# Skew category algebras and modules on ringed finite sites

Mawei Wu  and  Fei Xu<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Mathematics
Shantou University
Shantou, Guangdong 515063, China

###### Abstract.

Let ${\mathcal{C}}$ be a small category. We investigate ringed sites
$(\operatorname{\mathbf{C}},\mathfrak{R})$ on ${\mathcal{C}}$ and the
resulting module categories $\operatorname{\mathfrak{M}od-}\mathfrak{R}$. When
${\mathcal{C}}$ is finite, based on Grothendieck and Verdier’s classification
of finite topoi, we prove that each
$\operatorname{\mathfrak{M}od-}\mathfrak{R}$ is equivalent to
$\operatorname{Mod-}\mathfrak{R}|_{{\mathcal{D}}}[{\mathcal{D}}]$, where
$\mathfrak{R}|_{{\mathcal{D}}}[{\mathcal{D}}]$ is the skew category algebra,
canonically defined on $(\operatorname{\mathbf{C}},\mathfrak{R})$, for a
uniquely determined full subcategory ${\mathcal{D}}\subset{\mathcal{C}}$ and
the restriction $\mathfrak{R}|_{{\mathcal{D}}}$ of $\mathfrak{R}$ to
${\mathcal{D}}$.

###### Key words and phrases:

finite category, topos, modules on site, Grothendieck construction, skew
category algebra, subcategory topology

The authors gbsn(吴马威、徐斐) are supported by the NSFC grants No.12171297 and
No.11671245

###### Contents

  1. 1 Introduction
  2. 2 Prelimenaries
    1. 2.1 Grothendieck topologies and sheaves
      1. 2.1.1 Grothendieck topologies
      2. 2.1.2 Sheaves on sites
    2. 2.2 Geometric morphisms and the Comparison Lemma
    3. 2.3 Ringed sites, ringed topoi and their modules
    4. 2.4 Special categories
  3. 3 Modules over presheaves of algebras
    1. 3.1 Grothendieck construction
    2. 3.2 Skew category algebras
    3. 3.3 Presheaves of modules
  4. 4 Subcategory topologies and module categories
  5. 5 Further remarks
    1. 5.1 On dense EI-sites
    2. 5.2 Vertices and sources
    3. 5.3 Sipp-topology on $G$-Set
    4. 5.4 Hochschild cohomology of $\operatorname{\mathfrak{M}od-}\mathfrak{R}$

##  1\. Introduction

Let ${\mathcal{C}}$ be a small category. It is said to be finite if
Mor${\mathcal{C}}$ is finite, and it is object-finite if $\mathop{\rm
Ob}\nolimits{\mathcal{C}}$ is finite. One may put a Grothendieck topology
${\mathcal{J}}$ on ${\mathcal{C}}$ to obtain a site
$\operatorname{\mathbf{C}}=({\mathcal{C}},{\mathcal{J}})$, which is said to be
finite if the underlying category ${\mathcal{C}}$ is. If $\mathfrak{R}$ is a
sheaf of rings on $\operatorname{\mathbf{C}}$, one can consider the right
modules on the ringed site $(\operatorname{\mathbf{C}},\mathfrak{R})$. They
form an Abelian category, written as
$\operatorname{\mathfrak{M}od-}\mathfrak{R}$, which is our major subject of
investigation. In this paper, we mainly focus on finite categories, because
all of their Grothendieck topologies, as well as the resulting topoi, are
classified in [1]. The classification enables us to characterize module
categories on ringed finite sites.

Our motivation comes from the representation theory of finite groups and
finite-dimensional algebras. This is the main reason that we want to focus on
sites rather than topoi. On the one hand, in group representations, one often
looks into various finite categories associated to the group in question, and
consider their representations (i.e. presheaves) and cohomology. Recent
progresses showed that sheaves on these categories are interesting [4, 15]. On
the other hand, representations of quivers, posets and general small
categories have been under intensive investigations, in both algebra and
topology, see for instance [2, 3, 11]. This work might shed new light on these
subjects.

Suppose $k$ is a commutative ring with identity. We shall denote by $k$-Alg
the category of unital associative algebras and unital $k$-algebra
homomorphisms. Let $\mathfrak{R}$ be a presheaf of unital $k$-algebras on
${\mathcal{C}}$. Motivated by the Grothendieck construction
$Gr_{{\mathcal{C}}}\mathfrak{R}$ of $\mathfrak{R}$ on ${\mathcal{C}}$ [6,
VI.8], we introduce the skew category algebra $\mathfrak{R}[{\mathcal{C}}]$,
as a “linearization” of $Gr_{{\mathcal{C}}}\mathfrak{R}$, which includes skew
group algebras and category algebras as special cases. The right modules of
$\mathfrak{R}[{\mathcal{C}}]$ form the module category ${\rm
Mod}\text{-}\mathfrak{R}[{\mathcal{C}}]$. When ${\mathcal{C}}$ is a poset, the
above construction has been studied by Gerstenhaber and Schack [5], in
disguise, as what they called a diagram ring $\mathfrak{R}!$.

We establish in Section 3 the following category equivalence.

######  Theorem A (Theorem 3.5).

Let ${\mathcal{C}}$ be a small category and let
$\mathfrak{R}:{\mathcal{C}}^{op}\to k\mbox{\rm-Alg}$ be a presheaf of unital
$k$-algebras on $\operatorname{\mathbf{C}}$. If $\mathop{\rm
Ob}\nolimits{\mathcal{C}}$ is finite, then we have a category equivalence

$\operatorname{\mathfrak{M}od-}\mathfrak{R}\simeq{\rm
Mod\text{-}}\mathfrak{R}[{\mathcal{C}}].$

One can continue to put a Grothendieck topology ${\mathcal{J}}$ on
${\mathcal{C}}$. If it is the minimal topology ${\mathcal{J}}_{\min}$, then
sheaves on $({\mathcal{C}},{\mathcal{J}}_{\min})$ and presheaves on
${\mathcal{C}}$ are the same. When we consider another topology
${\mathcal{J}}$ on ${\mathcal{C}}$, we have to employ theories of sheaves and
topoi. Under the circumstance, the sheaf category (a Grothendieck topos)
$\mathop{\rm Sh}\nolimits({\mathcal{C}},{\mathcal{J}})$ becomes a subtopos of
the presheaf topos $\mathop{\rm PSh}\nolimits({\mathcal{C}})=\mathop{\rm
Sh}\nolimits({\mathcal{C}},{\mathcal{J}}_{\min})$. Grothendieck and Verdier
[1, IV.9] pointed out that ${\mathcal{J}}={\mathcal{J}}^{{\mathcal{D}}}$ is
uniquely determined by a strictly full Karoubian subcategory ${\mathcal{D}}$
of ${\mathcal{C}}$. Moreover $\mathop{\rm
Sh}\nolimits({\mathcal{C}},{\mathcal{J}}^{{\mathcal{D}}})\simeq\mathop{\rm
PSh}\nolimits({\mathcal{D}})$. Here being Karoubian means that the category is
idempotent-complete [9] or Cauchy-complete [7]. It allows us to deduce the
following result, when ${\mathcal{C}}$ is finite.

######  Theorem B (Theorem 4.5).

Let $\mathbf{C}=({\mathcal{C}},\mathcal{J})$ be a finite site and
$\mathfrak{R}:{\mathcal{C}}^{op}\to k\mbox{\rm-Alg}$ be a sheaf of unital
$k$-algebras on $\operatorname{\mathbf{C}}$. Then
${\mathcal{J}}={\mathcal{J}}^{{\mathcal{D}}}$ is determined by a strictly full
subcategory ${\mathcal{D}}$, and there is a category equivalence

$\operatorname{\mathfrak{M}od-}\mathfrak{R}\simeq\operatorname{Mod-}\mathfrak{R}|_{{\mathcal{D}}}[{\mathcal{D}}],$

where $\mathfrak{R}|_{{\mathcal{D}}}$ is the restriction of $\mathfrak{R}$ to
${\mathcal{D}}$.

If, for each $x\in\mathop{\rm Ob}\nolimits{\mathcal{D}}$, $\mathfrak{R}(x)$ is
Noetherian, then $\mathfrak{R}|_{{\mathcal{D}}}[{\mathcal{D}}]$ is finite-
dimensional and the coherent
$\mathfrak{R}|_{{\mathcal{D}}}[{\mathcal{D}}]$-modules are the finitely
generated ones.

######  Theorem C (Corollary 4.6).

Let $\mathbf{C}=({\mathcal{C}},\mathcal{J}^{{\mathcal{D}}})$ be a finite site
and $\mathfrak{R}:{\mathcal{C}}^{op}\to k\mbox{\rm-Alg}$ be a sheaf of
$k$-algebras on $\operatorname{\mathbf{C}}$. If for each $x\in\mathop{\rm
Ob}\nolimits{\mathcal{D}}$, $\mathfrak{R}(x)$ is Noetherian, then there is a
category equivalence

${\mathop{\rm
coh}\nolimits}\text{-}\mathfrak{R}\simeq\operatorname{mod-}\mathfrak{R}|_{{\mathcal{D}}}[{\mathcal{D}}].$

The paper is organized as follows. In Section 2, we recall basics on
Grothendieck topologies, topoi, ringed sites and their modules, and present
some concrete examples. Then we characterize the modules on object-finite
categories in Section 3. We recall Grothendieck and Verdier’s classification
of finite topoi in Section 4, which allows us to reduce the characterizations
of general module categories to the situation in the preceding section.
Afterwards, we can present the category equivalences that we have stated
above. We end in Section 5 with several remarks.

##  2\. Prelimenaries

In this section, we will recall the definitions of a Grothendieck topology, a
sheaf, as well as the concepts of a ringed site, a ringed topos and their
modules.

Convention: We shall use curly upper-case letters
${\mathcal{C}},{\mathcal{D}},{\mathcal{E}}$ etc for categories (underlying
sites); lower-case letters $w,x,y,z$ etc for the objects of categories; scan
Serif letters $\textsf{f},\textsf{g},\textsf{h},\textsf{u},\textsf{v}$ etc for
the morphisms of categories. Functors among these categories will be denoted
by Greek letters. Presheaves and sheaves are written as Fraktur letters
$\mathfrak{F},\mathfrak{G},\mathfrak{M},\mathfrak{N},\mathfrak{R}$ etc.

If f is a morphism in ${\mathcal{C}}$, we denote by ${\rm dom}(\textsf{f})$
and ${\rm cod}(\textsf{f})$ its domain and codomain.

In this paper, we shall let $k$ to be a commutative ring with identity, and
denote by $k$-Alg the category of unital associative algebras and unital
$k$-algebra homomorphisms. All algebras and algebra morphisms will be unital.
All modules, either over an algebra or a ringed site, are _right_ modules.

###  2.1. Grothendieck topologies and sheaves

Our main references on topos and sheaf theories are [1, 8, 9, 12]. Here we
only recall necessary ingredients.

Let ${\mathcal{C}}$ be a small category. We shall denote by $\mathop{\rm
PSh}\nolimits({\mathcal{C}})$ the category of presheaves of sets, or
equivalently the contra-variant functors from ${\mathcal{C}}$ to Set (the
category of sets). Given the coefficient ring $k$, we write $\mathop{\rm
PSh}\nolimits({\mathcal{C}},k)$ for the category of presheaves of $k$-modules
(contra-variant functors from ${\mathcal{C}}$ to Mod-$k$). If $k=\mathbb{Z}$,
this is often dubbed as
$\text{PSh}({\mathcal{C}},\mathbb{Z})=\text{PSh}({\mathcal{C}},{\rm Ab})$. If
$G$ is a group, regarded as a category with a single object, $\text{PSh}(G)$
and $\text{PSh}(G,k)$ are canonically identified $\operatorname{Set-}G$ and
Mod-$kG$, the categories of right $G$-sets and of right $kG$-modules,
respectively.

####  2.1.1. Grothendieck topologies

One may put a Grothendieck topology $\mathcal{J}_{{\mathcal{C}}}$ on
${\mathcal{C}}$, via the concept of a sieve. By definition a _sieve_ $S$ on
$x\in\text{Ob}{\mathcal{C}}$ is simply a subfunctor of the representable
functor $\text{Hom}_{{\mathcal{C}}}(-,x)$. It can be identified with a set
(still written as $S$) of morphisms with codomain $x$, satisfying the
condition that if $\textsf{u}\in S$ and uv exists then $\textsf{uv}\in S$. For
instance the _maximal sieve_ $\text{Hom}_{{\mathcal{C}}}(-,x)$ is given by the
set of all morphisms with codomain $x$. Meanwhile there is always an empty
sieve on $x$.

######  Definition 2.1.

A _(Grothendick) topology_ on a category ${\mathcal{C}}$ is a function
$\mathcal{J}$ which assigns to each object $x$ of ${\mathcal{C}}$ a non-empty
collection $\mathcal{J}(x)$ of sieves on ${\mathcal{C}}$, in such a way that

  1. (1)

the maximal sieve $\text{\rm Hom}_{{\mathcal{C}}}(-,x)$ is in
$\mathcal{J}(x)$;

  2. (2)

(stability axiom) if $S\in\mathcal{J}(x)$, then
$\textsf{f}^{*}(S)=\\{\textsf{g}~{}|~{}\textsf{fg}\in S\\}\in\mathcal{J}(y)$
for any arrow $\textsf{f}:y\rightarrow x$;

  3. (3)

(transitivity axiom) if $S_{1}\in\mathcal{J}(x)$, and $S_{2}$ is any sieve on
$x$ such that $\textsf{f}^{*}(S_{2})\in\mathcal{J}(y)$ for all
$\textsf{f}:y\rightarrow x$ in $S_{1}$, then $S_{2}\in\mathcal{J}(x)$.

Any sieve in $\mathcal{J}(x)$ is called a _covering sieve_ on $x$. A small
category ${\mathcal{C}}$ equipped with a Grothendieck topology $\mathcal{J}$
is called a _site_ $\textbf{C}=({\mathcal{C}},\mathcal{J})$.

Here are some examples of Grothendieck topologies that we will encounter.

######  Example 2.2.

  1. (1)

(_The minimal topology_) Given any category ${\mathcal{C}}$, one can define
the minimal topology $\mathcal{J}_{\rm min}$ such that $\mathcal{J}_{\rm
min}(x)=\\{\text{\rm Hom}_{{\mathcal{C}}}(-,x)\\}$, for each $x\in\mathop{\rm
Ob}\nolimits{\mathcal{C}}$.

  2. (2)

(_The dense topology_) The dense topology can be defined for an arbitrary
category ${\mathcal{C}}$: for a sieve $S$, let $S\in\mathcal{J}_{\rm den}(x)$
if and only if for any $\textsf{f}:y\rightarrow x$ there is a morphism
$\textsf{g}:z\rightarrow y$ such that $\textsf{fg}\in S$.

It can be seen from the definition that, the covering sieves of a dense
topology are always non-empty.

  3. (3)

(_The maximal topology_) Let ${\mathcal{C}}$ be any category. To define the
maximal topology ${\mathcal{J}}_{\rm max}$, we declare all sieves to be
covering sieves.

####  2.1.2. Sheaves on sites

Given a Grothendieck topology, one can consider sheaves on ${\mathcal{C}}$.
Roughly speaking, a sheaf of sets on ${\mathcal{C}}$ is a presheaf of sets on
${\mathcal{C}}$, satisfying certain gluing properties mandated by
$\mathcal{J}$. A concise definition of a sheaf is the following.

######  Definition 2.3.

A presheaf $\mathfrak{F}\in\mathop{\rm PSh}\nolimits({\mathcal{C}})$ is a
($\mathcal{J}$-)_sheaf_ of sets if, for every $x\in\mathop{\rm
Ob}\nolimits{\mathcal{C}}$ and every $S\in{\mathcal{J}}(x)$, the inclusion
$S\hookrightarrow\mathop{\rm Hom}\nolimits_{{\mathcal{C}}}(-,x)$ induces an
isomorphism

$\mathop{\rm Nat}\nolimits(\mathop{\rm
Hom}\nolimits_{{\mathcal{C}}}(-,x),\mathfrak{F})\overset{\cong}{\longrightarrow}\mathop{\rm
Nat}\nolimits(S,\mathfrak{F}).$

The category of sheaves of sets on $\mathbf{C}=({\mathcal{C}},\mathcal{J})$ is
the full subcategory of $\mathop{\rm PSh}\nolimits({\mathcal{C}})$, consisting
of all ($\mathcal{J}$-)sheaves, denoted by $\mathop{\rm
Sh}\nolimits(\mathbf{C})$.

Let ${\mathfrak{F}}_{\bullet}$ be the presheaf that sends every object $x$ to
a fixed singleton set $\bullet$, and every morphism to the identity. It is a
terminal object of $\mathop{\rm PSh}\nolimits({\mathcal{C}})$, and one quickly
verifies that it is always a sheaf, under any topology.

Given a presheaf, there is a natural way to build a sheaf. By standard
procedures of construction (see for instance [12, III.5] or [8, 7.10]), given
a presheaf $\mathfrak{F}$, we shall first construct the half-sheafification
(terminology by P. Johnstone)
$\mathfrak{F}^{\dagger}\in\text{PSh}({\mathcal{C}})$ which, for each
$x\in\text{Ob}{\mathcal{C}}$, we define

$\mathfrak{F}^{\dagger}(x)=\mathop{\varinjlim}\nolimits_{{\mathcal{J}}(x)}\text{Nat}(-,\mathfrak{F}),$

where ${\mathcal{J}}(x)$ is deemed as a poset by inclusion. If
$\mathfrak{F}^{\dagger}$ is not a sheaf, then we repeat the construction and
$\mathfrak{F}^{a}=(\mathfrak{F}^{\dagger})^{\dagger}\in\text{Sh}(\mathbf{C})$
is the _sheafification_ of $\mathfrak{F}$ or the sheaf associated to
$\mathfrak{F}$. The sheafification functor is exact and is left adjoint to the
forgetful functor $\text{Sh}(\mathbf{C})\rightarrow\text{PSh}({\mathcal{C}})$,
which is left exact.

######  Remark 2.4.

From the adjunction between the forgetful functor and the sheafification
functor, one realizes that $\mathop{\rm Sh}\nolimits(\mathbf{C})$ is a
localization of $\mathop{\rm PSh}\nolimits({\mathcal{C}})$ [1, 7, 9]. In fact
the morphisms that become isomorphisms after sheafification form a
multiplicative system $\Xi$ in $\mathop{\rm PSh}\nolimits({\mathcal{C}})$. The
morphisms in $\Xi$ are called local isomorphisms [9] or morphismes bicouvrant
[1], giving a category equivalence

$\mathop{\rm Sh}\nolimits(\operatorname{\mathbf{C}})\simeq\Xi^{-1}\mathop{\rm
PSh}\nolimits({\mathcal{C}}).$

######  Example 2.5.

  1. (1)

If ${\mathcal{C}}$ is given the minimal topology, then all presheaves are
sheaves. The local isomorphisms are exactly all the natural isomorphisms.

  2. (2)

If ${\mathcal{C}}$ is given the maximal topology, then the only sheaf is
${\mathfrak{F}}_{\bullet}$. The local isomorphisms are all the morphisms. (Be
aware of set-theoretic issues!)

Let $k$ be the coefficient ring. A presheaf of $k$-modules is a contra-
covariant functor ${\mathfrak{M}}$ from ${\mathcal{C}}$ to
$\operatorname{Mod-}k$. It is a sheaf of $k$-modules if the composite
${\mathcal{C}}{\buildrel{\mathfrak{M}}\over{\to}}\operatorname{Mod-}k\hookrightarrow$Set
is a sheaf of sets. The resulting sheaf category $\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{C}},k)$ is Abelian, while the zero object
$0$(=${\mathfrak{F}}_{\bullet}$) maps every object to $0$. The forgetful and
sheafification functors restrict to a pair of adjoint functors between
$\mathop{\rm Sh}\nolimits(\operatorname{\mathbf{C}},k)$ and $\mathop{\rm
PSh}\nolimits({\mathcal{C}},k)$. It follows that

$\mathop{\rm Sh}\nolimits(\operatorname{\mathbf{C}},k)\simeq\mathop{\rm
PSh}\nolimits({\mathcal{C}},k)/\mathcal{S},$

where $\mathcal{S}$ is the Serre subcategory consisting of
${\mathfrak{M}}\in\mathop{\rm PSh}\nolimits({\mathcal{C}},k)$ satisfying
${\mathfrak{M}}^{a}=0$.

In the Abelian category $\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{C}},k)$, one may consider its coherent
objects. They form an Abelian full subcategory ${\mathop{\rm
coh}\nolimits}(\operatorname{\mathbf{C}},k)$, see [14], as well as [1, 8].

######  Example 2.6.

On a group $G$, considered as a category with one object $\bullet$, there are
only two topologies. The minimal topology(=dense topology) contains only one
sieve $\text{\rm Hom}_{{\mathcal{C}}}(-,\bullet)$, while the maximal topology
has two sieves $\\{\emptyset,\text{\rm Hom}_{{\mathcal{C}}}(-,\bullet)\\}$.

Under the minimal topology, we have $\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{G}})=\operatorname{Set-}G$ and $\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{G}},k)=\operatorname{Mod-}kG$. When $k$ is
Noetherian, ${\mathop{\rm
coh}\nolimits}(\operatorname{\mathbf{G}},k)=\operatorname{mod-}kG$, the
category of finitely generated $kG$-modules.

###  2.2. Geometric morphisms and the Comparison Lemma

To compare sheaves on two sites, we need the concepts of continuous functors
[1]. Let $\alpha:{\mathcal{C}}\to{\mathcal{D}}$ be a (covariant) functor. Then
it induces a restriction along $\alpha$, $Res_{\alpha}:\mathop{\rm
PSh}\nolimits({\mathcal{D}})\to\mathop{\rm PSh}\nolimits({\mathcal{C}})$,
which has two adjoint functors $LK_{\alpha},RK_{\alpha}:\mathop{\rm
PSh}\nolimits({\mathcal{C}})\to\mathop{\rm PSh}\nolimits({\mathcal{D}})$,
called the left and right Kan extensions along $\alpha$.

######  Definition 2.7.

Let $\operatorname{\mathbf{C}}=({\mathcal{C}},{\mathcal{J}})$ and
$\operatorname{\mathbf{D}}=({\mathcal{D}},{\mathcal{J}}^{\prime})$ be two
sites. A functor $\alpha:{\mathcal{C}}\to{\mathcal{D}}$ is called _continuous_
if the restriction $Res_{\alpha}:\mathop{\rm
PSh}\nolimits({\mathcal{D}})\to\mathop{\rm PSh}\nolimits({\mathcal{C}})$
preserves sheaves.

When $\alpha$ is continuous, it gives rise to a functor
$Res_{\alpha}:\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{D}})\to\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{C}})$.

######  Definition 2.8.

Let $\operatorname{\mathbf{C}}=({\mathcal{C}},{\mathcal{J}})$ and
$\operatorname{\mathbf{D}}=({\mathcal{D}},{\mathcal{J}}^{\prime})$ be two
sites. A _morphism of sites_
$\Lambda:\operatorname{\mathbf{D}}\to\operatorname{\mathbf{C}}$ is given by a
continuous functor $\alpha:{\mathcal{C}}\to{\mathcal{D}}$ such that

$LK_{\alpha}^{a}=(-)^{a}\circ LK_{\alpha}:\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{C}})\to\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{D}})$

is left exact.

In topos theory, the notion of a topos is more important than that of a site.
Although we would like to focus on sites, certain relevant key results are
formulated in terms of topoi.

######  Definition 2.9.

A _(Grothendieck) topos_ is a category that is equivalent to some $\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{C}})$, a category of sheaves of sets on a
site $\operatorname{\mathbf{C}}$. A _(geometric) morphism of topoi_

$\Psi=(\Psi^{-1},\Psi_{*}):\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{D}})\to\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{C}})$

consists of a pair of functors $\Psi_{*}:\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{D}})\to\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{C}})$ and $\Psi^{-1}:\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{C}})\to\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{D}})$ such that $\Psi^{-1}$ is left exact
and is left adjoint to $\Psi_{*}$.

If $\Psi_{*}$ happens to be an inclusion which is fully faithful, then
$\mathop{\rm Sh}\nolimits(\operatorname{\mathbf{D}})$ is said to be a
_subtopos_ of $\mathop{\rm Sh}\nolimits(\operatorname{\mathbf{C}})$.

A morphism of sites
$\Lambda:\operatorname{\mathbf{D}}\to\operatorname{\mathbf{C}}$ always induces
a morphism of topoi $\Lambda=(\Lambda^{-1},\Lambda_{*})$, where
$\Lambda_{*}=Res_{\alpha}:\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{D}})\to\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{C}})$ and
$\Lambda^{-1}=LK_{\alpha}^{a}:\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{C}})\to\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{D}})$.

The Comparison Lemma will be used in Section 4, in order to establish our main
results.

######  Definition 2.10.

Let $({\mathcal{C}},{\mathcal{J}})$ be a site. A full subcategory
${\mathcal{D}}$ of ${\mathcal{C}}$ is said to be _${\mathcal{J}}$ -dense_ if
for every object $x$ of ${\mathcal{C}}$ the sieve generated by the family of
arrows from objects in ${\mathcal{D}}$ to $x$ is ${\mathcal{J}}$-covering.

Given such a subcategory ${\mathcal{D}}$, the Grothendieck topology
${\mathcal{J}}|_{{\mathcal{D}}}$ on ${\mathcal{D}}$ induced by ${\mathcal{J}}$
is defined as follows: for any sieve $S$ in ${\mathcal{D}}$ on an object $x$,
$S\in{\mathcal{J}}|_{{\mathcal{D}}}(x)$ if and only if
$\bar{S}\in{\mathcal{J}}(x)$, where $\bar{S}$ is the sieve in ${\mathcal{C}}$
generated by the arrows in $S$.

######  Theorem 2.11 (Comparison Lemma).

Let $({\mathcal{C}},{\mathcal{J}})$ be a site and ${\mathcal{D}}$ be a full
subcategory of ${\mathcal{C}}$ which is ${\mathcal{J}}$-dense. Then the
restriction $Res_{\iota}$, along
$\iota:{\mathcal{D}}\hookrightarrow{\mathcal{C}}$, induce a functor
$Res_{\iota}:\mathop{\rm
Sh}\nolimits({\mathcal{C}},{\mathcal{J}})\to\mathop{\rm
Sh}\nolimits({\mathcal{D}},{\mathcal{J}}|_{{\mathcal{D}}})$, which is an
equivalence. The quasi-inverse is the right Kan extension $RK_{\iota}$ along
$\iota$.

It is illuminating to have a concrete yet more sophiscated example other than
groups.

######  Example 2.12.

Let ${\mathcal{P}}$ be the category
$x{\buildrel\textsf{f}\over{\to}}y{\buildrel\textsf{g}\over{\to}}z$, which is
essentially a poset. Since it is finite and Karoubian, there are exactly eight
Grothendieck topologies on ${\mathcal{P}}$, corresponding to the eight
subposets (full subcategories, including the empty one) of ${\mathcal{P}}$
(See Propositions 4.1 and 4.4). Since the subposets are determined by the
subsets of ${\mathcal{P}}$, we shall write for instance $xy$ for the subposet
$x\to y$, for brevity. In fact, the poset of subposets of ${\mathcal{P}}$ is
opposite isomorphic to that of topologies on ${\mathcal{P}}$.

$\textstyle{{\mathcal{J}}^{\emptyset}}$$\textstyle{{\mathcal{J}}^{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}^{y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}^{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}^{xy}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}^{xz}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}^{yz}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}^{xyz}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Here ${\mathcal{J}}^{\emptyset}={\mathcal{J}}_{\max}$,
${\mathcal{J}}^{xyz}={\mathcal{J}}^{{\mathcal{P}}}={\mathcal{J}}_{\min}$, and
${\mathcal{J}}^{x}={\mathcal{J}}_{\rm den}$. As to the rest, because there
exists the minimal covering sieve on each object (for any given topology), we
only need to list them since all sieves larger than the minimal ones must be
covering. The minimal covering sieve of ${\mathcal{J}}^{{\mathcal{Q}}}(w)$,
for an object $w\in\mathop{\rm Ob}\nolimits{\mathcal{P}}$ and a subposet
${\mathcal{Q}}$, is formed by all morphisms originated from an object of
${\mathcal{Q}}$ and ending at $w$.

Table 1. The MINIMAL covering sieve on each object |  x |  y |  z
---|---|---|---
${\mathcal{J}}^{xyz}$ |  $\mathop{\rm Hom}\nolimits_{{\mathcal{P}}}(-,x)$ |  $\mathop{\rm Hom}\nolimits_{{\mathcal{P}}}(-,y)$ |  $\mathop{\rm Hom}\nolimits_{{\mathcal{P}}}(-,z)$
${\mathcal{J}}^{x}$ |  $\mathop{\rm Hom}\nolimits_{{\mathcal{P}}}(-,x)$ |  $\\{\textsf{f}\\}$ |  $\\{\textsf{g}\textsf{f}\\}$
${\mathcal{J}}^{y}$ |  $\emptyset$ |  $\mathop{\rm Hom}\nolimits_{{\mathcal{P}}}(-,y)$ |  $\\{\textsf{g},\textsf{g}\textsf{f}\\}$
${\mathcal{J}}^{z}$ |  $\emptyset$ |  $\emptyset$ |  $\mathop{\rm Hom}\nolimits_{{\mathcal{P}}}(-,z)$
${\mathcal{J}}^{xy}$ |  $\mathop{\rm Hom}\nolimits_{{\mathcal{P}}}(-,x)$ |  $\mathop{\rm Hom}\nolimits_{{\mathcal{P}}}(-,y)$ |  $\\{\textsf{g},\textsf{g}\textsf{f}\\}$
${\mathcal{J}}^{xz}$ |  $\mathop{\rm Hom}\nolimits_{{\mathcal{P}}}(-,x)$ |  $\\{\textsf{f}\\}$ |  $\mathop{\rm Hom}\nolimits_{{\mathcal{P}}}(-,z)$
${\mathcal{J}}^{yz}$ |  $\emptyset$ |  $\mathop{\rm Hom}\nolimits_{{\mathcal{P}}}(-,y)$ |  $\mathop{\rm Hom}\nolimits_{{\mathcal{P}}}(-,z)$
${\mathcal{J}}^{\emptyset}$ |  $\emptyset$ |  $\emptyset$ |  $\emptyset$

Let ${\mathcal{Q}}$ be a subposet of ${\mathcal{Q}}^{\prime}$. Then
${\mathcal{J}}^{{\mathcal{Q}}^{\prime}}\subset{\mathcal{J}}^{{\mathcal{Q}}}$
and the identity functor $\mathop{\rm Id}\nolimits_{{\mathcal{P}}}$ induces a
morphism of sites
$\operatorname{\mathbf{P}}_{{\mathcal{Q}}^{\prime}}=({\mathcal{P}},{\mathcal{J}}^{{\mathcal{Q}}^{\prime}})\to\operatorname{\mathbf{P}}_{{\mathcal{Q}}}=({\mathcal{P}},{\mathcal{J}}^{{\mathcal{Q}}})$,
and consequently a morphism of topoi $\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{{\mathcal{Q}}})\to\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{{\mathcal{Q}}^{\prime}})$ which is
fully faithful and makes the former a subtopos of the latter. Indeed they form
a lattice of subtopoi of $\mathop{\rm PSh}\nolimits({\mathcal{P}})$ as follows

$\textstyle{\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{\emptyset})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{x})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{xy})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{xz})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{yz})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{xyz})}$

By direct calculations via Definition 2.3, one readily deduces that
$\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{{\mathcal{Q}}})\simeq\mathop{\rm
PSh}\nolimits({\mathcal{Q}})$. For instance, $\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{\emptyset})=\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{\max})=\\{{\mathfrak{F}}_{\bullet}\\}$,
$\mathop{\rm Sh}\nolimits(\operatorname{\mathbf{P}}_{xyz})=\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{{\mathcal{P}}})=\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{\min})=\mathop{\rm
PSh}\nolimits({\mathcal{P}})$, and $\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{xy})$ consists of (pre)sheaves
${\mathfrak{F}}$, satisfying ${\mathfrak{F}}(y)={\mathfrak{F}}(z)$ and
${\mathfrak{F}}(\textsf{g})=\mathop{\rm Id}\nolimits$.

As a subcategory of $\mathop{\rm PSh}\nolimits({\mathcal{P}})$, each sheaf
category $\mathop{\rm Sh}\nolimits(\operatorname{\mathbf{P}}_{{\mathcal{Q}}})$
can be obtained by localization with respect to a set of local isomorphisms,
that is, $\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{{\mathcal{Q}}})\simeq\Xi^{-1}_{{\mathcal{Q}}}\mathop{\rm
PSh}\nolimits({\mathcal{P}})$. The set $\Xi_{{\mathcal{Q}}}$ can be explicitly
written out. For instance, $\Xi_{xy}$ consists of natural transformations that
become isomorphisms after sheafification. It contains exactly the natural
transformations of the following form

$\textstyle{{\mathfrak{F}}(x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathop{\rm
Id}\nolimits_{{\mathfrak{F}}(x)}}$$\textstyle{{\mathfrak{F}}(y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathfrak{F}}(\textsf{f})}$$\scriptstyle{\mathop{\rm
Id}\nolimits_{{\mathfrak{F}}(y)}}$$\textstyle{{\mathfrak{F}}(z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathfrak{F}}(\textsf{g})}$$\scriptstyle{{\mathfrak{F}}(\textsf{g})}$$\textstyle{{\mathfrak{F}}(x)}$$\textstyle{{\mathfrak{F}}(y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathfrak{F}}(\textsf{f})}$$\textstyle{{\mathfrak{F}}(y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathop{\rm
Id}\nolimits_{{\mathfrak{F}}(y)}}$

The upper row depicts a presheaf on ${\mathcal{P}}$, and the lower one is its
sheafification with respect to ${\mathcal{J}}^{xy}$, a sheaf on
$\operatorname{\mathbf{P}}_{xy}$.

In the end, for the coefficient ring $k$, $\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{P}}_{{\mathcal{Q}}},k)\simeq\mathop{\rm
PSh}\nolimits({\mathcal{Q}},k)\simeq\operatorname{Mod-}k{\mathcal{Q}}$, where
$k{\mathcal{Q}}$ is the incidence algebra of ${\mathcal{Q}}$.

We hope the above example can be useful for the reader whose background is in
the representation theory of finite-dimensional algebras, or of finite groups.

###  2.3. Ringed sites, ringed topoi and their modules

Let $\operatorname{\mathbf{C}}$ be a site. We recall the definitions of a
ringed site, a ringed topos, and their modules see [1, 8, 9].

######  Definition 2.13.

Let $\operatorname{\mathbf{C}}$ be a site.

  1. (1)

A _sheaf of $k$-algebras_ on $\operatorname{\mathbf{C}}$ is a presheaf
$\mathfrak{R}:{\mathcal{C}}^{op}\to k$-Alg which becomes a sheaf of sets after
composing with the canonical functor $k\text{-}{\rm Alg}\hookrightarrow{\rm
Set}$.

  2. (2)

A _sheaf of $\mathfrak{R}$-modules_ is a presheaf of right
$\mathfrak{R}$-modules $\mathfrak{M}$, such that the underlying presheaf of
$k$-modules $\mathfrak{M}$ is a sheaf of sets.

  3. (3)

A _morphism of sheaves of right $\mathfrak{R}$-modules_ is a morphism of
presheaves of $\mathfrak{R}$-modules.

  4. (4)

The _category of sheaves of $\mathfrak{R}$-modules_ is denoted by
$\operatorname{\mathfrak{M}od-}\mathfrak{R}$.

We will encounter both ringed sites and ringed topoi later on. From
representation-theorists’ perspective, we prefer working with ringed sites
whenever it is possible, since they are comparable with group algebras etc.

######  Definition 2.14.

  1. (1)

A _ringed site_ is a pair $(\operatorname{\mathbf{C}},\mathfrak{R})$ where
$\operatorname{\mathbf{C}}$ is a site and $\mathfrak{R}$ is a sheaf of rings
on $\operatorname{\mathbf{C}}$. The sheaf $\mathfrak{R}$ is called the
_structure sheaf_ of the ringed site.

  2. (2)

Let $(\operatorname{\mathbf{C}},\mathfrak{R})$ and
$(\operatorname{\mathbf{C}}^{\prime},\mathfrak{R}^{\prime})$ be two ringed
sites. A _morphism of ringed sites_
$(\Lambda,\Lambda^{\sharp}):(\operatorname{\mathbf{C}},\mathfrak{R})\to(\operatorname{\mathbf{C}}^{\prime},\mathfrak{R}^{\prime})$
is given by a morphism of sites
$\Lambda:\operatorname{\mathbf{C}}\to\operatorname{\mathbf{C}}^{\prime}$ and a
map of sheaves of $k$-algebras
$\Lambda^{\sharp}:\Lambda^{-1}(\mathfrak{R}^{\prime})\to\mathfrak{R}$, which
by adjunction is the same thing as a map of sheaves of $k$-algebras
$\Lambda^{\sharp}:\mathfrak{R}^{\prime}\to\Lambda_{*}(\mathfrak{R})$.

  3. (3)

A _ringed topos_ is a pair $(\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{C}}),\mathfrak{R})$ where
$\operatorname{\mathbf{C}}$ is a site and $\mathfrak{R}$ is a sheaf of rings
on $\operatorname{\mathbf{C}}$. The sheaf $\mathfrak{R}$ is the _structure
sheaf_ of the ringed topos.

  4. (4)

Let $(\mathop{\rm Sh}\nolimits(\operatorname{\mathbf{C}}),\mathfrak{R})$,
$(\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{C}}^{\prime}),\mathfrak{R}^{\prime})$ be
ringed topoi. A _morphism of ringed topoi_ $(\Psi,\Psi^{\sharp}):(\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{C}}),\mathfrak{R})\to(\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{C}}^{\prime}),\mathfrak{R}^{\prime})$ is
given by a morphism of topoi $\Psi=(\Psi^{-1},\Psi_{*}):\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{C}})\to\mathop{\rm
Sh}\nolimits(\operatorname{\mathbf{C}}^{\prime})$ together with a map of
sheaves of $k$-algebras
$\Psi^{\sharp}:\Psi^{-1}(\mathfrak{R}^{\prime})\to\mathfrak{R}$, which by
adjunction is the same thing as a map of sheaves of $k$-algebras
$\Psi^{\sharp}:\mathfrak{R}^{\prime}\to\Psi_{*}(\mathfrak{R})$.

A ringed topos is just a ringed site. The difference is on their morphisms. A
morphism between ringed sites always induces one between corresponding ringed
topoi. For the experienced reader, it is possible to present our contents in
this paper with ringed sites only. However we choose to use both ringed sites
and ringed topoi in order to be consistent with references such as [1].

The following example may give the reader some feeling of the sheaves of
modules.

######  Example 2.15.

Let ${\mathcal{C}}$ be the following category

$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textsf{1}_{x}}$hfg$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\textsf{1}_{y}}$

such that $\textsf{h}^{2}=\textsf{1}_{x}$ and
$\textsf{f}\textsf{h}=\textsf{g}$. By Propositions 4.1 and 4.4, there are four
topologies.

A presheaf of $k$-algebras $\mathfrak{R}$ on ${\mathcal{C}}$ is represented by
the following picture

$\textstyle{\mathfrak{R}(x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{R}(\textsf{1}_{x})}$$\scriptstyle{\mathfrak{R}(\textsf{h})}$$\textstyle{\mathfrak{R}(y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\mathfrak{R}(\textsf{f})}$$\scriptstyle{\mathfrak{R}(\textsf{g})}$$\scriptstyle{\mathfrak{R}(\textsf{1}_{y})}$

which consists of two algebras and five compatible algebra homomorphisms. One
can analogously draw any right $\mathfrak{R}$-module like this.

If ${\mathcal{C}}$ is given a topology and $\mathfrak{R}$ becomes a sheaf of
$k$-algebras, the module category $\operatorname{\mathfrak{M}od-}\mathfrak{R}$
is a subcategory of $\mathop{\rm
PSh}\nolimits({\mathcal{C}},k)\simeq\operatorname{Mod-}k{\mathcal{C}}$, where
$k{\mathcal{C}}$ is the category algebra [16].

###  2.4. Special categories

A full subcategory ${\mathcal{D}}$ of ${\mathcal{C}}$ is said to be strictly
full if an object $x$ belongs to ${\mathcal{D}}$ then every object isomorphic
to $x$ in ${\mathcal{C}}$ must lie in ${\mathcal{D}}$.

A category is _EI_ if all of its endomorphisms are isomorphisms [11]. Groups
and partially ordered sets are EI-categories. Given an EI-category
${\mathcal{C}}$, there is a pre-order defined on Ob${\mathcal{C}}$, that is
$x\leq y$ if and only if $\text{Hom}_{{\mathcal{C}}}(x,y)\neq\emptyset$. Let
$[x]$ be the isomorphism class of an object $x\in$ Ob${\mathcal{C}}$. This
pre-order induces a partial order on the set Iso${\mathcal{C}}$ of isomorphism
classes of Ob${\mathcal{C}}$ (specified by $[x]\leq[y]$ if and only if
$\text{Hom}_{{\mathcal{C}}}(x,y)\neq\emptyset$). For an EI-category
${\mathcal{C}}$ and an object $x\in$ Ob${\mathcal{C}}$, we define a full
subcategory ${\mathcal{C}}_{\leq x}\subset{\mathcal{C}}$, consisting of all
$y\in\text{Ob}{\mathcal{C}}$ such that
$\text{Hom}_{{\mathcal{C}}}(y,x)\neq\emptyset$. Similarly we can define
several other full subcategories of ${\mathcal{C}}$: ${\mathcal{C}}_{<x}$,
${\mathcal{C}}_{\geq x}$ and ${\mathcal{C}}_{>x}$.

######  Definition 2.16.

Let ${\mathcal{C}}$ be an EI-category.

  1. (1)

We denote by ${\mathcal{C}}_{\min}$ the full subcategory consisting of all
minimal objects.

  2. (2)

A (strictly) full subcategory ${\mathcal{D}}$ is called a _co-ideal_ , if
$x\in\mathop{\rm Ob}\nolimits{\mathcal{D}}$ implies that ${\mathcal{C}}_{\leq
x}\subset{\mathcal{D}}$.

Here is an EI-category related to a finite group.

######  Example 2.17.

Let $G$ be a finite group. The _orbit category_ ${\mathcal{O}}(G)$ has objects
the orbits $G/H$, where $H$ is a subgroup, with morphism sets

${\rm Hom}_{{\mathcal{O}}(G)}(G/H,G/K)=\\{c_{g}:G/H\to G/K\bigm{|}g\in G\\}.$

Here $c_{g}$ is the $G$-map given by $H\mapsto gK$ (different group elements
may induce the same map).

In practice, one often takes a certain collection of subgroups and then
creates smaller orbit categories. For instance, one can focus on the $p$-orbit
category ${\mathcal{O}}_{p}(G)$ on $p$-subgroups, and the smaller orbit
category ${\mathcal{O}}^{\circ}_{p}(G)$ on non-identity $p$-subgroups of $G$.
More examples of EI-categories, including the fusion systems, can be found in
[2]. To understand the representations (that is, functors) and cohomology of
such categories (and their applications to group representation theory) is our
motivation to examine sheaves over finite categories.

######  Definition 2.18.

A category is _Karoubian_ [1, IV, Exercise 7.5] (i.e. _idempotent-complete_
[9, Exercise 2.9] or _Cauchy-complete_ [7]), if every idempotent endomorphism
in ${\mathcal{C}}$ has an image.

If ${\mathcal{C}}$ is an EI-category, then the only idempotent endomorphisms
are the identities. Therefore EI-categories are always Karoubian.

##  3\. Modules over presheaves of algebras

Under the minimal topology, all presheaves on ${\mathcal{C}}$ are sheaves.
This section only deals with presheaves of algebras and their modules. We will
see that it is interesting in its own right, and moreover it is necessary for
upcoming developments.

###  3.1. Grothendieck construction

Let ${\mathcal{C}}$ be a small category and
$\mathfrak{R}:{\mathcal{C}}^{op}\to k\text{-Alg}$ be a presheaf of
$k$-algebras. Since every unital ring can be regarded as a small additive
category with a single object [13], $\mathfrak{R}$ becomes a contravariant
functor from ${\mathcal{C}}$ to ${\rm Cat}$, the category of small categories.

There is a device, called the Grothendieck construction (see [6, VI.8]), to
build a fibred category over a small category ${\mathcal{C}}$ equipped with a
pseudo-functor to ${\rm Cat}$. Since strict functors are pseudo, we can
examine the construction for $\mathfrak{R}$.

######  Definition 3.1.

Let $\mathfrak{F}:{\mathcal{C}}^{op}\to{\rm Cat}$ be a (strict) functor. The
_Grothendieck construction_ $Gr_{{\mathcal{C}}}\mathfrak{F}$ is a small
category whose objects are $\\{(x,c)~{}|~{}x\in\mathop{\rm
Ob}\nolimits{\mathcal{C}},c\in\mathop{\rm Ob}\nolimits\mathfrak{F}(x)\\}$, and
whose morphisms are

${\rm
Hom}_{Gr_{{\mathcal{C}}}\mathfrak{F}}((x,c),(y,d))=\\{(\textsf{f},\textsf{u})~{}|~{}\textsf{f}:x\to
y\in{\rm Mor}{\mathcal{C}},\textsf{u}:c\to\mathfrak{F}(\textsf{f})(d)\in{\rm
Mor}~{}\mathfrak{F}(x)\\}.$

The composition of morphisms is given by
$(\textsf{g},\textsf{v})\circ(\textsf{f},\textsf{u})=(\textsf{g}\textsf{f},\mathfrak{F}(\textsf{f})(\textsf{v})\circ\textsf{u})$.

Since $\mathfrak{R}$ admits an “action” by ${\mathcal{C}}$, our category
$Gr_{{\mathcal{C}}}\mathfrak{R}$ carries certain algebraic structures on its
morphism sets. To explain, we present $Gr_{{\mathcal{C}}}\mathfrak{R}$ as
follows

  1. (1)

Objects: $\\{(x,\bullet_{x})~{}|~{}x\in\mathop{\rm
Ob}\nolimits{\mathcal{C}}\;\mbox{and}\,\bullet_{x}\,\mbox{is the unique object
of}\;\mathfrak{R}(x)\\}$,

  2. (2)

Morphisms:

${\rm
Hom}_{Gr_{{\mathcal{C}}}\mathfrak{R}}((x,\bullet_{x}),(y,\bullet_{y}))=\\{(\textsf{f},r)~{}|~{}\textsf{f}:x\to
y,r:\bullet_{x}\to\mathfrak{R}(\textsf{f})(\bullet_{y}),r\in\mathfrak{R}(x)\\},$

  3. (3)

Composition of morphisms:
$(\textsf{g},s)\circ(\textsf{f},r)=(\textsf{g}\textsf{f},\mathfrak{R}(\textsf{f})(s)r)$.

Note that we always have $\mathfrak{R}(\textsf{f})(\bullet_{y})=\bullet_{x}$
for any morphism $\textsf{f}:x\to y$.

######  Proposition 3.2.

Let $Gr_{{\mathcal{C}}}\mathfrak{R}$ be the above Grothendieck construction.
Suppose $\textsf{f}:x\to y$ is a morphism in ${\mathcal{C}}$. Then

  1. (1)

The following subset, of ${\rm
Hom}_{Gr_{{\mathcal{C}}}\mathfrak{R}}((x,\bullet_{x}),(y,\bullet_{y}))$,

$\mathop{\rm
Hom}\nolimits^{\textsf{f}}((x,\bullet_{x}),(y,\bullet_{y}))=\\{(\textsf{f},r)~{}|~{}r:\bullet_{x}\to\mathfrak{R}(\textsf{f})(\bullet_{y}),r\in\mathfrak{R}(x)\\}$

has a natural $k$-module structure.

  2. (2)

The set $\mathop{\rm
Aut}\nolimits^{\textsf{1}_{x}}(x,\bullet_{x})=\\{(\textsf{1}_{x},r)~{}|~{}r:\bullet_{x}\to\mathfrak{R}(\textsf{1}_{x})(\bullet_{x}),r\in\mathfrak{R}(x)\\}$
has a natural ring structure and is isomorphic to $\mathfrak{R}(x)$.

  3. (3)

The set $\mathop{\rm
Hom}\nolimits^{\textsf{f}}((x,\bullet_{x}),(y,\bullet_{y}))$ is a rank one
free right $\mathop{\rm Aut}\nolimits^{\textsf{1}_{x}}(x,\bullet_{x})$-module,
with $(\textsf{f},1_{\mathfrak{R}(x)})$ as a base element.

###### Proof.

On $\mathop{\rm Hom}\nolimits^{\textsf{f}}((x,\bullet_{x}),(y,\bullet_{y}))$,
one can define an addition by asking $(\textsf{f},r)+(\textsf{f},r^{\prime})$
to be $(\textsf{f},r+r^{\prime})$. For $l\in k$, one sets
$l(\textsf{f},r)=(\textsf{f},lr)$ and then the operations do give a $k$-module
structure.

Specializing to $\mathop{\rm
Aut}\nolimits^{\textsf{1}_{x}}(x,\bullet_{x})=\mathop{\rm
Hom}\nolimits^{\textsf{1}_{x}}((x,\bullet_{x}),(x,\bullet_{x}))$, it has an
multiplicative structure on top of the $k$-module structure. In fact, we can
set $(1_{x},r)(1_{x},r^{\prime})=(1_{x},rr^{\prime})$ and verify that it is
compatible with the $k$-module structure. It is easy to see that there exists
a natural isomorphism $\mathop{\rm
Aut}\nolimits^{\textsf{1}_{x}}(x,\bullet_{x})\cong\mathfrak{R}(x)$.

Finally, from $(\textsf{f},1_{\mathfrak{R}(x)})(1_{x},r)=(\textsf{f},r)$, we
can readily establish the third statement. ∎

Since there is no reasonable way to define a “sum” of $(\textsf{f},r)$ and
$(\textsf{f}^{\prime},r^{\prime})$, the set ${\rm
Hom}_{Gr_{{\mathcal{C}}}\mathfrak{R}}((x,\bullet_{x}),(y,\bullet_{y}))$ is not
an Abelian group itself. Nonetheless, motivated by the above analyse and the
work of Mitchell [13], we introduce an algebra on the Grothendieck
construction.

###  3.2. Skew category algebras

The idea is to consider the coproduct of all $k$-modules constructed in
Proposition 3.2 (1). However for future applications, we shall slightly
simplify the notations.

######  Definition 3.3.

Let ${\mathcal{C}}$ be a (non-empty) small category. Let
$\mathfrak{R}:{\mathcal{C}}^{op}\to k\text{-}{\rm Alg}$ be a presheaf of
$k$-algebras. The _skew category algebra_ $\mathfrak{R}[{\mathcal{C}}]$ on
${\mathcal{C}}$ with respect to $\mathfrak{R}$ is a $k$-module spanned over
elements of the form $r\textsf{f}$, where $\textsf{f}\in\mathop{\rm
Mor}\nolimits{\mathcal{C}}$ and $r\in\mathfrak{R}(\mathop{\rm
dom}(\textsf{f}))$. We define the multiplication on two base elements by the
rule

$\displaystyle s\textsf{g}\ast
r\textsf{f}=\begin{cases}(\mathfrak{R}(\textsf{f})(s)r)\textsf{g}\textsf{f},&\text{if}~{}{\rm
dom}(\textsf{g})={\rm cod}(\textsf{f});\\\ 0,&{\rm otherwise}.\end{cases}$

Extending this product linearly to two arbitrary elements,
$\mathfrak{R}[{\mathcal{C}}]$ becomes an associative $k$-algebra.

The algebra $\mathfrak{R}[{\mathcal{C}}]$ has an identity
$\Sigma_{x\in\mathop{\rm
Ob}\nolimits{\mathcal{C}}}1_{\mathfrak{R}(x)}\textsf{1}_{x}$ if
${\mathcal{C}}$ is object-finite.

If ${\mathcal{C}}$ is empty, the skew category algebra is understood to be the
null ring. The above construction will be used to characterize the category of
right $\mathfrak{R}$-modules over a presheaf $\mathfrak{R}$ of $k$-algebras.

######  Remark 3.4.

  1. (1)

If $\mathfrak{R}=\underline{k}:{\mathcal{C}}^{op}\to k\text{-}{\rm Alg}$ is
the constant presheaf such that $\underline{k}(x)=k$ for all $x\in\mathop{\rm
Ob}\nolimits{\mathcal{C}}$, then the skew category algebra
$\underline{k}[{\mathcal{C}}]$ on ${\mathcal{C}}$ with respect to
$\underline{k}$ is just the category algebra $k{\mathcal{C}}$ [16].

  2. (2)

If ${\mathcal{C}}$ be a group $G$, then $\mathfrak{R}=\mathfrak{R}(\bullet)$
($\bullet$ being the unique object of $G$ as a category) is simply a
$k$-algebra with a designated $G$-action, and the skew category algebra is
just a skew group algebra.

  3. (3)

If ${\mathcal{C}}={\mathcal{P}}$ is a poset, we recover the construction
$\mathfrak{R}!$ of Gerstenhaber and Schack [5].

We note that due to our convention the above constructions are slightly
different from their common forms in the literature. For example, in (2) the
group $G$ acts on $\mathfrak{R}$ on the right, while one usually sets
$\mathfrak{R}$ to be a left $G$-module, see for instance [3, III.4]. If we
consider the Grothendieck construction for covariant functors and the
resulting skew category algebra, then the difference will disappear.

###  3.3. Presheaves of modules

For a presheaf of $k$-algebras $\mathfrak{R}$ on an object-finite small
category ${\mathcal{C}}$, the following theorem characterizes the category
$\operatorname{\mathfrak{M}od-}\mathfrak{R}$ of right $\mathfrak{R}$-modules.

######  Theorem 3.5.

Let ${\mathcal{C}}$ be a small category and
$\mathfrak{R}:{\mathcal{C}}^{op}\to k\mbox{-}{\rm Alg}$ be a presheaf of
$k$-algebras on $\operatorname{\mathbf{C}}$. If $\mathop{\rm
Ob}\nolimits{\mathcal{C}}$ is finite, then we have the following equivalence

$\operatorname{\mathfrak{M}od-}\mathfrak{R}\simeq\operatorname{Mod-}\mathfrak{R}[{\mathcal{C}}].$

###### Proof.

We will prove by constructing a pair of functors, and then show that they give
rise to an equivalence between the categories. Recall that the category
$\operatorname{\mathfrak{M}od-}\mathfrak{R}$ is just the category of
preasheaves of $\mathfrak{R}$-modules.

Firstly, let us define a functor
$\Theta:\operatorname{\mathfrak{M}od-}\mathfrak{R}\to{\rm
Mod\text{-}}\mathfrak{R}[{\mathcal{C}}]$ by

$\mathfrak{M}\mapsto\bigoplus_{y\in{\mathcal{C}}}\mathfrak{M}(y).$

Here $\bigoplus_{y\in{\mathcal{C}}}\mathfrak{M}(y)$ is a right
$\mathfrak{R}[{\mathcal{C}}]$-module: for any $m\in\mathfrak{M}(y)$ and any
$r\textsf{f}\in\mathfrak{R}[{\mathcal{C}}]$, we can define an action

$\displaystyle m\cdot
r\textsf{f}=\begin{cases}\mathfrak{M}(\textsf{f})(m)\cdot
r,&\text{if}~{}\mathop{\rm cod}(\textsf{f})=y;\\\ 0,&\text{otherwise.}\\\
\end{cases}$

We are now at the position to verify that the operation defined above does
give ${\mathfrak{M}}$ a right $\mathfrak{R}[{\mathcal{C}}]$-module structure.
It is clear that the operation is closed. For the identity
$1_{\mathfrak{R}[{\mathcal{C}}]}\in\mathfrak{R}[{\mathcal{C}}]$, since

$m\cdot 1_{\mathfrak{R}(y)}\textsf{1}_{y}=\mathfrak{M}(\textsf{1}_{y})(m)\cdot
1_{\mathfrak{R}(y)}=m,$

it follows that

$m\cdot
1_{\mathfrak{R}[{\mathcal{C}}]}=m\cdot\sum_{y}1_{\mathfrak{R}(y)}\textsf{1}_{y}=m.$

It remains to show that

$(m\cdot s\textsf{g})\cdot r\textsf{f}=m\cdot(s\textsf{g}\ast r\textsf{f})$

for any two elements $s\textsf{g},r\textsf{f}\in\mathfrak{R}[{\mathcal{C}}]$.
We only need to consider the case that
$x\stackrel{{\scriptstyle\textsf{f}}}{{\to}}y\stackrel{{\scriptstyle\textsf{g}}}{{\to}}z$
are composable in ${\mathcal{C}}$, $m\in\mathfrak{M}(z)$,
$s\in\mathfrak{R}(y)$ and $r\in\mathfrak{R}(x)$. Then we have

$\displaystyle(m\cdot s\textsf{g})\cdot r\textsf{f}$ $\displaystyle=$
$\displaystyle(\mathfrak{M}(\textsf{g})(m)\cdot s)\cdot r\textsf{f}$
$\displaystyle=$
$\displaystyle\mathfrak{M}(\textsf{f})(\mathfrak{M}(\textsf{g})(m)\cdot
s)\cdot r,$

and

$\displaystyle m\cdot(s\textsf{g}\ast r\textsf{f})$ $\displaystyle=$
$\displaystyle m\cdot(\mathfrak{R}(\textsf{f})(s)r)\textsf{g}\textsf{f}$
$\displaystyle=$
$\displaystyle\mathfrak{M}(\textsf{g}\textsf{f})(m)\cdot(\mathfrak{R}(\textsf{f})(s)r)$
$\displaystyle=$
$\displaystyle\mathfrak{M}(\textsf{f})(\mathfrak{M}(\textsf{g})(m))\cdot(\mathfrak{R}(\textsf{f})(s)r)$
$\displaystyle=$
$\displaystyle(\mathfrak{M}(\textsf{f})(\mathfrak{M}(\textsf{g})(m))\cdot\mathfrak{R}(\textsf{f})(s))\cdot
r.$

Due to the right $\mathfrak{R}$-module structure on $\mathfrak{M}$, it follows
that

$\mathfrak{M}(\textsf{f})(\mathfrak{M}(\textsf{g})(m)\cdot
s)=\mathfrak{M}(\textsf{f})(\mathfrak{M}(\textsf{g})(m))\cdot\mathfrak{R}(\textsf{f})(s).$

Hence

$(m\cdot s\textsf{g})\cdot r\textsf{f}=m\cdot(s\textsf{g}\ast r\textsf{f}).$

With obvious map on morphisms, it is not difficult to see that $\Theta$ is a
functor.

Secondly, let us define another functor

$\Omega:{\rm
Mod\text{-}}\mathfrak{R}[{\mathcal{C}}]\to\operatorname{\mathfrak{M}od-}\mathfrak{R}$

by $N\mapsto\mathfrak{N}$. The presheaf
$\mathfrak{N}:{\mathcal{C}}^{op}\to\operatorname{Mod-}k$ is defined as follows

${\mathfrak{N}}(y)=N\cdot 1_{\mathfrak{R}(y)}\textsf{1}_{y},\forall
y\in\mathop{\rm Ob}\nolimits{\mathcal{C}}$

and

${\mathfrak{N}}(\textsf{f})=-\cdot 1_{\mathfrak{R}(x)}\textsf{f}:N\cdot
1_{\mathfrak{R}(y)}\textsf{1}_{y}\to N\cdot
1_{\mathfrak{R}(x)}\textsf{1}_{x},\forall\textsf{f}:x\to y.$

Our task now is to verify that $\mathfrak{N}$ is a right
$\mathfrak{R}$-module. Recall that, a presheaf of $\mathfrak{R}$-module
$\mathfrak{N}$ is a presheaf
$\mathfrak{N}:{\mathcal{C}}^{op}\to\operatorname{Mod-}k$ such that, for each
$x\in\mathop{\rm Ob}\nolimits{\mathcal{C}}$, $\mathfrak{N}(x)$ has a right
$\mathfrak{R}(x)$ action.

Let us first check that $\mathfrak{N}$ is a functor. For any $n\cdot
1_{\mathfrak{R}(y)}\textsf{1}_{y}\in N\cdot
1_{\mathfrak{R}(y)}\textsf{1}_{y}$, observe that

$(n\cdot 1_{\mathfrak{R}(y)}\textsf{1}_{y})\cdot
1_{\mathfrak{R}(x)}\textsf{f}=n\cdot(1_{\mathfrak{R}(x)}\textsf{f}\ast
1_{\mathfrak{R}(x)}\textsf{1}_{x})=(n\cdot 1_{\mathfrak{R}(x)}\textsf{f})\cdot
1_{\mathfrak{R}(x)}\textsf{1}_{x}\in N\cdot
1_{\mathfrak{R}(x)}\textsf{1}_{x},$

so $\mathfrak{N}(\textsf{f})=-\cdot 1_{\mathfrak{R}(x)}\textsf{f}$ is well
defined. Obviously,

$\mathfrak{N}(\textsf{1}_{y})=-\cdot 1_{\mathfrak{R}(y)}\textsf{1}_{y}={\rm
Id}_{N\cdot 1_{\mathfrak{R}(y)}\textsf{1}_{y}},$

and

$\displaystyle\mathfrak{N}(\textsf{f})\circ\mathfrak{N}(\textsf{g})$
$\displaystyle=$ $\displaystyle-\cdot 1_{\mathfrak{R}(y)}\textsf{g}\cdot
1_{\mathfrak{R}(x)}\textsf{f}$ $\displaystyle=$
$\displaystyle-\cdot(1_{\mathfrak{R}(y)}\textsf{g}\ast
1_{\mathfrak{R}(x)}\textsf{f})$ $\displaystyle=$
$\displaystyle-\cdot(\mathfrak{R}(\textsf{f})(1_{\mathfrak{R}(y)})1_{\mathfrak{R}(x)})\textsf{g}\textsf{f}$
$\displaystyle=$ $\displaystyle-\cdot 1_{\mathfrak{R}(x)}\textsf{g}\textsf{f}$
$\displaystyle=$ $\displaystyle\mathfrak{N}(\textsf{g}\textsf{f}).$

According to the fact that $\mathfrak{R}(\textsf{f})$ is a unital algebra
homomorphism, the fourth equality above holds. Thus we have checked that
$\mathfrak{N}$ is a functor.

The next thing to do is to check that $\mathfrak{N}$ has a right
$\mathfrak{R}$-module structure. For each $y\in{\rm Ob}{\mathcal{C}}$, there
is a right action of $\mathfrak{R}(y)$ on $N\cdot
1_{\mathfrak{R}(y)}\textsf{1}_{y}$ as follows. For any $n\cdot
1_{\mathfrak{R}(y)}\textsf{1}_{y}\in N\cdot 1_{\mathfrak{R}(y)}\textsf{1}_{y}$
and $r\in\mathfrak{R}(y)$, $r$ acts on $n\cdot
1_{\mathfrak{R}(y)}\textsf{1}_{y}$ by $r\textsf{1}_{y}$ from the right:
$(n\cdot 1_{\mathfrak{R}(y)}\textsf{1}_{y})\cdot r\textsf{1}_{y}=(n\cdot
r\textsf{1}_{y})\cdot 1_{\mathfrak{R}(y)}\textsf{1}_{y}\in N\cdot
1_{\mathfrak{R}(y)}\textsf{1}_{y}$. It is straightforward to verify that this
makes $\mathfrak{N}$ a right $\mathfrak{R}$-module.

From the definitions of $\Theta$ and $\Omega$, it is not difficult to see that

$\Omega\Theta\cong{\rm Id}_{\operatorname{\mathfrak{M}od-}\mathfrak{R}},$

and, moreover if ${\rm Ob}{\mathcal{C}}$ is finite,

$\Theta\Omega\cong{\rm Id}_{{\rm Mod\text{-}}\mathfrak{R}[{\mathcal{C}}]}.$

Thus we arrive at the conclusion that if ${\mathcal{C}}$ is object-finite then

$\operatorname{\mathfrak{M}od-}\mathfrak{R}\simeq{\rm
Mod\text{-}}\mathfrak{R}[{\mathcal{C}}].$

∎

######  Remark 3.6.

If ${\mathcal{C}}$ is not object-finite, then we only have a fully faithful
functor
$\Theta:\operatorname{\mathfrak{M}od-}\mathfrak{R}\hookrightarrow\operatorname{Mod-}\mathfrak{R}[{\mathcal{C}}]$.
For example, if $X$ is a topological space and ${\mathcal{P}}_{X}$ is the
poset of open subsets of $X$, then the classical sheaf theory can be recovered
as $\mathop{\rm Sh}\nolimits({\mathcal{P}}_{X})$, by giving
${\mathcal{P}}_{X}$ a canonical Grothendieck topology. In this case,
${\mathcal{P}}_{X}$ is not object-finite in general.

##  4\. Subcategory topologies and module categories

In [1, Exposé IV], Grothendieck and Verdier outlined a classification of
“topos finis”. A topos is said to be finite, if it is equivalent to the
presheaf topos $\mathop{\rm PSh}\nolimits({\mathcal{C}})$ over a finite
category ${\mathcal{C}}$.

######  Proposition 4.1 ([1] IV, Exercise 9.1.12).

Let ${\mathcal{C}}$ be a finite category.

  1. (1)

Every subtopos of a finite topos is again a finite topos.

  2. (2)

There is an order-reversing bijection between the poset of Grothendieck
topologies on ${\mathcal{C}}$, and that of the strictly full Karoubian
subcategories of ${\mathcal{C}}$.

  3. (3)

There is an order-reversing bijection between the poset of Grothendieck
topologies on ${\mathcal{C}}$, and that of the subtopoi of $\mathop{\rm
PSh}\nolimits({\mathcal{C}})$.

###### Proof.

The proof is a combination of several exercises in [1, Exposé IV], in Sections
7 and 9. It relies on considering the points of topoi, which are geometric
morphisms from Set to the presheaf topos in our case. Since it is involved and
is far from our purposes, we will reorganize the existing approaches and
describe another way to partially demonstrate the facts, which can be extended
to ringed finite sites and their module categories. ∎

The reader can revisit Examples 2.12 and 2.15, where every subcategory is a
strictly full Karoubian subcategory, to better understand the statements.

To establish the above statements, [1, IV Exercise 9.1.12] introduced a
topology on ${\mathcal{C}}$, for each strictly full subcategory
${\mathcal{D}}$. For reference, we recode it below and give the topology a
name. Suppose $x\in\mathop{\rm Ob}\nolimits{\mathcal{C}}$ and $\\{x_{i}\to
x\\}_{i}$ is a set of morphisms in ${\mathcal{C}}$. For any $w\in\mathop{\rm
Ob}\nolimits{\mathcal{C}}$, there is a natural set map

$\coprod_{x_{i}\to x}\mathop{\rm
Hom}\nolimits_{{\mathcal{C}}}(w,x_{i})\to\mathop{\rm
Hom}\nolimits_{{\mathcal{C}}}(w,x).$

######  Definition 4.2.

Let ${\mathcal{C}}$ be a finite category and ${\mathcal{D}}$ be a strictly
full subcategory. The _subcategory topology_ ${\mathcal{J}}^{{\mathcal{D}}}$
is given by defining ${\mathcal{J}}^{{\mathcal{D}}}(x)$, $\forall
x\in\mathop{\rm Ob}\nolimits{\mathcal{C}}$, to be the following

$\\{\mbox{sieves}\ S\ \mbox{on}\ x\bigm{|}\coprod_{y\to x\in S}\mathop{\rm
Hom}\nolimits_{{\mathcal{C}}}(w,y)\to\mathop{\rm
Hom}\nolimits_{{\mathcal{C}}}(w,x)\ \mbox{is surjective for all}\
w\in\mathop{\rm Ob}\nolimits{\mathcal{D}}\\}.$

To verify that the above is truly a Grothendieck topology on ${\mathcal{C}}$,
one can check that the three conditions in Definition 2.1 are satisfied. This
is tedious but routine.

######  Remark 4.3.

Let ${\mathcal{C}}$ be a finite category and ${\mathcal{D}}$ be a strictly
full subcategory. For each object $x\in\mathop{\rm Ob}\nolimits{\mathcal{C}}$,
the minimal covering sieve (possibly empty) in
${\mathcal{J}}^{{\mathcal{D}}}(x)$ is the one generated by all morphisms from
objects of ${\mathcal{D}}$ to $x$.

We are interested in the following result, stated in [1, Exposé IV, Exercise
9.1.12 (e)], which makes it possible to classify module categories over ringed
finite sites.

######  Proposition 4.4.

[1, Exposé IV, Exercise 9.1.12 (e)] Let ${\mathcal{C}}$ be a finite category.

  1. (1)

The subcategory topologies exhausts all possible topologies on a finite
category ${\mathcal{C}}$.

  2. (2)

Let ${\mathcal{D}}$ be a strictly full subcategory. Under the subcategory
topology ${\mathcal{J}}^{{\mathcal{D}}}$, there is an equivalence

$\mathop{\rm
Sh}\nolimits({\mathcal{C}},{\mathcal{J}}^{{\mathcal{D}}})\simeq\mathop{\rm
PSh}\nolimits({\mathcal{D}}).$

###### Proof.

The first part depends on a discussion on points of topoi, which we shall not
dwell into. A relatively elementary proof for Karoubian categories in one
direction can be found in [7, C 2.2 Lemma 2.2.21]. As to the second part, by
Definition 2.10, the full subcategory ${\mathcal{D}}$ equipped with
${\mathcal{J}}^{{\mathcal{D}}}|_{{\mathcal{D}}}$ is a dense subsite of
$({\mathcal{C}},{\mathcal{J}}^{{\mathcal{D}}})$. Moreover, since we must have
$1_{w}\in S$ for any $S\in{\mathcal{J}}^{{\mathcal{D}}}(w)$, $\forall
w\in\mathop{\rm Ob}\nolimits{\mathcal{D}}$,
${\mathcal{J}}^{{\mathcal{D}}}|_{{\mathcal{D}}}$ is the minimal topology on
${\mathcal{D}}$. Therefore we have an equivalence $\mathop{\rm
Sh}\nolimits({\mathcal{C}},{\mathcal{J}}^{{\mathcal{D}}})\simeq\mathop{\rm
PSh}\nolimits({\mathcal{D}})$, by the Comparison Lemma. ∎

The above equivalence can be extended to one between suitable ringed topoi.
The later will give us a classification of module categories on ringed finite
sites, together with the main result in the preceding section on presheaf
topoi.

######  Theorem 4.5.

Let ${\mathcal{C}}$ be a finite category and ${\mathcal{D}}$ be a strictly
full subcategory. Let $\mathfrak{R}$ be a sheaf of $k$-algebras on the site
$({\mathcal{C}},{\mathcal{J}}^{{\mathcal{D}}})$. Then we have category
equivalences

$\operatorname{\mathfrak{M}od-}\mathfrak{R}\simeq\operatorname{Mod-}\mathfrak{R}|_{{\mathcal{D}}}[{\mathcal{D}}].$

Here $\mathfrak{R}|_{{\mathcal{D}}}$ is the restriction of $\mathfrak{R}$ to
${\mathcal{D}}$.

###### Proof.

Given the equivalence of topoi

$\Psi=(\Psi^{-1},\Psi_{*})=(RK_{\iota},Res_{\iota}):\mathop{\rm
Sh}\nolimits({\mathcal{C}},{\mathcal{J}}^{{\mathcal{D}}}){\buildrel\simeq\over{\to}}\mathop{\rm
PSh}\nolimits({\mathcal{D}}),$

and a structure sheaf $\mathfrak{R}$ on $\mathop{\rm
Sh}\nolimits({\mathcal{C}},{\mathcal{J}}^{{\mathcal{D}}})$, we introduce a
morphism of ringed topoi

$(\Psi,\Psi^{\sharp}):(\mathop{\rm
Sh}\nolimits({\mathcal{C}},{\mathcal{J}}^{{\mathcal{D}}}),\mathfrak{R})\to(\mathop{\rm
PSh}\nolimits({\mathcal{D}}),\mathfrak{R}^{\prime}),$

where $\mathfrak{R}^{\prime}=\Psi_{*}(\mathfrak{R})$ and $\Psi^{\sharp}$ is
the identity. This is an equivalence of ringed topoi. Hence we obtain an
equivalence of module categories

$\operatorname{\mathfrak{M}od-}\mathfrak{R}\simeq\operatorname{\mathfrak{M}od-}\mathfrak{R}^{\prime}.$

But $\mathfrak{R}^{\prime}=\Psi_{*}(\mathfrak{R})=Res_{\iota}\mathfrak{R}$ is
the restriction of $\mathfrak{R}$ along the inclusion functor. Thus
$\mathfrak{R}^{\prime}=\mathfrak{R}|_{{\mathcal{D}}}$, and we have desired
equivalences

$\operatorname{\mathfrak{M}od-}\mathfrak{R}\simeq\operatorname{\mathfrak{M}od-}\mathfrak{R}|_{{\mathcal{D}}}\simeq\operatorname{Mod-}\mathfrak{R}|_{{\mathcal{D}}}[{\mathcal{D}}],$

by Theorem 3.5. ∎

If, for each $x\in\mathop{\rm Ob}\nolimits{\mathcal{D}}$, $\mathfrak{R}(x)$ is
Noetherian, then $\mathfrak{R}|_{{\mathcal{D}}}[{\mathcal{D}}]$ is finite-
dimensional and the coherent
$\mathfrak{R}|_{{\mathcal{D}}}[{\mathcal{D}}]$-modules are the finitely
generated ones [14].

######  Corollary 4.6.

Let $\mathbf{C}=({\mathcal{C}},\mathcal{J})$ be a finite site and
$\mathfrak{R}:{\mathcal{C}}^{op}\to k\mbox{\rm-Alg}$ be a sheaf of
$k$-algebras on $\operatorname{\mathbf{C}}$. If for each $x\in\mathop{\rm
Ob}\nolimits{\mathcal{D}}$, $\mathfrak{R}(x)$ is Noetherian, then there is a
category equivalence

${\mathop{\rm
coh}\nolimits}\text{-}\mathfrak{R}\simeq\operatorname{mod-}\mathfrak{R}|_{{\mathcal{D}}}[{\mathcal{D}}],$

for some full subcategory ${\mathcal{D}}$ uniquely determined by
${\mathcal{J}}$.

##  5\. Further remarks

From our perspective, it is important to understand the role of various finite
EI-categories in finite group representations. We end this paper with several
remarks, demonstrating promising connections between the present paper and
some other works.

###  5.1. On dense EI-sites

Finite EI-categories frequently occur in group representations and cohomology.
We consider sheaves on finite EI-categories under the dense topology, as an
example to illustrate methods developed earlier. Moreover, we will see that
the sheafification may be used to replace the right Kan extension in
Proposition 4.4(2) and Theorem 4.5. The following lemma will help us to
compute the sheafifications of presheaves.

######  Lemma 5.1.

Let ${\mathcal{C}}$ be a finite EI-category and ${\mathcal{J}}_{\rm den}$ the
dense topology. Then, for each object $x$, the minimal non-empty covering
sieve $S^{\min}_{x}$ is given by all the morphisms from the minimal objects to
$x$, namely

$S^{\min}_{x}=\\{\textsf{f}\in{\rm
Hom}_{{\mathcal{C}}}(y,x)~{}|~{}y\in\mathop{\rm
Ob}\nolimits({\mathcal{C}}_{\leq x})_{\min}\\}.$

Subsequently ${\mathcal{J}}_{\rm den}={\mathcal{J}}^{{\mathcal{C}}_{\min}}$.

###### Proof.

Firstly, we verify that $S_{x}$ is a sieve. For any morphism $\textsf{f}:y\to
x\in S_{x}$ and any morphism $\textsf{g}:z\to y$ with ${\rm
cod}(\textsf{g})={\rm dom}(\textsf{f})$, we have to show that $\textsf{fg}\in
S_{x}$. Note that $z\leq y$ and $y$ is minimal, we infer that $z\in\mathop{\rm
Ob}\nolimits({\mathcal{C}}_{\leq x})_{\min}$. It follows that $\textsf{fg}\in
S_{x}$.

Secondly, we shall show that $S_{x}$ is a covering sieve. For any
$\textsf{f}:y\to x\in{\rm Hom}_{{\mathcal{C}}}(-,x)$, we have to prove that
there exists a morphism $\textsf{g}:z\to y$ such that $\textsf{fg}\in S_{x}$.
Since ${\mathcal{C}}$ is finite, we can always find such a morphism
$\textsf{g}:z\to y$.

Finally, we will show that $S_{x}$ is minimal in ${\mathcal{J}}_{\rm den}(x)$.
That is, we have to check that $S_{x}\subseteq T$ for any
$T\in{\mathcal{J}}_{\rm den}(x)$. For any $\textsf{f}:y\to x\in S_{x}$, by
definition, $y\in\mathop{\rm Ob}\nolimits({\mathcal{C}}_{\leq x})_{\min}$.
Since $T\in{\mathcal{J}}_{\rm den}(x)$, according to the definition of the
dense topology, there exists a morphism $\textsf{g}:z\to y$ such that
$\textsf{fg}:z\to x\in T$. Now $y$ is minimal and clearly $[z]\leq[y]$, it
follows that $z\cong y$. Thus there exists a morphism $\textsf{h}:y\to z$. It
follows immediately that $\textsf{fgh}:y\to x\in T$ as $T$ is a sieve.
Evidently, $\textsf{gh}\in{\rm Aut}_{{\mathcal{C}}}(y)$, therefore
$\textsf{f}=\textsf{fgh}(\textsf{gh})^{-1}\in T$. This proves that
$S_{x}\subseteq T$. The non-emptiness of sieves follows from the definition of
the dense topology. ∎

Equivalently, the unique minimal covering sieve of ${\mathcal{J}}_{\rm
den}(x)$ can be characterized as a functor:

$\displaystyle S_{x}^{\rm
min}(y)=\begin{cases}\text{Hom}_{{\mathcal{C}}}(y,x),&\text{if}~{}y\in\mathop{\rm
Ob}\nolimits({\mathcal{C}}_{\leq x})_{\min};\\\ \emptyset,&{\rm
otherwise}.\end{cases}$

######  Proposition 5.2.

Let $\mathbf{C}=({\mathcal{C}},\mathcal{J}_{\rm den})$ be the dense site on a
finite EI-category. Then, for each presheaf
$\mathfrak{F}:{\mathcal{C}}^{op}\rightarrow{\rm Set}$ and each $x\in{\rm
Ob}{\mathcal{C}}$, the half-sheafification of $\mathfrak{F}$ can be obtained
as follows (with obvious maps)

$\mathfrak{F}^{\dagger}(x)\cong\underset{[y]\subset\mathop{\rm
Ob}\nolimits({\mathcal{C}}_{\leq
x})_{\min}}{\prod}\underset{i}{\biguplus}~{}~{}\mathfrak{F}(y)^{H_{i}}.$

where $H_{i}$’s are the stabilizers of the transitive $\mathop{\rm
Aut}\nolimits_{{\mathcal{C}}}(y)$-sets in the morphism set

${\rm
Hom}_{{\mathcal{C}}}(y,x)=\underset{i}{\biguplus}~{}~{}H_{i}\backslash\text{\rm
Aut}_{{\mathcal{C}}}(y).$

Moreover, $\mathfrak{F}^{a}\cong\mathfrak{F}^{\dagger}$. Consequently,
$\mathfrak{F}^{\dagger}$ is a sheaf.

Let $\iota:{\mathcal{C}}_{\min}\hookrightarrow{\mathcal{C}}$ be the inclusion.
Any presheaf ${\mathfrak{G}}$ on ${\mathcal{C}}_{\min}$ can be naturally
extended to a presheaf ${\mathfrak{G}}^{\prime}$ on ${\mathcal{C}}$ satisfying
$({\mathfrak{G}}^{\prime})^{a}\cong RK_{\iota}{\mathfrak{G}}$.

###### Proof.

We proceed to prove this proposition by direct computation. Given a presheaf
$\mathfrak{F}$, let us first compute its half-sheafification. On an object
$x\in{\rm Ob}{\mathcal{C}}$,

$\displaystyle\mathfrak{F}^{\dagger}(x)$ $\displaystyle=$
$\displaystyle\mathop{\varinjlim}\nolimits_{S\in\mathcal{J}(x)}\text{Nat}(S,\mathfrak{F})$
$\displaystyle\cong$ $\displaystyle\text{Nat}(S_{x}^{\rm min},\mathfrak{F})$
$\displaystyle\cong$ $\displaystyle\underset{[y]\subset\mathop{\rm
Ob}\nolimits({\mathcal{C}}_{\leq x})_{\min}}{\prod}\text{Hom}(S_{x}^{\rm
min}(y),\mathfrak{F}(y))$ $\displaystyle\cong$
$\displaystyle\underset{[y]\subset\mathop{\rm Ob}\nolimits({\mathcal{C}}_{\leq
x})_{\min}}{\prod}\text{Hom}_{\text{Aut}_{{\mathcal{C}}}(y)}(\text{Hom}_{{\mathcal{C}}}(y,x),\mathfrak{F}(y))$
$\displaystyle\cong$ $\displaystyle\underset{[y]\subset\mathop{\rm
Ob}\nolimits({\mathcal{C}}_{\leq
x})_{\min}}{\prod}\text{Hom}_{\text{Aut}_{{\mathcal{C}}}(y)}\left(\underset{i}{\biguplus}~{}~{}H_{i}\backslash\text{Aut}_{{\mathcal{C}}}(y),\mathfrak{F}(y)\right)$
$\displaystyle\cong$ $\displaystyle\underset{[y]\subset\mathop{\rm
Ob}\nolimits({\mathcal{C}}_{\leq
x})_{\min}}{\prod}\underset{i}{\biguplus}~{}~{}\mathfrak{F}(y)^{H_{i}}.$

Here $[y]$ is the isomorphism class of the object $y$ in ${\mathcal{C}}$. The
third isomorphism above follows from the characterization of the minimal
sieve. Particularly if $y\in\mathop{\rm Ob}\nolimits{\mathcal{C}}_{\min}$,
then $S_{y}^{\rm min}(y)=\text{Aut}_{{\mathcal{C}}}(y)$. Thus

$\mathfrak{F}^{\dagger}(y)\cong\text{Hom}_{\text{Aut}_{{\mathcal{C}}}(y)}(\text{Aut}_{{\mathcal{C}}}(y),\mathfrak{F}(y))\cong\mathfrak{F}(y).$

For a morphism $\textsf{f}:w\to x$, one can see that
${\mathfrak{F}}^{\dagger}(\textsf{f}):{\mathfrak{F}}^{\dagger}(x)\to{\mathfrak{F}}^{\dagger}(w)$
is given by the restriction on fixed points, because f induces an $\mathop{\rm
Aut}\nolimits_{{\mathcal{C}}}(y)$-map

$\mathop{\rm Hom}\nolimits_{{\mathcal{C}}}(y,w)\to\mathop{\rm
Hom}\nolimits_{{\mathcal{C}}}(y,x)$

and subsequently an injective group homomorphism between their stabilizers in
$\mathop{\rm Aut}\nolimits_{{\mathcal{C}}}(y)$.

Repeating the above steps, we get the sheafification of $\mathfrak{F}$:

$\displaystyle\mathfrak{F}^{a}(x)$ $\displaystyle=$
$\displaystyle(\mathfrak{F}^{\dagger})^{\dagger}(x)$ $\displaystyle\cong$
$\displaystyle\text{Nat}(S_{x}^{\rm min},\mathfrak{F}^{\dagger})$
$\displaystyle\cong$ $\displaystyle\underset{[y]\subset\mathop{\rm
Ob}\nolimits({\mathcal{C}}_{\leq
x})_{\min}}{\prod}\text{Hom}_{\text{Aut}_{{\mathcal{C}}}(y)}(S_{x}^{\rm
min}(y),\mathfrak{F}^{\dagger}(y))$ $\displaystyle\cong$
$\displaystyle\underset{[y]\subset\mathop{\rm Ob}\nolimits({\mathcal{C}}_{\leq
x})_{\min}}{\prod}\text{Hom}_{\text{Aut}_{{\mathcal{C}}}(y)}(S_{x}^{\rm
min}(y),\mathfrak{F}(y))$ $\displaystyle\cong$
$\displaystyle\mathfrak{F}^{\dagger}(x).$

One readily checks that $\mathfrak{F}^{\dagger}$ and $\mathfrak{F}^{a}$
coincide on morphisms as well. Therefore
$\mathfrak{F}^{a}\cong\mathfrak{F}^{\dagger}$, it follows that
$\mathfrak{F}^{\dagger}$ is a sheaf.

In the end, given a presheaf ${\mathfrak{G}}$ on ${\mathcal{C}}_{\min}$, one
can define a presheaf ${\mathfrak{G}}^{\prime}$ on ${\mathcal{C}}$ by asking
${\mathfrak{G}}^{\prime}(y)={\mathfrak{G}}(y)$ if $y$ is minimal, and
${\mathfrak{G}}^{\prime}(y)=\emptyset$ otherwise. Now by our previous
discussions, both $({\mathfrak{G}}^{\prime})^{a}$ and
$RK_{\iota}{\mathfrak{G}}$ are sheaves on $\operatorname{\mathbf{C}}_{\rm
den}=({\mathcal{C}},{\mathcal{J}}_{\rm den})$. By Theorem 4.5, we must have
$({\mathfrak{G}}^{\prime})^{a}\cong RK_{\iota}{\mathfrak{G}}$ because they
agree on ${\mathcal{C}}_{\min}$. ∎

######  Remark 5.3.

Let $\mathbf{C}=({\mathcal{C}},\mathcal{J}_{\rm den})$ be the dense site on a
finite EI-category.

  1. (1)

We emphasize that the last statement of Proposition 5.2 works for an arbitrary
co-ideal ${\mathcal{D}}$ of ${\mathcal{C}}$. For such a strictly full
subcategory, any presheaf on ${\mathcal{D}}$ can be extended to a presheaf on
${\mathcal{C}}$, as in the above proof. Therefore, the right Kan extension may
be computed by the sheafification, which provide another way to compare two
sheaf categories.

  2. (2)

From Proposition 5.2, one can quickly deduce the following equivalence

$\mathop{\rm Sh}\nolimits(\operatorname{\mathbf{C}}_{\rm
den})\simeq\underset{[y]\subset\mathop{\rm
Ob}\nolimits{\mathcal{C}}_{\min}}{\prod}\operatorname{Set-}{\rm
Aut}_{{\mathcal{C}}}(y),$

which matches with Proposition 4.4(2). Let $\mathfrak{R}$ be a sheaf of
$k$-algebras on $\operatorname{\mathbf{C}}_{\rm den}$. Passing to the module
categories, we obtain an equivalence
$\operatorname{\mathfrak{M}od-}\mathfrak{R}\simeq\prod_{[y]\subset\mathop{\rm
Ob}\nolimits{\mathcal{C}}_{\min}}\operatorname{Mod-}\mathfrak{R}(y)[\mathop{\rm
Aut}\nolimits_{{\mathcal{C}}}(y)]\simeq\operatorname{Mod-}\prod_{[y]\subset\mathop{\rm
Ob}\nolimits{\mathcal{C}}_{\min}}\mathfrak{R}(y)[\mathop{\rm
Aut}\nolimits_{{\mathcal{C}}}(y)]$.

The preceding result provides some kind of “block decomposition” of
$\operatorname{\mathfrak{M}od-}\mathfrak{R}$.

######  Example 5.4.

Let $G$ be a finite group and $p$ be a prime number such that $p\bigm{|}|G|$.
Consider
$\mathbf{O}_{p}^{\circ}(G)=({\mathcal{O}}_{p}^{\circ}(G),\mathcal{J}_{\rm
den})$ (see Section 2.4). Let $\underline{k}$ be the constant sheaf given by
$k$. Then

$\mathop{\rm
Sh}\nolimits(\mathbf{O}_{p}^{\circ}(G),k)=\operatorname{\mathfrak{M}od-}\underline{k}\simeq\prod_{P}\operatorname{Mod-}k(N_{G}(P)/P),$

where $P$ runs over the set of conjugacy classes of order $p$-subgroups.

###  5.2. Vertices and sources

In [16], given a finite EI-category ${\mathcal{C}}$, one can introduce a
vertex (which is a unique full subcategory of ${\mathcal{C}}$) and a source
for each finite-dimensional indecomposable $k{\mathcal{C}}$-module (covariant
functor from ${\mathcal{C}}$ to $\operatorname{Mod-}k$) ${\mathfrak{M}}$. Let
${\mathfrak{N}}$ be a presheaf on ${\mathcal{C}}$. In light of sheaf theory,
there is a (unique) finest topology ${\mathcal{J}}_{{\mathfrak{N}}}$, for
which ${\mathfrak{N}}:{\mathcal{C}}^{op}\to\operatorname{Mod-}k$ becomes a
sheaf on $({\mathcal{C}},{\mathcal{J}}_{{\mathfrak{N}}})$. Now we know that
${\mathcal{J}}_{{\mathfrak{N}}}={\mathcal{J}}^{{\mathcal{D}}}$ for a unique
strictly full subcategory ${\mathcal{D}}$. It is interesting to compare
${\mathcal{D}}$ with the vertex of ${\mathfrak{N}}$ (when it is
indecomposable), because they seem to be closely related. Plus, the
construction of ${\mathcal{D}}$ for an indecomposable module works for any
finite category, not just the finite EI-categories. Moreover it is defined for
arbitrary modules, even the infinite-dimensional ones.

###  5.3. Sipp-topology on $G$-Set

There is a sipp-topology introduced on $G$-Set by Balmer [4]. It may be
restricted to the finite EI-category ${\mathcal{O}}(G)$, the orbit category of
$G$. One can verify that the sipp-topology is a subcategory topology on
${\mathcal{O}}(G)$, given by the strictly full subcategory
${\mathcal{O}}_{p}(G)$, the $p$-orbit category (see Section 2.4). As a
consequence $\mathop{\rm Sh}\nolimits(G\mbox{-Set},{\mathcal{J}}_{\rm
sipp})\simeq\mathop{\rm Sh}\nolimits({\mathcal{O}}(G),{\mathcal{J}}_{\rm
sipp})\simeq\mathop{\rm PSh}\nolimits({\mathcal{O}}_{p}(G))$. Since every
strictly full subcategory of ${\mathcal{O}}(G)$ defines a topology, it would
be interesting to learn how it can be used to understand group
representations.

###  5.4. Hochschild cohomology of
$\operatorname{\mathfrak{M}od-}\mathfrak{R}$

Let $\operatorname{\mathbf{C}}$ be a finite site and
${\mathcal{J}}^{{\mathcal{D}}}$ be the subcategory topology. Suppose
$\mathfrak{R}$ is a sheaf of $k$-algebras on $\operatorname{\mathbf{C}}$.
Since $\mathfrak{R}|_{{\mathcal{D}}}[{\mathcal{D}}]$ is an associative
algebra, one can consider its Hochschild cohomology. Due to the category
equivalence we established earlier, Hochschild cohomology theory for
$\mathfrak{R}|_{{\mathcal{D}}}[{\mathcal{D}}]$ can be considered as one for
the Abelian category $\operatorname{\mathfrak{M}od-}\mathfrak{R}$ on the
ringed site $(\operatorname{\mathbf{C}},\mathfrak{R})$, in which
$\operatorname{\mathbf{C}}=({\mathcal{C}},{\mathcal{J}}^{{\mathcal{D}}})$. In
light of Remark 3.6, it is curious to know weather this coincides with the one
introduced by Lowen and Van den Bergh [10] for Abelian categories, where they
had module categories over ringed spaces in mind.

## References

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#  Vignat: Vulnerability Identification by Learning Code Semantics via Graph
Attention NeTworks

Shuo Liu
Columbia University
New York, NY
<EMAIL_ADDRESS>
Gail Kaiser
Columbia University
New York, NY
<EMAIL_ADDRESS>

###### Abstract

Vulnerability identification is crucial to protect software systems from
attacks for cyber-security. However, huge projects have more than millions of
lines of code, and the complex dependencies make it hard to carry out
traditional static and dynamic methods. Furthermore, the semantic structure of
various types of vulnerabilities differs greatly and may occur simultaneously,
making general rule-based methods difficult to extend. In this paper, we
propose Vignat, a novel attention-based framework for identifying
vulnerabilities by learning graph-level semantic representations of code. We
represent codes with code property graphs (CPGs) in fine grain and use graph
attention networks (GATs) for vulnerability detection. The results show that
Vignat is able to achieve $57.38\%$ accuracy on reliable datasets derived from
popular C libraries. Furthermore, the interpretability of our GATs provides
valuable insights into vulnerability patterns.

_K_ eywords Vunerability identification $\cdot$ Graph attention networks
$\cdot$ Code property graph

##  1 Introduction

Vulnerability identification is crucial in ensuring the security, integrity,
and resilience of systems, networks, and applications. By proactively
identifying and addressing vulnerabilities, organizations can reduce the risk
of security breaches, disruption of services and malware propagation.
Vulnerability identification has promising applications in fields such as
software development, where it can be used to improve code quality and reduce
the likelihood of introducing vulnerabilities in the first place, and in
security research, where it can be used to develop new defensive technologies
and techniques.

Some works identify vulnerabilities by executing testing programs and
analyzing runtime behaviors, such as [sutton2007fuzzing], [cadar2008klee], and
[stephens2016driller]. However, due to the limited code coverage, execution
dependencies, and substantial overhead associated with dynamic approaches,
researchers start to focus on more efficient static vulnerability detection
methods. There are several challenges that make static vulnerability detection
difficult. Since large-scale projects often consist of millions of lines and
exhibit complex calling relationships within and between projects, it is hard
to effectively feature and analyze the code. Moreover, real-world
vulnerabilities are sparse, which can cause bias when confronted with new and
unseen patterns.

Some rule-based static analyzers have been developed to locate vulnerabilities
using defined patterns or signatures that match known vulnerabilities or
programming errors, [lawall2008coccinelle], [clang2019analyzer], and
[wheeler2021flawfinder]. While these methods are useful in detecting common
types of vulnerabilities, they heavily rely on predefined rules and struggle
to handle complex projects, making them unsatisfactory in real-world
scenarios. Besides, some transformer-based models are also applied to this
task because of their sequence processing ability and interpretability,
[BERT], [Sanh2019DistilBERTAD], [RoBERTA], and [feng2020codebert].
Nonetheless, these studies are difficult to obtain a comprehensive
understanding of program semantics solely. Because, unlike natural language
sequences, source codes are more structured, and the vulnerabilities can often
be subtle flaws that need thorough examination from various semantic
perspectives. Therefore, it seems a more reasonable way to find the patterns
of vulnerabilities by analyzing the complex relationships in various code
representations.

In this paper, we address these challenges by constructing graph embeddings of
code functions and analyzing the logical relationships between tokens to
predict code vulnerability. Using the self-attention mechanism
[vaswani2017attention], our models exhibit excellent explainability, enabling
us to infer vulnerability patterns instead of relying on experience to detect
vulnerabilities. Our main contributions are:

  * •

We obtain a comprehensive representation of code functions by embedding them
into CPGs to capture code syntactic structure, control flow, and data
dependencies.

  * •

We propose a GAT framework, Vignat, for capturing and modeling complex
relationships among nodes in CPGs, where higher attention edges can reveal
patterns of vulnerabilities.

  * •

We evaluate the effectiveness of Vignat on manually labeled datasets collected
from 4 large-scale C projects. Vignat achieves better results on the dataset
with various attention-based models, up to 10% accuracy and 5% F-score
improvement over baseline methods.

Figure 1: An overview of Vignat framework.

The rest of this paper is organized as follows. Section 2 presents Vignat
framework. Section LABEL:sec:evaluation demonstrates the performance and
illustrates the interpretability of our model. Section LABEL:sec:conclu
concludes this paper.

##  2 Vignat Framework

This section provides an overview of our Vignat framework, as shown in Fig. 1.
Within the Vignat framework, we first tokenize the source code functions and
construct CPGs, a composite representation of code semantics. Following this,
we employ various embedding methods to obtain node embeddings. In conjunction
with graph connectivity, the graphs are inputted into a GAT for graph-level
predictions. By extracting attention weights from the output of the attention
layer, we can identify salient edges, revealing patterns associated with
vulnerabilities.

###  2.1 Graph Embedding of Code

Aside from the semantics conveyed by code tokens and the logic present in
natural code sequencing (NCS), highly structured graph representations of code
obscure a significant amount of logical information. The abstract syntax tree
(AST) is a tree-like representation that encodes how statements and
expressions are nested to produce a program. Inner nodes denote operators,
leaf nodes denote operands, and edges specify container and content
relationships. The control flow graph (CFG) describes the order in which code
statements are executed. It also shows the conditions that must be met for a
particular execution path. In a CFG, nodes represent statements or predicates,
and edges denote the paths the program can traverse. The program dependence
graph (PDG) represents dependencies among statements and predicates in a
program. Data dependence edges show how a node’s outcome impacts a variable in
another node, while control dependence edges reveal the effect of predicates
on variable values. Integrating elements of ASTs, CFGs, PDGs, a CPG provides a
unified representation for program analysis and enables us to simultaneously
reason about all perspectives of code properties. A CPG for a vulnerable code
snippet is shown in Fig. LABEL:fig:cpg.

⬇

1 int x = source();

2 if (isEven(x)) {

3 proceed(10 / x);

4 }

5}\end{lstlisting}

6 % \caption{}

7 % \label{subfig:code}

8 \end{subfigure}

9 \hfill

10 \begin{subfigure}[b]{0.72\textwidth}

11 \centering

12 \includegraphics[width=\textwidth]{cpg.pdf}

13 % \caption{}

14 % \label{subfig:cpg}

15 \end{subfigure}

16 \caption{A vulnerable code snippet and its corresponding CPG.}

17 \label{fig:cpg}

18\end{figure*}

19

20In this paper, we make use of an open-source code analysis platform
\texttt{Joern} to parse source code functions and construct CPGs in batches.
In a CPG, nodes represent program constructs, including variables, methods,
control structure, etc. Each node contains several tokens and has some
attributes according to its type, such as the name of local variables, the
signature of the method, and the type of control structure. Nodes are
connected by directed edges to represent relationships in the program
structure. In \textit{Vignat}, Considering that a node may have different
numbers of tokens in it, we obtain its embedding by averaging its token
embeddings. To make the graph embedding tractable for graph neural networks
(GNNs), we simplify the CPGs by disregarding the heterogeneity of various
relationships and eliminating duplicate edges. Only the connectivity
information is passed to the subsequent GAT model.

21

22\subsection{GAT}

23

24Graph Neural Networks (GNNs) have emerged as a family of models for learning
representations of graph-structured data. A graph $\mathcal{G} = (\mathcal{V},
\mathcal{E})$ consists of a set of nodes $\mathcal{V}$ and edges
$\mathcal{E}$, where each edge $(i, j)$ represents a connection between nodes
$i$ and $j$. The key idea behind GNNs is to iteratively update node
representations by aggregating information from neighboring nodes. Let
$\mathbf{X}$ be the initial node feature matrix, where each row corresponds to
the feature vector of a node. Mathematically, the propagation of a GNN can be
expressed as:

25

26\begin{equation}

27 \mathbf{h}^{(l)} = \sigma\left(\mathbf{A} \cdot
ReLU\left(\mathbf{W}^{(l-1)} \cdot \mathbf{h}^{(l-1)}\right)\right)

28\end{equation}

29

30where $\mathbf{h}^{(l)}$ is the node representation matrix at layer $l$,
$\mathbf{W}^{(l)}$ is the learnable weight matrix, $\mathbf{A}$ is the
adjacency matrix capturing graph connectivity, and $\sigma$ denotes an
activation function. This layer-wise aggregation process allows GNNs to
capture complex relationships in graph-structured data and learn expressive
node representations.

31

32GAT was introduced by \cite{velickovic2018graph} as a novel approach for
graph representation learning, which can naturally generalize convolutions to
irregular graph structures. GAT incorporates self-attention mechanisms, which
enables the model to weigh the importance of neighboring nodes for the target
one, thus capturing the local code structure of the CPG.

33

34The GAT model consists of a series of graph attention layers. Each layer
computes a new set of node embeddings based on the input node features and the
learned attention coefficients. The attention coefficients between nodes $i$
and $j$ in layer $l$ can be computed as follows:

35\begin{equation}

36 e_{ij}^{(l)} =
LeakyReLU\left(\textbf{a}^{(l)T}\left[\textbf{W}^{(l)}\textbf{h}_i^{(l-1)}
\parallel \textbf{W}^{(l)}\textbf{h}_j^{(l-1)}\right]\right),

37\end{equation}

38

39where $\parallel$ denotes the concatenation operation, and
$\textbf{a}^{(l)}$ is a learnable vector.

40

41To ensure that the coefficients are normalized across the neighboring nodes,
we apply the softmax function:

42\begin{equation}

43 \alpha_{ij}^{(l)} = softmax_j\left(e_{ij}^{(l)}\right) =
\frac{exp(e_{ij}^{(l)})}{\sum_{k \in \mathcal{N}(i)}^{} exp(e_{ik}^{(l)})},

44\end{equation}

45

46where $\mathcal{N}(i)$ denotes the set of neighboring nodes of node $i$. The
new node embeddings for layer $l$ are computed as a weighted sum of the input
features of the neighboring nodes:

47\begin{equation}

48 \textbf{h}_i^{(l)} = \sigma\left(\sum_{j \in \mathcal{N}(i)}^{}
\alpha_{ij}^{(l)}\textbf{W}^{(l)}\textbf{h}_j^{(l-1)}\right),

49\end{equation}

50

51where $\sigma$ is an activation function, such as the rectified linear unit
(ReLU).

52

53One of the key advantages of GAT is its explainability. In contrast to graph
convolutional networks (GCNs) \cite{kipf2016semi}, where the aggregation of
neighboring node features is often performed uniformly, making it difficult to
discern the influence of specific nodes, GAT provides an interpretable measure
of the importance of neighboring nodes for a target node. By investigating the
attention coefficients $e_{ij}^{(l)}$, we are able to gain insights into the
rationale of our model, which is useful for identifying influential paths and
addressing potential biases.

54

55\section{Experiments} \label{sec:evaluation}

56

57We evaluate \textit{Vignat} with the following research questions.

58

59\begin{enumerate}

60 \item How does \textit{Vignat} compare to the transformers that are based
on the NCS?

61 \item How does \textit{Vignat} perform on different kinds of code
representations?

62 \item How do different kinds of word embedding methods affect the
performance of \textit{Vignat}?

63 \item How does attention-mechanism powered \textit{Vignat} compare to GCNs?

64 \item How to infer patterns of code vulnerabilities through attention
coefficients?

65\end{enumerate}

66

67\subsection{Setup}

68

69We use the \textit{Devign} dataset to evaluate our models
\cite{zhou2019devign}. Compared to datasets with labels generated by static
analyzers and artificial ones, it is more reliable and maintains a sparsity of
vulnerabilities and is more reliable by manually labeling vulnerabilities in
large-scale, real-world C projects. Specifically, we focused on two projects,
\texttt{FFmpeg} and \texttt{QEMU}, and selected medium-length functions
containing less than $1200$ tokens as samples for model training and testing.
As the DFG edges are labeled with the variables involved, it tremendously
complicates embedded graphs. Therefore, we only extracted ASTs and CFGs and
combined them into composite graph as CPGs for our dataset using
\texttt{Joern}. We calculate the average embedding of tokens within a node to
obtain its embedding and standardize its embedding size to $768$ using
pretrained models, BERT, DistilBERT, RoBERTa, and CodeBERT. If a CPG has more
than $225$ nodes, we cut the first $225$ ones; otherwise, we use padding zeros
to keep the size of input code features same.

70

71We configure the GAT models as follows. Models are configured with the input
hidden dimension of $225$ and learning rate $0.0001$. We implement our models
using PyTorch 2.0.0, CUDA 11.8, and train them $100$ epochs with batch size
$8$ using 16GB Tesla V100-SXM2 GPU. It takes us $0.0358$ seconds to generate
each CPG in average. And we takes $116$ minutes to train For \textit{Vignat}
with Roberta embedding on \texttt{FFmpeg}, we spend $116$ minutes to train; It
takes us $127$ minutes to train \textit{Vignat} with Roberta embedding on
\texttt{QEMU}.

72

73\subsection{Result Analysis}

74

75To evaluate the performance of \textit{Vignat}, we compare it with some
baseline methods, including state-of-the-art transformers based on the NCS and
GCNs based on graph representation of codes. To avoid the resulting gap caused
by data heterogeneity, we provide these baselines with the same training set
and test set after shuffle and split. We measure the performance of different
methods from 4 metrics, accuracy, precision, recall, and F1 score. The
comparison results are shown in Table. \ref{table:compare}.

76

77\renewcommand{\arraystretch}{1.3}

78\begin{table*}[ht]

79{

80\scriptsize

81%\fontsize{8}{1}\selectfont

82 \centering

83 \begin{tabular}{c|c|c|cccc}

84 \hline \hline

85 \multirow{2}{*}{\makecell{Class}} & \multirow{2}{*}{\makecell{Embedding}} &
\multirow{2}{*}{\makecell{Representation}} & \multicolumn{4}{c}{FFmpeg} \\\

86 \cline{4-7} & & & Acc & Prec & Rec & F1 \\\

87 \hline

88 \multirow{4}{*}{Transformer} & BERT & \multirow{4}{*}{NCS} & $53.33\%$ &
$55.77\%$ & $90.63\%$ & $68.89\%$

89 \\\

90 & DistilBERT & & $56.67\%$ & $54.45\%$ & $96.88\%$ & $69.05\%$

91 \\\

92 & RoBERTa & & $53.33\%$ & $53.85\%$ & $87.50\%$ & $66.67\%$

93 \\\

94 & CodeBERT & & $51.67\%$ & $52.83\%$ & $87.50\%$ & $65.88\%$

95 \\\

96 \hline

97 \multirow{15}{*}{GCN} &\multirow{3}{*}{Word2Vec} & AST & $54.09\%$ &
$53.70\%$ & $90.63\%$ & $67.44\%$

98 \\\

99 & & CFG &$52.46\%$ & $52.46\%$ & $100.00\%$ & $68.82\%$

100 \\\

101 & & CPG & $55.74\%$ & $56.10\%$ & $71.88\%$ & $63.10\%$

102 \\\

103 \cline{2-7} &\multirow{3}{*}{BERT} & AST & $50.82\%$ & $52.00\%$ &
$81.25\%$ & $63.41\%$

104 \\\

105 & & CFG & $52.46\%$ & $52.46\%$ & $100.00\%$ & $68.82\%$

106 \\\

107 & & CPG & $57.38\%$ & $56.82\%$ & $78.13\%$ & $65.79\%$

108 \\\

109 \cline{2-7} &\multirow{3}{*}{DistilBERT} & AST & $50.82\%$ & $51.92\%$ &
$84.37\%$ & $64.29\%$

110 \\\

111 & & CFG & $54.10\%$ & $53.57\%$ & $93.75\%$ & $68.18\%$

112 \\\

113 & & CPG & $50.82\%$ & $51.92\%$ & $84.38\%$ & $64.29\%$

114 \\\

115 \cline{2-7} &\multirow{3}{*}{\textbf{RoBERTa}} & AST & $50.82\%$ &
$51.92\%$ & $84.37\%$ & $64.29\%$

116 \\\

117 & & CFG & $52.46\%$ & $52.46\%$ & $100.00\%$ & $68.82\%$

118 \\\

119 & & \textbf{CPG} & $\textbf{57.38\%}$ & $\textbf{56.52\%}$ &
$\textbf{81.25\%}$ & $\textbf{66.67\%}$

120 \\\

121 \cline{2-7} &\multirow{3}{*}{CodeBERT} & AST & $52.46\%$ & $52.46\%$ &
$56.25\%$ & $55.38\%$

122 \\\

123 & & CFG & $52.46\%$ & $52.46\%$ & $100.00\%$ & $68.82\%$

124 \\\

125 & & CPG & $57.38\%$ & $57.50\%$ & $71.88\%$ & $63.89\%$

126 \\\

127 \hline

128 \multirow{6}{*}{\textbf{GAT}} &\multirow{3}{*}{Word2Vec} & AST & $55.74\%$
& $56.10\%$ & $71.86\%$ & $63.01\%$

129 \\\

130 & & CFG &$52.46\%$ & $52.46\%$ & $100.00\%$ & $68.82\%$

131 \\\

132 & & \textbf{CPG} & $\textbf{63.93\%}$ & $\textbf{63.16\%}$ &
$\textbf{75.00\%}$ & $\textbf{68.58\%}$

133 \\\

134 \cline{2-7} &\multirow{3}{*}{BERT} & AST & $59.02\%$ & $57.14\%$ &
$87.50\%$ & $69.14\%$ \\\

135 & & CFG & $52.46\%$ & $52.46\%$ & $100.00\%$ & $68.82\%$

136 \\\

137 & & CPG & $60.66\%$ & $59.09\%$ & $81.25\%$ & $68.42\%$ \\\

138 \hline \hline

139 \end{tabular}

140 \caption{Comparison of different methods. Acc, Prec, Rec, F1 represent
accuracy, precision, recall, and F1 score respectively.}

141 \label{table:compare}}

142\end{table*}

143

144As shown in Table. \ref{table:compare}, on \texttt{FFmpeg} dataset,
\textit{Vignat} has the best performance over all models. Compared with
transformer baselines, it improves $4.05\%$ accuracy over BERT and RoBERTa,
and $5.71\%$ over CodeBERT. We find that CPGs provide the most comprehensive
representation among the graphs, as the accuracy improves $6.56\%$ over ASTs,
and $4.92\%$ over CFGs. Besides, experiments also show that different
embedding methods have an impact on model performance. RoBERTa performs well
on all data sets. For \textit{Vignat} using GCNs, though it does not show a
substantial enhancement compared to alternative embedding techniques on
accuracy, it has a better overall performance with higher precision and
recall.

145

146\subsection{Pattern Explanation}

147

148Fig. \ref{fig:cpg_marked} illustrates 5 CPG edges with the largest values
of attention coefficients $e^{(l)}_{ij}$ for the source code in Fig.
\ref{fig:cpg}. In this code example predicted to be vulnerable, the type and
operator of a variable are highlighted with red arrows. These edges reveal the
vulnerabilities that could occur in this code snippet. Specifically, the
integer may overflow when declaring the variable and the divisor can be zero,
and these vulnerabilities will affect the following proceeding functions.

149

150\begin{figure*}[htb]

151 \centering

152 \includegraphics[scale=.5]{cpg-marked.pdf}

153 \caption{5 edges with highest attention in Fig. \ref{fig:cpg}.}

154 \label{fig:cpg_marked}

155\end{figure*}

156

157\section{Conclusion}

158\label{sec:conclu}

159

160This paper presents \textit{Vignat}, an innovative attention-based
framework for vulnerability identification. By leveraging CPGs to
comprehensively capture the syntactic structure, control flow, and data
dependencies of code functions, and employing GATs to model complex
relationships among nodes, \textit{Vignat} effectively predicts and reasons
the vulnerabilities in source codes. Our method demonstrates remarkable
performance, achieving up to $57.38\%$ accuracy on reliable datasets, and at
least $4\%$ accuracy improvement over transformers and $1\%$-$5\%$ over other
GNNs. Furthermore, by employing the interpretability of GATs in
\textit{Vignat}, we are able to get valuable insights into vulnerability
patterns, paving the way for future advances in vulnerability identification
and prevention in cybersecurity.

161

162\section{Acknowledgements}

163

164Kaiser is supported in part by DARPA/NIWC-Pacific N66001-21-C-4018 and in
part by NSF CNS–2247370 and CCF-2313055.

165

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168\bibliography{references}

169

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171\end{document}