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\begin{document}

\title{Sites and Sheaves}

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\tableofcontents

\section{Introduction}
\label{section-introduction}

\noindent
The notion of a site was introduced by Grothendieck to be able to study
sheaves in the \'etale topology of schemes. The basic reference for this
notion is perhaps \cite{SGA4}. 

\section{Topologies}
\label{section-topologies}

\noindent
In this section we define what sites are and how to think about sheaves
on sites. See \cite{ArtinTopologies}.

\subsection{Definitions}
\label{subsection-sites-definitions}

\noindent
Let $C$ be a category, see Conventions, 
\autoref{conventions-section-categories}. 
In the following the notation $\{U_i \to U\}_{i\in I}$ means that 
$U \in \text{Ob}(C)$, that $I$ is a set and that for each $i\in I$ 
we are given a morphism $U_i \to U$ of $C$ with target $U$. This
notation suggests an open covering as in topology.

\begin{definition}
\label{definition-site}
A {\it site} is given by a category $\mathcal{C}$ and a set 
$\text{Cov} \mathcal{C}$ consisting of coverings 
$\{U_i \to U\}_{i \in I}$ satisfying the following axioms
\begin{enumerate}
\item If $V \to U$ is an isomorphism then $\{V \to U\} \in 
\text{Cov} \mathcal{C}$.
\item If $\{U_i \to U\} \in \text{Cov} \mathcal{C}$ and for each
$i$ we have $\{V_{ij} \to U_i\} \in \text{Cov} \mathcal{C}$, then
$\{V_{ij} \to U\} \in \text{Cov} \mathcal{C}$.
\item If $\{U_i \to U\}\in \text{Cov} \mathcal{C}$ and $V \to U$ is a
morphism of $\mathcal{C}$ then $U_i \times_U V$ exists for all
$i$ and $\{U_i \times_U V \to V \} \in \text{Cov} \mathcal{C}$.
\end{enumerate}
\end{definition}

\begin{example}
Let $X$ be a topological space.  Let $T_X$ be the category whose
objects consist of all the open sets $U$ in $X$ and whose morphisms
are just the inclusion maps.  That is, there is at most one morphism
between any two objects in $T_X$.  Now define a site on this
category by defining $\{U_i \to U\}\in \text{Cov} T_X$ if $\bigcup
U_i = U$. Conditions (1) and (2) above are clear, and (3) is also
clear once we realize that in $T_X$ we have $U \times V = U \bigcap V$.
Presheaves and sheaves (as defined below) on the site $T_X$ will
agree exactly with the usual notion of a (pre)sheaf on a topological
space.
\end{example}

\begin{example}
\label{example-site on group} Every category (with products) has a
canonical topology associated to it. (Note: this is the finest
topology where all representable presheaves are sheaves). Here is one
example. Let $G$ be a group. Consider the category whose objects are 
sets $X$ with a left $G$-action, with $G$-equivariant maps as the
morphisms.  We define a topology by declaring $\{U_i \to U\}$ to be 
a covering if $\bigcup U_i = U$. To verify that fibred products do 
exist in this category, suppose $f: S \rightarrow U$ and 
$h: T \rightarrow U$ are morphisms in the category. Then let 
$W = \{(s, t) \in S \times T \mid f(s) = h(t)\}$. This is a $G$-set. 
The action is given by $g \bullet(s,t) = (g \bullet s, g \bullet t)$.
Projections onto $S$ and $T$ are clearly $G$-maps and so 
$W = S \times_U T$.  Conditions (1), (2), and (3) are now easily 
verified. This site will be denoted $T_G$.
\end{example}

\begin{example}
FIXME. We can have a lot of examples linked from here.
\end{example}

\noindent
In this context a presheaf of sets is a contravariant functor
$\mathcal{F}$ from $\mathcal{C}$ to $\text{Sets}$ (see Categories, 
\hyperref[categories-remark-functor-into-sets]%
{Remark~\ref*{categories-remark-functor-into-sets}}). 
So for every object $U$ of $\mathcal{C}$ we have a set 
$\mathcal{F}(U)$ and for every morphism
$f : V \to U$ a ``restricton map'' $f^\ast : \mathcal{F}(U) \to 
\mathcal{F}(V)$. Sometimes we say an element
of $\mathcal{F}(U)$ is a section of $\mathcal{F}$ over $U$ , and sometimes
we use the notation $f^\ast(s) =: s|_V$. 

\smallskip\noindent
Similarly, we can define the notion of a presheaf of abelian groups, rings,
or a presheaf with values in a category. Note also that a presheaf is defined
without reference to the topology.

\begin{definition}
\label{definition-presheaf}
A presheaf $\mathcal{F}$ on a category $\mathcal{C}$ with values in a category
$\mathcal{A}$ is a contravariant functor from $\mathcal{C}$ to $\mathcal{A}$,
i.e., $\mathcal{F} : \mathcal{C}^\circ \to \mathcal{A}$.
\end{definition}

\noindent
The sheaf condition is taken with respect to the coverings of the 
topology as follows.

\begin{definition}
\label{definition-sheaf}
Let $\mathcal{F} : \mathcal{C}^\circ \to \mathcal{A}$ be a presheaf 
with values in a category $\mathcal{A}$. We say that $\mathcal{F}$ 
is a sheaf if for every covering $\{U_i \to U\} \in \text{Cov} \mathcal{C}$ 
the diagram
$$
\xymatrix{
\mathcal{F}(U) \ar[r] 
& 
\prod\nolimits_i \mathcal{F}(U_i) \ar@/^/[r] \ar@/_/[r] 
&
\prod\nolimits_{ij} \mathcal{F}(U_i\times_U U_j)
}
\leqno{(*)}
$$
is exact. The meaning of this will be made clear below.
\end{definition}

\noindent
First, we explain what this means if $\mathcal{F}$ is a presheaf of sets:
It means that given sections $s_i \in F(U_i)$ such that
$\text{pr}_1^\ast(s_i) = \text{pr}_2^\ast(s_j)$ in 
$\mathcal{F}(U_i \times_U U_j)$ for all $i,j$ there exists a unique
$s \in \mathcal{F}(U)$ such that $s_i = s|_{U_i}$.

\smallskip\noindent
If $\mathcal{A}$ is arbitrary, the condition means that for any
object $X\in \text{Ob}(\mathcal{A})$ the diagram of sets
$$
\xymatrix{
\text{Mor}_\mathcal{A}(X,\mathcal{F}(U)) \ar[r] 
& 
\prod\nolimits_i \text{Mor}_\mathcal{A}(X,\mathcal{F}(U_i)) 
\ar@/^/[r] \ar@/_/[r] 
&
\prod\nolimits_{ij} \text{Mor}_\mathcal{A}(X,\mathcal{F}(U_i\times_U U_j))
}
$$
is exact as before. If the products in $(*)$ exist then this condition just
means that the first arrow is the equalizer of the other two.

FIXME: A little about the canonical topology, and in particular some examples.

\begin{example}
As an example, suppose $\mathcal{F}$ is a sheaf of sets on $T_G$
(see Example \ref{example-site on group}).  First we note that $G$ 
itself is an object in the category (the action given by left 
multiplication). As a $G$-set, denote it ${}_GG$. Next, remark that
the map
$$
\text{Hom}_G({}_GG,{}_GG) \longrightarrow G^{opp}, 
\varphi \longmapsto \varphi(1)
$$
is an isomorphism of groups. The inverse map sends $g \in G$
to the map $ s \mapsto s \cdot g$ (i.e.\ right multiplication). Then
$\mathcal{F}({}_GG)$ is also a $G$-set where the action $g \bullet s$
for $g \in G$ and $s \in \mathcal{F}({}_GG)$ is given by
$\mathcal{F}(\cdot g)(s)$. Claim: If $\mathcal{F}$ is a sheaf then we
can recover $\mathcal{F}$ from the $G$-set $\mathcal{F}({}_GG)$
and vice versa. That is, there is an equivalence of categories
between left $G$-sets and sheaves of sets on $T_G$. We will show a
quasi-inverse of the functor $\mathcal{F} \mapsto \mathcal{F}({}_GG)$ is
given by $U \mapsto \text{Hom}_G(\cdot, U)$ where $U$ is a $G$-set. 
Since $T_G$ has the canonical topology, the presheaves  
$\text{Hom}_G(\cdot, U)$ are sheaves.
Composing $U \rightarrow \text{Hom}_G(\cdot, U)$ with 
$\mathcal{F} \rightarrow \mathcal{F}({}_GG)$ we get 
$U \mapsto \text{Hom}_G({}_GG,U)$ which is
canonically isomorphic to $U$ (namely, a $G$-equivariant map of ${}_GG$ 
into $U$ is uniquely determined by the image of $1$ in the exact same way 
as above). Composing in the reverse direction $\mathcal{F} \mapsto 
\mathcal{F}({}_GG)$ with $U \mapsto \text{Hom}_G(\cdot, U)$ we have to 
show that the presheaf $\text{Hom}_G(\cdot, \mathcal{F}({}_GG))$ is naturally
isomorphic to $\mathcal{F}$, provided that $\mathcal{F}$ is a sheaf.
Suppose $U$ is another $G$-set. Then
$\{{}_GG \stackrel{\phi_u}{\rightarrow} U\}_{u \in U}$ (where
$\phi_u(g) = g \bullet u$) is a covering of $U$. Since $\mathcal{F}$ is a 
sheaf we have the exact sequence:
$$
\xymatrix{ \mathcal{F}(U) \ar[r] & \prod\nolimits_{u \in U}
\mathcal{F}({}_GG) \ar@/^/[r] \ar@/_/[r] & \prod\nolimits_{u, v \in
U} \mathcal{F}({}_GG \times_U{}_GG) } \leqno{(*)}
$$
Now we note that the middle term is exactly 
$\text{Mor}(U,\mathcal{F}({}_GG))$ (maps of sets). Since the sequence is 
exact, we have that $\mathcal{F}(U)$ is the equalizer of the second two
arrows. This means it is exactly isomorphic to the subset of morphisms in
$\text{Mor}(U, \mathcal{F}(_G G))$ that commute with the $G$-action (FIXME?),
i.e., $\mathcal{F}(U) \cong \text{Hom}_G(U, \mathcal{F}({}_GG))$.
This isomorphisms is clearly functorial in $U$ so we have an
isomorphism of sheaves, as desired. Note that in the special case that 
$U$ is a left $G$-module rather than just a set, then this process gives 
an equivalence between left $G$-modules and sheaves of abelian groups on $T_G$.
\end{example}

\subsection{More about coverings}
\label{subsection-coverings}

\noindent
Let $\mathcal{C}$ be a site. A morphism of coverings of $\mathcal{C}$ from 
$\mathcal{U}=\{U_i \to U\}_{i\in I}$ to $\mathcal{V}=\{V_j \to V\}_{j\in J}$
is given by a morphism $U \to V$, a map of sets $\alpha : I \to J$ and
for each $i\in I$ a morphism $U_i \to V_{\alpha(i)}$ such that
the diagram
$$
\xymatrix{
U_i \ar[r] \ar[d] & V_{\alpha(i)} \ar[d] \\
U \ar[r] & V
}
$$
is commutative. In the special case that $U=V$ and $U\to V$ is the identity
we call $\mathcal{U}$ a refinement of the covering $\mathcal{V}$.

\smallskip\noindent
Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$, and let
$\mathcal{U}$ be a covering in $\mathcal{C}$ as above. Let us use the
notation $\mathcal{F}(\mathcal{U})$ to indicate the equalizer
$$
\mathcal{F}(\mathcal{U}) = \{ (s_i)_i \in \prod_i \mathcal{F}(U_i)
\mid  \text{pr}_1^\ast s_i = \text{pr}_2^\ast s_j \forall i,j \in I\}.
$$
There is a canonical map $\mathcal{F}(U) \to \mathcal{F}(\mathcal{U})$.
It is clear that a morphism of coverings $\mathcal{U} \to \mathcal{V}$
induces commutative diagrams
$$
\xymatrix{
& U_i \ar[rr] & & V_{\alpha(i)} \\
U_i \times_U U_j \ar[rr] \ar[ur] \ar[dr] & & 
V_{\alpha(i)}\times_V V_{\alpha(j)} \ar[ur] \ar[dr] & \\
& U_j \ar[rr] & & V_{\alpha(j)}
}.
$$
This in turn produces a map $\mathcal{F}(\mathcal{V}) \to 
\mathcal{F}(\mathcal{U})$, compatible with the map $\mathcal{F}(V) 
\to \mathcal{F}(U)$. 

\begin{lemma}
\label{lemma-indepent-refinement}
Any two morphisms $f,g: \mathcal{U} \to \mathcal{V}$ of coverings
inducing the same morphism $U \to V$ induce the same
map $\mathcal{F}(\mathcal{V}) \to \mathcal{F}(\mathcal{U})$.
\end{lemma}

\begin{proof}
Let $\mathcal{U}=\{U_i \to U\}_{i\in I}$ and 
$\mathcal{V}=\{V_j \to V\}_{j\in J}$.
The morphism~$f$ consists of a map $U\to V$, a map $\alpha\colon I\to J$ and
maps $f_i\colon U_i\to V_{\alpha(i)}$.
Likewise, $g$~determines a map $\beta\colon I\to J$ and maps
$g_i\colon U_i\to V_{\beta(i)}$.
As $f$~and~$g$ induce the same map $U\to V$, the diagram
$$\xymatrix{&V_{\alpha(i)}\ar[dr]\\
  U_i\ar[ur]^{f_i}\ar[dr]_{g_i}&&V\\
  &V_{\beta(i)}\ar[ur]}$$
is commutative for every~$i\in I$. Hence $f$~and~$g$ factor through 
the fibre product
$$\xymatrix{&V_{\alpha(i)}\\
  U_i\ar[r]^-\varphi\ar[ur]^{f_i}\ar[dr]_{g_i}&
  V_{\alpha(i)}\times_VV_{\beta(i)}\ar[u]_{\text{pr}_1}\ar[d]^{\text{pr}_2}\\
  &V_{\beta(i)}.}$$
Now let $s=(s_j)_j\in\mathcal{F}(\mathcal{V})$.
Then for all~$i\in I$:
 $$(f^*s)_i=f_i^*(s_{\alpha(i)})=\varphi^*\text{pr}_1^*(s_{\alpha(i)})
   =\varphi^*\text{pr}_2^*(s_{\beta(i)})=g_i^*(s_{\beta(i)})=(g^*s)_i,$$
where the middle equality is given by the definition 
of~$\mathcal{F}(\mathcal{V})$.
This shows that the maps $\mathcal{F}(\mathcal{V})\to\mathcal{F}(\mathcal{U})$
induced by $f$~and~$g$ are equal.
\end{proof}

\smallskip\noindent
Suppose that $\mathcal{F}$ is a presheaf of abelian groups on a
fixed site $T$.  We would like to canonically associate a sheaf
$\mathcal{F}^\#$ to $\mathcal{F}$ such that there exists a
functorial morphism $\mathcal{F} \rightarrow \mathcal{F}^\#$ and such that for
any morphism from $\mathcal{F}$ to an abelian sheaf $\mathcal{G}$
there is a unique factorization $\mathcal{F} \rightarrow
\mathcal{F}^\# \rightarrow \mathcal{G}$. The sheaf $\mathcal{F}^\#$ will be
called the sheafification of $\mathcal{F}$.

\smallskip\noindent
FIXME: Move the following stuff on limits to another file, and make it
more general. Find nice lim symbol anyone?

\begin{definition}
A directed set is a set $S$ together with a relation $\geq$ which is
transitive and reflexive such that for $a, b \in S$ there exists another
element $c \in S$ such that $c \geq a$ and $c \geq b$.
\end{definition}

\noindent
A directed system over a directed set $S$, is given by a set $M_s$ for
every $s\in S$ and a map $M_a \to M_b$ for every pair $b\geq a$ such
that all the composition $M_a \to M_b \to M_c$ equals the map
$M_a \to M_c$ whenever $c \geq b \geq a$. The limit of the directed system
is the set $\lim_{s\in S} M_s = \big(\coprod_{a\in S} M_a\big)/\sim$. Here,
if $m\in M_a$ and $m'\in M_{a'}$, then $m \sim m'$ if and only if $m$ and $m'$
map to the same element in some $M_b$ for some $b$ with $b\geq a$ and
$b \geq a'$. If the system is in the category of abelian groups then the
limit has the structure of an abelian group.

\noindent
Let $\mathcal{J}_U$ be the set of all coverings of $U$. It is not hard to
check that $\mathcal{J}_U$ with morphisms being morphisms of coverings
over $U$, is a category. It is also clear that given any two coverings of $U$,
$\mathcal{U}_1$ and $\mathcal{U}_2$, there is another covering refining them
both. That is, the covering $\{U_{1i} \times_U U_{2j} \rightarrow U\}$ is a
cover of $U$ and the natural projection maps give the refinements: it is
exactly conditions (2) and (3) in \ref{definition-site} that allow us to conclude
that this is a cover. Now, by the above remarks, we see that $\mathcal{J}_U$
is a directed set, where we say that $\{U_i \to U\} \geq \{V_j \to U\}$ if 
and only if $\{U_i \to U\}$ is a refinement of $\{V_j \to U\}$. Lemma
\ref{lemma-indepent-refinement} tells us that $\mathcal{U} \mapsto
\mathcal{F}(\mathcal{U})$ is a directed system over $\mathcal{J}(U)$, if
we define, for $\mathcal{U} \geq \mathcal{V}$ the map 
$\mathcal{F}(\mathcal{U}) \to \mathcal{F}(\mathcal{V})$ to be induced from any
morphism of coverings of $U$. Hence we can take the direct limit over the set
of coverings of $U$. Thus we define 
$$
\mathcal{F}^\dagger(U) = \lim_{\mathcal{U}}\mathcal{F}(\mathcal{U})
$$
where the limit is over the directed set of coverings $\mathcal{J}_U$.
This is sometimes denoted $\check{H}^0(U, F)$, ie, the $0$th \v{C}ech
cohomology group. 

\smallskip\noindent
Finally, we say that $\mathcal{F}$ is separated if, for all coverings
of $U$, $\{U_i \rightarrow U$ the map 
$\mathcal(F) \rightarrow \prod \mathcal(F)(U_i)$ is injective.

\begin{theorem}
With $\mathcal{F}$ as above
\begin{enumerate}
\item $\mathcal{F}^\dagger$ is separated
\item If $\mathcal{F}$ is separated, then $\mathcal{F}^\dagger$ is a sheaf.
\item $\mathcal{F}^{\dagger\dagger}$ is always a sheaf.
\end{enumerate}
\end{theorem}

\begin{proof}
FIXME.
\end{proof}

\noindent
FIXME. Discuss the more general case when $\mathcal{F}$ may not be a sheaf with
values in $\mathcal{A}b$.

\section{Representable sheaves}
\label{section-representable-sheaves}

\noindent
FIXME. Talk about representable presheaves, canonical topology and
representable sheaves.

\section{Morphisms of sites}
\label{section-morphism-sites}

\noindent
FIXME. Talk about continuous functors, and explain the condition that leads to
the correct functoriality on sheaves (i.e., exactness of the pullback functor).
It makes sense to not always assume this holds.

\section{Topoi}

\noindent
The topos associated to a site $\mathcal{C}$ is its ``category'' of sheaves of
sets. Conversely, any topos is equivalent to such a ``category'' of sheaves.
Our conventions do not allow us to talk about topoi. Of course we can choose a
large cardinal $\alpha$ and consider the category of sheaves of sets
$\text{Sh}_\alpha(\mathcal{C})$ contained in $\alpha$, but this does not have
the same flavor.

\smallskip\noindent
FIXME. What are topoi? What are morphisms of topoi? Do we need them? (Yes, in a
way.)

\smallskip\noindent
As a result some of the discussion in this project uses sites in places where
it might be more convenient to use the language of topoi. We discuss a few
of these ``inconveniences'' in this section.

\subsection{Sites and points}
\label{subsection-points}

\noindent
A point of a topos $\mathcal{S}$ is a morphism of topoi from $\text{Sets}$ to
$\mathcal{S}$. As discussed above we do not use this definition. In stead, we
somewhat akwardly define a point as follows. A point is a functor
$p : \mathcal{C} \to \text{Sets}$ such that
\begin{enumerate}
\item if $V\times_U W$ exists then $p(V\times_U W)=p(V)\times_{p(U)}p(W)$,
\item if $\{U_i \to U\}$ is a covering, then $\coprod_i p(U_i) \to p(U)$ is
surjective,
\item for any $x\in p(U)$ and $y\in p(V)$ there exists a $z\in p(W)$ and
morphisms $\alpha:W \to U$, $\beta:W \to V$ such that $p(\alpha)(z)=x$,
and $p(\beta)(z)=y$, and
\item for any pair of morphisms $f,g : V \to U$, and $y\in p(V)$ such
that $p(f)(x)=p(g)(x)$, there exists a $h: W \to V$, $z\in p(W)$ such that
$p(h)(z)=y$ and $g\circ h = f \circ h$.
\end{enumerate}
Once we have this, then we can define the stalk of a (pre)sheaf $\mathcal{F}$
at $p$ as follows
$$
\mathcal{F}_p = \lim_{(U,x)} \mathcal{F},
$$
where the limit is over the category of pairs 
$\{(U,x) \mid U \in \text{Ob}(\mathcal{C}), x\in p(U)$. The conditions
above imply this is a FIXME filtered limit. This implies that taking
stalks is an exact functor. FIXME: Need a section on limits.

\begin{lemma}
\label{lemma-points-recover}
In the situation above we have $p(U) = (U^{++})_p$. FIXME: notation.
\end{lemma}

\begin{proof}
FIXME.
\end{proof}

\noindent
We say that a site $\mathcal{C}$ has enough points if the following equivalence
is true for every morphism of sheaves of sets 
$\phi : \mathcal{F} \to \mathcal{G}$:
$$
\phi\ \text{is}\ \text{injective} 
\Leftrightarrow 
\forall p, \phi_p\ \text{is}\ \text{injective} 
$$
This will then imply the same thing for ``bijective'' and ``surjective'', and
it allows you to check exactness of sequences of sheaves of abelian groups
on stalks. (FIXME: explain?) Often sites that we work with have enough points
and it is easier to work with them, e.g., it is fairly easy to construct
injective sheaves of abelian groups on such a site.

\smallskip\noindent
To continue reading,
\begin{enumerate}

\item visit the next section: Flat descent for quasi-coherent sheaves,
\autoref{flat-section-introduction}, or

\item go back to the
table of contents: \url{index.html#contents}.

\end{enumerate}


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