{proof}


\subsection{The category $\VI$ and the infinite general linear group}

In this subsection let $R$ be a field of characteristic 0, and let $G = \varinjlim_n \mathrm{GL}_n(\mathbb{F}_q)$. Let $\C$ be the category with objects $\mathbb{F}_q^n$, $n \in \N$, and morphisms linear injections. Then one has the following equivalence:
\[
RG \DMod \cong \Sh(\BC^{\op}, R) \cong \C \Mod / \C \Mod^{\tor}.
\]
To obtain the equivalences on the ``finitely generated" version, we need to introduce a few special discrete representations of $RG$.

It is easy to see that under the above equivalence, the projective $\C$-module $P(n) = R\C(\mathbb{F}_q^n, -)$ corresponds to the $RG$-module $F(n)$ spanned by (as an $R$-module) all $\mathbb{F}_q$-linear injections from $\mathbb{F}_q^n$ to the $\mathbb{F}_q^{\N}$. It is easy to check that $F(n)$ is a discrete representation of $G$. We say a discrete $RG$-module is finitely generated if it is a quotient of $\oplus_{n \in \N} F(n)^{d_n}$ such that $\sum_{n \in \N} d_n$ is finite. Then the category of finitely generated discrete $RG$-module is abelian (see for instance \cite{GL0}), denoted by $RG \dmodule$. We thus have (by Proposition \ref{equivalence of Z}):
\[
RG \dmodule \cong \mathrm{sh}(\BC^{\op}, R) \cong \C \module / \C \module^{\tor} \cong \C \fdmod.
\]

By (4) of Corollary \ref{finitely generated sheaves}, isomorphism classes of simple discrete $RG$-modules, or simple saturated $\C$-modules, are parameterized by isomorphism classes of simple $\C$-modules, which in turn are parameterized $\sqcup_{n \in \N} \mathrm{Irr}(\mathrm{GL}_n(\mathbb{F}_q))$, where $\mathrm{Irr}(\mathrm{GL}_n(\mathbb{F}_q))$ is the set of isomorphism classes of irreducible representations of $\mathrm{GL}_n(\mathbb{F}_q)$ (See for instance \cite[Section 9]{Ze} for a classification of these irreducible representations). Consequently, the Grothendieck group
\[
K_0(R\varinjlim_n \mathrm{GL}_n(\mathbb{F}_q) \DMod) \cong \bigoplus_{n \in \N} K_0(R\mathrm{GL}_n(\mathbb{F}_q) \Mod).
\]
By \cite{Ze}, $\mathscr{H} = \bigoplus_{n \in \N} K_0(\mathrm{GL}_n(\mathbb{F}_q) \Mod)$ has a natural positive self-adjoint Hopf algebra structure, whose multiplication and comultiplication are induced by the induction and restriction on representations of finite general linear groups respectively. By this isomorphism, one can also impose a positive self-adjoint Hopf algebra structure on $K_0(R\varinjlim_n \mathrm{GL}_n(\mathbb{F}_q) \DMod)$.

Gan and Watterlond describe in \cite{GW} the representation stability result for $\C$-modules over $R$. Relying on their works as well as the results in \cite[Section 9]{Ze}, one can give a complete description of simple objects in $\Sh(\BC^{\op}, R)$. For this purpose, we introduce a few notations, most of which follow \cite{GW}.

Suppose that $R$ is algebraically closed. Let $\mathcal{C}_n$ be the set of isomorphism classes of irreducible cuspidal representations of $\mathrm{GL}_n(\mathbb{F}_q)$, and let $\mathcal{C} = \sqcup_{n \in \N} \mathcal{C}_n$. Let $\mathcal{P}_n$ be the set of partitions of $[n]$, and let $\mathcal{P} = \sqcup_{n \in \N}\mathcal{P}_n$. For $\rho \in \mathcal{C}_n$, we set $d(\rho) = n$. For a partition $\lambda: \lambda_1 \geqslant \lambda_2 \geqslant \ldots$, we set $|\lambda| = \lambda_1 + \lambda_2 + \ldots$. For any function $\boldsymbol{\lambda}: \mathcal{C} \to \mathcal{P}$, we let
\[
\| \boldsymbol{\lambda} \| = \sum _{\rho \in \mathcal{C}} d(\rho) |\boldsymbol{\lambda}(\rho)|.
\]
By \cite[Section 9]{Ze}, elements in $\mathrm{Irr}(\mathrm{GL}_n(\mathbb{F}_q))$ are parameterized by functions $\boldsymbol{\lambda}: \mathcal{C} \to \mathcal{P}$ such that $\| \boldsymbol{\lambda} \| = n$.

Now let $\iota$ be the trivial representation of $\mathrm{GL}_1(\mathbb{F}_q)$, and for any $\boldsymbol{\lambda}: \mathcal{C} \to \mathcal{P}$, we let $\boldsymbol{\lambda}(\iota)$ be the partition $\lambda_1 \geqslant \lambda_2 \geqslant \ldots.$ For any integer $n \geqslant \| \boldsymbol{\lambda} \| + \lambda_1$, we define a new function
\[
\boldsymbol{\lambda}[n]: \mathcal{C} \to \mathcal{P}
\]
such that $\boldsymbol{\lambda}[n](\iota)$ is the partition $n -\lambda_1 \geqslant \lambda_1 \geqslant \lambda_2 \geqslant \ldots$ and $\boldsymbol{\lambda}[n](\rho) = \boldsymbol{\lambda}(\rho)$ for $\rho \neq \iota$. Note that $\boldsymbol{\lambda}[n]$ corresponds to an irreducible representation of $\mathrm{GL}_n(\mathbb{F}_q)$.

With these notations, we have:

\begin{proposition}
There is a bijection between the set of functions $\boldsymbol{\lambda}: \mathcal{C} \to \mathcal{P}$ such that $\| \boldsymbol{\lambda} \| < \infty$ and the set of isomorphism classes of simple objects in $\Sh(\BC^{\op}, R)$. Explicitly, given such a function $\boldsymbol{\lambda}$, the corresponding simple object $V(\boldsymbol{\lambda})$ in $\Sh(\BC^{\op}, R)$ has the following structure: for $n < \| \boldsymbol{\lambda} \| + \lambda_1$, one has $V(\boldsymbol{\lambda})_n = 0$; and for $n \geqslant \| \boldsymbol{\lambda} \| + \lambda_1$, $V(\boldsymbol{\lambda})_n$ is the irreducible representation of $\mathrm{GL}_n(\mathbb{F}_q)$ corresponding to $\boldsymbol{\lambda}[n]$.
\end{proposition}

\begin{proof}
The first statement is clear by (4) of Corollary \ref{finitely generated sheaves} and the parametrization of irreducible representations of $\mathrm{GL}_n(\mathbb{F}_q)$. For the second statement, one can mimic the techniques in Subsection 6.1, replacing results on representation theory of symmetric groups by corresponding results on representation theory of general linear groups over a finite field.
\end{proof}

\begin{remark} \normalfont
By Example \ref{SL}, one also has
\[
\Sh(\VI^{\op}, R) \cong R\varinjlim_n \mathrm{SL}_n(\mathbb{F}_q) \DMod.
\]
Consequently, one has
\[
R \varinjlim_n \mathrm{GL}_n(\mathbb{F}_q) \DMod \cong R\varinjlim_n \mathrm{SL}_n(\mathbb{F}_q) \DMod
\]
and
\[
R \varinjlim_n \mathrm{GL}_n(\mathbb{F}_q) \dmodule \cong R\varinjlim_n \mathrm{SL}_n(\mathbb{F}_q) \dmodule
\]
as expected. Thus all results described in this subsection also hold for discrete representations of $\varinjlim_n \mathrm{SL}_n(\mathbb{F}_q)$.
\end{remark}

\subsection{The categories $\mathcal{Z}(p^n)$ and $\mathcal{Z}(p^{\infty})$}

Let $p$ be a fixed prime number, and let $R$ be a field whose characteristic is distinct from $p$. Recall that $\mathcal{Z}(p^n)$ is the category of finitely generated $\mathbb{Z}/p^n \mathbb{Z}$-modules, and $\mathcal{Z}(p^{\infty})$ is the category of finite abelian $p$-groups and conjugacy classes of surjective group homomorphisms. Furthermore, by Proposition \ref{equivalence of Z}, if $\C$ is one of these two categories, one has
\[
\mathrm{sh}(\BC, R) \cong \C^{\op} \fdmod.
\]
Note that simple objects in $\C^{\op} \fdmod$ are parameterized by the set of irreducible representations of $\Aut(G)$ when $G$ ranges over all finite abelian groups of exponent dividing $p^n$ (for $\mathcal{Z}(p^n))$ or finite abelian $p$-groups (for $\mathcal{Z}(p^{\infty})$). By the above equivalence, this set also parameterizes all simple objects in $\Sh(\BC, R)$.

\subsection{The category $\FI^m$ and the product group of $\Aut(\N)$}

Let $R$ be a field of characteristic 0, $\C$ be the category with objects $[n_1] \times \ldots \times [n_m]$, $n_i \in \N$, and morphisms $(f_1, \ldots, f_m)$ such that each $f_i$ is injective, and let $G$ be the product group of $m$ copies of $\Aut(\N)$. By Proposition \ref{equivalence of Z}, one has $\mathrm{sh}(\BC^{\op}, R) \cong RG \dmodule$. Consequently, one has:

\begin{proposition}
Isomorphism classes of simple objects in $\mathrm{sh}(\BC^{\op}, R)$ or in $RG \dmodule$ are parameterized by the set $\mathcal{P}^m$, the Cartesian product of $m$ copies of $\mathcal{P}$.
\end{proposition}

\begin{proof}
Indeed, by (4) of Corollary \ref{finitely generated sheaves}, simple objects in these two categories are parameterized by the disjoint union of isomorphism classes of irreducible $R(\Aut([n_1]) \times \ldots \times \Aut([n_m]))$-modules with $n_i \in \N$. But isomorphism classes of irreducible $R(\Aut([n_1]) \times \ldots \times \Aut([n_m]))$-modules are parameterized by $\mathcal{P}_{n_1} \times \ldots \times \mathcal{P}_{n_m}$. The conclusion follows from this observation.
\end{proof}

Furthermore, one can easily classify all simple objects or indecomposable injective objects in $RG \dmodule$. For instance, to obtain all simple objects in $RG \dmodule$, one only needs to exhaust the products of $m$ simple objects in $R\Aut(\N) \dmodule$. One can obtain all indecomposable injective objects in $RG \dmodule$ in the same way. Similarly, we can classify all simple objects and indecomposable injective objects in $\mathrm{sh} (\BC^{\op}, R)$.

