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\input{preamble}
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\begin{document}
\title{Sites and Sheaves}
\maketitle
\phantomsection
\label{section-phantom}
\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}. Our notion of a site differs from that
in \cite{SGA4}; what we call a site is called a category endowed with
a pretopology in \cite[Expos\'e II, D\'efinition 1.3]{SGA4}.
The reason we do this is that in algebraic geometry it is often convenient to
work with a given class of coverings, for example when defining when
a property of schemes is local in a given topology, see Descent,
Section \ref{descent-section-descending-properties}.
Our exposition will closely follow \cite{ArtinTopologies}.
We will not use universes.
\section{Presheaves}
\label{section-presheaves}
\noindent
Let $\mathcal{C}$ be a category.
A {\it presheaf of sets} is a contravariant functor $\mathcal{F}$
from $\mathcal{C}$ to $\textit{Sets}$ (see Categories, Remark
\ref{categories-remark-functor-into-sets}).
So for every object $U$ of $\mathcal{C}$ we have a set
$\mathcal{F}(U)$. The elements of this set are called
the {\it sections} of $\mathcal{F}$ over $U$. For every morphism
$f : V \to U$ the map $\mathcal{F}(f) : \mathcal{F}(U) \to \mathcal{F}(V)$
is called the {\it restriction map} and is often denoted
$f^\ast : \mathcal{F}(U) \to \mathcal{F}(V)$. Another way
of expressing this is to say that $f^*(s)$ is the {\it pullback}
of $s$ via $f$. Functoriality means that $g^* f^* (s) = (f \circ g)^*(s)$.
Sometimes we use the notation $s|_V := f^\ast(s)$.
This notation is consistent with the notion of restriction
of functions from topology because if $W \to V \to U$
are morphisms in $\mathcal{C}$ and $s$ is a section of
$\mathcal{F}$ over $U$ then $s|_W = (s|_V)|_W$ by the
functorial nature of $\mathcal{F}$. Of course we have to be
careful since it may very well happen
that there is more than one morphism $V \to U$ and it is
certainly not going to be the case that the corresponding
pullback maps are equal.
\begin{definition}
\label{definition-presheaves-sets}
A {\it presheaf of sets} on $\mathcal{C}$ is a contravariant
functor from $\mathcal{C}$ to $\textit{Sets}$. {\it Morphisms
of presheaves} are transformations of functors. The category
of presheaves of sets is denoted $\textit{PSh}(\mathcal{C})$.
\end{definition}
\noindent
Note that for any object $U$ of $\mathcal{C}$ the functor of
points $h_U$, see Categories, Example \ref{categories-example-hom-functor}
is a presheaf. These are called the {\it representable presheaves}.
These presheaves have the pleasing property that for any
presheaf $\mathcal{F}$ we have
\begin{equation}
\label{equation-map-representable-into-presheaf}
\Mor_{\textit{PSh}(\mathcal{C})}(h_U, \mathcal{F})
=
\mathcal{F}(U).
\end{equation}
This is the Yoneda lemma (Categories, Lemma \ref{categories-lemma-yoneda}).
\medskip\noindent
Similarly, we can define the notion of a presheaf of abelian groups,
rings, etc. More generally we may define a presheaf with values in a
category.
\begin{definition}
\label{definition-presheaf}
Let $\mathcal{C}$, $\mathcal{A}$ be categories.
A {\it presheaf} $\mathcal{F}$ on $\mathcal{C}$
with values in $\mathcal{A}$ is a contravariant
functor from $\mathcal{C}$ to $\mathcal{A}$,
i.e., $\mathcal{F} : \mathcal{C}^{opp} \to \mathcal{A}$.
A {\it morphism} of presheaves $\mathcal{F} \to \mathcal{G}$
on $\mathcal{C}$ with values in $\mathcal{A}$ is a transformation
of functors from $\mathcal{F}$ to $\mathcal{G}$.
\end{definition}
\noindent
These form the objects and morphisms of the category of presheaves
on $\mathcal{C}$ with values in $\mathcal{A}$.
\begin{remark}
\label{remark-big-presheaves}
As already pointed out we may consider the category of
presheaves with values in any of the ``big'' categories
listed in Categories, Remark \ref{categories-remark-big-categories}.
These will be ``big'' categories as well and they will be
listed in the above mentioned remark as we go along.
\end{remark}
\section{Injective and surjective maps of presheaves}
\label{section-injective-surjective}
\begin{definition}
\label{definition-presheaves-injective-surjective}
Let $\mathcal{C}$ be a category, and let $\varphi : \mathcal{F}
\to \mathcal{G}$ be a map of presheaves of sets.
\begin{enumerate}
\item We say that $\varphi$ is {\it injective} if for every object
$U$ of $\mathcal{C}$ the map $\varphi_U : \mathcal{F}(U)
\to \mathcal{G}(U)$ is injective.
\item We say that $\varphi$ is {\it surjective} if for every object
$U$ of $\mathcal{C}$ the map $\varphi_U : \mathcal{F}(U)
\to \mathcal{G}(U)$ is surjective.
\end{enumerate}
\end{definition}
\begin{lemma}
\label{lemma-mono-epi}
The injective (resp.\ surjective) maps defined above
are exactly the monomorphisms (resp.\ epimorphisms) of
$\textit{PSh}(\mathcal{C})$. A map is an isomorphism
if and only if it is both injective and surjective.
\end{lemma}
\begin{proof}
We shall show that $\varphi : \mathcal{F} \to
\mathcal{G}$ is injective if and only if it is a monomorphism
of $\textit{PSh}(\mathcal{C})$. Indeed, the ``only if''
direction is straightforward, so let us show the ``if''
direction. Assume that $\varphi$ is a monomorphism. Let
$U \in \Ob(\mathcal{C})$; we need to show that $\varphi_U$ is
injective. So let $a, b \in \mathcal{F}(U)$ be such that
$\varphi_U (a) = \varphi_U (b)$; we need to check that $a = b$.
Under the isomorphism
(\ref{equation-map-representable-into-presheaf}), the elements
$a$ and $b$ of $\mathcal{F}(U)$ correspond to two natural
transformations
$a', b' \in \Mor_{\textit{PSh}(\mathcal{C})}(h_U, \mathcal{F})$.
Similarly, under the analogous isomorphism
$\Mor_{\textit{PSh}(\mathcal{C})}(h_U, \mathcal{G})
= \mathcal{G}(U)$,
the two equal elements $\varphi_U (a)$ and $\varphi_U (b)$ of
$\mathcal{G}(U)$ correspond to the two natural transformations
$\varphi \circ a', \varphi \circ b'
\in \Mor_{\textit{PSh}(\mathcal{C})}(h_U, \mathcal{G})$,
which therefore must also be equal. So
$\varphi \circ a' = \varphi \circ b'$, and thus $a' = b'$
(since $\varphi$ is monic), whence $a = b$. This finishes (1).
\medskip\noindent
We shall show that $\varphi : \mathcal{F} \to
\mathcal{G}$ is surjective if and only if it is an epimorphism
of $\textit{PSh}(\mathcal{C})$. Indeed, the ``only if''
direction is straightforward, so let us show the ``if''
direction. Assume that $\varphi$ is an epimorphism.
\medskip\noindent
For any two morphisms $f : A \to B$ and $g : A \to C$ in the
category $\textit{Sets}$, we let $\operatorname{inl}_{f,g}$ and
$\operatorname{inr}_{f,g}$ denote the two canonical maps from
$B$ and $C$ to $B \coprod_A C$. (Here, the pushout is
evaluated in $\textit{Sets}$.)
\medskip\noindent
Now, we define a presheaf $\mathcal{H}$ of sets on $\mathcal{C}$
by setting $\mathcal{H}(U)
= \mathcal{G}(U) \coprod_{\mathcal{F}(U)} \mathcal{G}(U)$ (where
the pushout is evaluated in $\textit{Sets}$ and induced by
the map $\varphi_U : \mathcal{F}(U) \to \mathcal{G}(U)$) for
every $U \in \Ob(\mathcal{C})$; its action on morphisms is
defined in the obvious way (by the functoriality of pushout).
Then, there are two natural
transformations $i_1 : \mathcal{G} \to \mathcal{H}$ and
$i_2 : \mathcal{G} \to \mathcal{H}$ whose components at an object
$U \in \Ob(\mathcal{C})$ are given by the maps
$\operatorname{inl}_{\varphi_U, \varphi_U}$ and
$\operatorname{inr}_{\varphi_U, \varphi_U}$, respectively. The
definition of a pushout shows that $i_1 \circ \varphi
= i_2 \circ \varphi$, whence $i_1 = i_2$ (since $\varphi$ is an
epimorphism). Thus, for every $U \in \Ob(\mathcal{C})$, we have
$\operatorname{inl}_{\varphi_U, \varphi_U}
= \operatorname{inr}_{\varphi_U, \varphi_U}$. Thus, $\varphi_U$
must be surjective (since a simple combinatorial argument shows
that if $f : A \to B$ is a morphism in $\textit{Sets}$, then
$\operatorname{inl}_{f,f} = \operatorname{inr}_{f,f}$ if and
only if $f$ is surjective). In other words, $\varphi$ is
surjective, and (2) is proven.
\medskip\noindent
We shall show that $\varphi : \mathcal{F} \to
\mathcal{G}$ is both injective and surjective if and only if it
is an isomorphism of $\textit{PSh}(\mathcal{C})$. This time,
the ``if'' direction is straightforward. To prove the ``only if''
direction, it suffices to observe that if $\varphi$ is both
injective and surjective, then $\varphi_U$ is an invertible map
for every $U \in \Ob(\mathcal{C})$, and the inverses of these
maps for all $U$ can be combined to a natural transformation
$\mathcal{G} \to \mathcal{F}$ which is an inverse to $\varphi$.
\end{proof}
\begin{definition}
\label{definition-sub-presheaf}
We say $\mathcal{F}$ is a {\it subpresheaf} of $\mathcal{G}$
if for every object $U \in \Ob(\mathcal{C})$ the set
$\mathcal{F}(U)$ is a subset of $\mathcal{G}(U)$, compatibly
with the restriction mappings.
\end{definition}
\noindent
In other words, the inclusion
maps $\mathcal{F}(U) \to \mathcal{G}(U)$
glue together to give an (injective) morphism of
presheaves $\mathcal{F} \to \mathcal{G}$.
\begin{lemma}
\label{lemma-image}
Let $\mathcal{C}$ be a category.
Suppose that $\varphi : \mathcal{F} \to \mathcal{G}$ is a
morphism of presheaves of sets on $\mathcal{C}$.
There exists a unique subpresheaf $\mathcal{G}' \subset \mathcal{G}$
such that $\varphi$ factors as
$\mathcal{F} \to \mathcal{G}' \to \mathcal{G}$
and such that the first map is surjective.
\end{lemma}
\begin{proof}
To prove existence, just set
$\mathcal{G}'(U) = \varphi_U \left(\mathcal{F}(U)\right)$
for every $U \in \Ob(C)$ (and inherit the action on morphisms
from $\mathcal{G}$), and prove that this defines a
subpresheaf of $\mathcal{G}$ and that $\varphi$ factors as
$\mathcal{F} \to \mathcal{G}' \to \mathcal{G}$ with the
first map being surjective. Uniqueness is straightforward.
\end{proof}
\begin{definition}
\label{definition-image}
Notation as in Lemma \ref{lemma-image}. We
say that $\mathcal{G}'$ is the {\it image of $\varphi$}.
\end{definition}
\section{Limits and colimits of presheaves}
\label{section-limits-colimits-PSh}
\noindent
Let $\mathcal{C}$ be a category.
Limits and colimits exist in the category
$\textit{PSh}(\mathcal{C})$. In addition, for any
$U \in \text{ob}(\mathcal{C})$ the functor
$$
\textit{PSh}(\mathcal{C})
\longrightarrow
\textit{Sets}, \quad
\mathcal{F}
\longmapsto
\mathcal{F}(U)
$$
commutes with limits and colimits. Perhaps the easiest way to prove
these statements is the following. Given a diagram
$
\mathcal{F} :
\mathcal{I}
\to
\textit{PSh}(\mathcal{C})
$
define presheaves
$$
\mathcal{F}_{\lim} :
U
\longmapsto
\lim_{i \in \mathcal{I}} \mathcal{F}_i(U)
\text{ and }
\mathcal{F}_{\colim} :
U
\longmapsto
\colim_{i \in \mathcal{I}} \mathcal{F}_i(U)
$$
There are clearly projection maps $\mathcal{F}_{\lim} \to \mathcal{F}_i$
and canonical maps $\mathcal{F}_i \to \mathcal{F}_{\colim}$. These
maps satisfy the requirements of the maps of a limit (reps.\ colimit)
of Categories, Definition \ref{categories-definition-limit}
(resp.\ Categories, Definition \ref{categories-definition-colimit}).
Indeed, they clearly form a cone, resp. a cocone, over $\mathcal{F}$.
Furthermore, if $(\mathcal{G}, q_i : \mathcal{G} \to \mathcal{F}_i)$
is another
system (as in the definition of a limit), then we get for every
$U$ a system of maps $\mathcal{G}(U) \to \mathcal{F}_i(U)$
with suitable functoriality requirements. And thus a unique
map $\mathcal{G}(U) \to \mathcal{F}_{\lim}(U)$. It is easy
to verify these are compatible as we vary $U$ and arise from
the desired map $\mathcal{G} \to \mathcal{F}_{\lim}$.
A similar argument works in the case of the colimit.
\section{Functoriality of categories of presheaves}
\label{section-functoriality-PSh}
\noindent
Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between categories.
In this case we denote
$$
u^p :
\textit{PSh}(\mathcal{D})
\longrightarrow
\textit{PSh}(\mathcal{C})
$$
the functor that associates to $\mathcal{G}$ on $\mathcal{D}$ the presheaf
$u^p\mathcal{G} = \mathcal{G} \circ u$. Note that by the previous section
this functor commutes with all limits.
\medskip\noindent
For $V \in \text{ob}(\mathcal{D})$ let $\mathcal{I}^u_V$
denote the category with
\begin{equation}
\label{equation-colim-category}
\begin{matrix}
\Ob(\mathcal{I}^u_V)
&
=
&
\{
(U, \phi)
\mid
U \in \Ob(\mathcal{C}),
\phi : V \to u(U)
\}
\\
\Mor_{\mathcal{I}^u_V}((U, \phi), (U', \phi'))
&
=
&
\{
f : U \to U' \text{ in }\mathcal{C}
\mid
u(f) \circ \phi = \phi'
\}
\end{matrix}
\end{equation}
We sometimes drop the subscript ${}^u$ from the notation and we simply write
$\mathcal{I}_V$.
We will use these categories to define a left adjoint to the functor $u^p$.
Before we do so we prove a few technical lemmas.
\begin{lemma}
\label{lemma-almost-directed}
Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between categories.
Suppose that $\mathcal{C}$ has fibre products and equalizers, and that
$u$ commutes with them. Then the categories $(\mathcal{I}_V)^{opp}$
satisfy the hypotheses of
Categories, Lemma \ref{categories-lemma-split-into-directed}.
\end{lemma}
\begin{proof}
There are two conditions to check.
\medskip\noindent
First, suppose we are given three objects
$\phi : V \to u(U)$, $\phi' : V \to u(U')$, and $\phi'' : V \to u(U'')$
and morphisms $a : U' \to U$, $b : U'' \to U$ such that
$u(a) \circ \phi' = \phi$ and $u(b) \circ \phi'' = \phi$.
We have to show there exists another object $\phi''' : V \to u(U''')$
and morphisms $c : U''' \to U'$ and $d : U''' \to U''$ such that
$u(c) \circ \phi''' = \phi'$, $u(d) \circ \phi''' = \phi''$ and
$a \circ c = b \circ d$. We take $U''' = U' \times_U U''$
with $c$ and $d$ the projection morphisms. This works as $u$ commutes
with fibre products; we omit the verification.
\medskip\noindent
Second, suppose we are given two objects
$\phi : V \to u(U)$ and $\phi' : V \to u(U')$
and morphisms $a, b : (U, \phi) \to (U', \phi')$.
We have to find a morphism $c : (U'', \phi'') \to (U, \phi)$
which equalizes $a$ and $b$. Let $c : U'' \to U$ be the equalizer of
$a$ and $b$ in the category $\mathcal{C}$. As $u$ commutes
with equalizers and since $u(a) \circ \phi = u(b) \circ \phi = \phi'$
we obtain a morphism $\phi'' : V \to u(U'')$.
\end{proof}
\begin{lemma}
\label{lemma-directed}
Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between categories.
Assume
\begin{enumerate}
\item the category $\mathcal{C}$ has a final object $X$ and
$u(X)$ is a final object of $\mathcal{D}$ , and
\item the category $\mathcal{C}$ has fibre products and
$u$ commutes with them.
\end{enumerate}
Then the index categories $(\mathcal{I}^u_V)^{opp}$ are filtered (see
Categories, Definition \ref{categories-definition-directed}).
\end{lemma}
\begin{proof}
The assumptions imply that the assumptions of
Lemma \ref{lemma-almost-directed}
are satisfied (see the discussion in
Categories, Section \ref{categories-section-finite-limits}).
By
Categories, Lemma \ref{categories-lemma-split-into-directed}
we see that $\mathcal{I}_V$ is a (possibly empty) disjoint union of
directed categories.
Hence it suffices to show that $\mathcal{I}_V$ is connected.
\medskip\noindent
First, we show that $\mathcal{I}_V$ is nonempty.
Namely, let $X$ be the final object of $\mathcal{C}$,
which exists by assumption.
Let $V \to u(X)$ be the morphism coming from the fact
that $u(X)$ is final in $\mathcal{D}$ by assumption.
This gives an object of $\mathcal{I}_V$.
\medskip\noindent
Second, we show that $\mathcal{I}_V$ is connected.
Let $\phi_1 : V \to u(U_1)$ and $\phi_2 : V \to u(U_2)$ be
in $\Ob(\mathcal{I}_V)$. By assumption $U_1\times U_2$
exists and $u(U_1\times U_2) = u(U_1)\times u(U_2)$.
Consider the morphism $\phi : V \to u(U_1\times U_2)$
corresponding to $(\phi_1, \phi_2)$ by the universal property
of products. Clearly the object $\phi : V \to u(U_1\times U_2)$
maps to both $\phi_1 : V \to u(U_1)$ and $\phi_2 : V \to u(U_2)$.
\end{proof}
\noindent
Given $g : V' \to V$ in $\mathcal{D}$ we get a functor
$\overline{g} : \mathcal{I}_V \to \mathcal{I}_{V'}$
by setting $\overline{g}(U, \phi) = (U, \phi \circ g)$
on objects. Given a presheaf $\mathcal{F}$ on $\mathcal{C}$
we obtain a functor
$$
\mathcal{F}_V :
\mathcal{I}_V^{opp}
\longrightarrow
\textit{Sets}, \quad
(U, \phi)
\longmapsto
\mathcal{F}(U).
$$
In other words, $\mathcal{F}_V$ is a presheaf of sets on $\mathcal{I}_V$.
Note that we have $\mathcal{F}_{V'} \circ \overline{g} = \mathcal{F}_V$.
We define
$$
u_p\mathcal{F}(V) :=
\colim_{\mathcal{I}_V^{opp}} \mathcal{F}_V
$$
As a colimit we obtain for each $(U, \phi) \in \Ob(\mathcal{I}_V)$
a canonical map $\mathcal{F}(U)\xrightarrow{c(\phi)}u_p\mathcal{F}(V)$.
For $g : V' \to V$ as above there is a
canonical restriction map
$g^* : u_p\mathcal{F}(V) \to u_p\mathcal{F}(V')$
compatible with
$\mathcal{F}_{V'} \circ \overline{g} = \mathcal{F}_V$
by Categories, Lemma \ref{categories-lemma-functorial-colimit}.
It is the unique map so that for all $(U, \phi) \in \Ob(\mathcal{I}_V)$
the diagram
$$
\xymatrix{
\mathcal{F}(U) \ar[r]^{c(\phi)} \ar[d]_{\text{id}}
&
u_p\mathcal{F}(V) \ar[d]^{g^*}
\\
\mathcal{F}(U) \ar[r]^{c(\phi \circ g)}
&
u_p\mathcal{F}(V')
}
$$
commutes. The uniqueness of these maps implies that we obtain a
presheaf. This presheaf will be denoted $u_p\mathcal{F}$.
\begin{lemma}
\label{lemma-recover}
There is a canonical map
$\mathcal{F}(U) \to u_p\mathcal{F}(u(U))$,
which is compatible with restriction maps
(on $\mathcal{F}$ and on $u_p\mathcal{F}$).
\end{lemma}
\begin{proof}
This is just the map $c(\text{id}_{u(U)})$ introduced above.
\end{proof}
\noindent
Note that any map of presheaves $\mathcal{F} \to \mathcal{F}'$
gives rise to compatible systems of maps between functors
$\mathcal{F}_Y \to \mathcal{F}'_Y$, and hence to a map
of presheaves $u_p\mathcal{F} \to u_p\mathcal{F}'$. In other
words, we have defined a functor
$$
u_p :
\textit{PSh}(\mathcal{C})
\longrightarrow
\textit{PSh}(\mathcal{D})
$$
\begin{lemma}
\label{lemma-adjoints-u}
The functor $u_p$ is a left adjoint to the functor $u^p$.
In other words the formula
$$
\Mor_{\textit{PSh}(\mathcal{C})}(\mathcal{F}, u^p\mathcal{G})
=
\Mor_{\textit{PSh}(\mathcal{D})}(u_p\mathcal{F}, \mathcal{G})
$$
holds bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
\end{lemma}
\begin{proof}
Let $\mathcal{G}$ be a presheaf on $\mathcal{D}$ and let
$\mathcal{F}$ be a presheaf on $\mathcal{C}$.
We will show that the displayed formula holds
by constructing maps either way. We will leave
it to the reader to verify they are each others inverse.
\medskip\noindent
Given a map $\alpha : u_p \mathcal{F} \to \mathcal{G}$
we get $u^p\alpha : u^p u_p \mathcal{F} \to u^p \mathcal{G}$.
Lemma \ref{lemma-recover} says that there is a
map $\mathcal{F} \to u^p u_p \mathcal{F}$. The composition
of the two gives the desired map. (The good thing about this construction
is that it is clearly functorial in everything in sight.)
\medskip\noindent
Conversely, given a map $\beta : \mathcal{F} \to u^p\mathcal{G}$
we get a map $u_p\beta : u_p\mathcal{F} \to u_p u^p\mathcal{G}$.
We claim that the functor $u^p\mathcal{G}_Y$ on $\mathcal{I}_Y$
has a canonical map to the constant functor with value $\mathcal{G}(Y)$.
Namely, for every object $(X, \phi)$ of $\mathcal{I}_Y$,
the value of $u^p\mathcal{G}_Y$ on this object is $\mathcal{G}(u(X))$
which maps to $\mathcal{G}(Y)$ by $\mathcal{G}(\phi) = \phi^* $.
This is a transformation of functors because $\mathcal{G}$ is a functor
itself. This leads to a map $u_p u^p \mathcal{G}(Y) \to \mathcal{G}(Y)$.
Another trivial verification shows that this is functorial in $Y$
leading to a map of presheaves $u_p u^p \mathcal{G} \to \mathcal{G}$.
The composition $u_p\mathcal{F} \to u_p u^p\mathcal{G} \to
\mathcal{G}$ is the desired map.
\end{proof}
\begin{remark}
\label{remark-functoriality-presheaves-values}
Suppose that $\mathcal{A}$ is a category such that
any diagram $\mathcal{I}_Y \to \mathcal{A}$ has a
colimit in $\mathcal{A}$. In this case it is clear
that there are functors $u^p$ and $u_p$, defined in
exactly the same way as above, on the categories
of presheaves with values in $\mathcal{A}$.
Moreover, the adjointness of the pair
$u^p$ and $u_p$ continues to hold in this setting.
\end{remark}
\begin{lemma}
\label{lemma-pullback-representable-presheaf}
Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between categories.
For any object $U$ of $\mathcal{C}$ we have $u_ph_U = h_{u(U)}$.
\end{lemma}
\begin{proof}
By adjointness of $u_p$ and $u^p$ we have
$$
\Mor_{\textit{PSh}(\mathcal{D})}(u_ph_U, \mathcal{G})
=
\Mor_{\textit{PSh}(\mathcal{C})}(h_U, u^p\mathcal{G})
=
u^p\mathcal{G}(U) =
\mathcal{G}(u(U))
$$
and hence by Yoneda's lemma we see that $u_ph_U = h_{u(U)}$ as
presheaves.
\end{proof}
\section{Sites}
\label{section-sites-definitions}
\noindent
Our notion of a site uses the following type of structures.
\begin{definition}
\label{definition-family-morphisms-fixed-target}
Let $\mathcal{C}$ be a category, see
Conventions, Section \ref{conventions-section-categories}.
A {\it family of morphisms with fixed target} in $\mathcal{C}$ is
given by an object $U \in \Ob(\mathcal{C})$, a set $I$ and
for each $i\in I$ a morphism $U_i \to U$ of $\mathcal{C}$ with target $U$.
We use the notation $\{U_i \to U\}_{i\in I}$ to indicate this.
\end{definition}
\noindent
It can happen that the set $I$ is empty! This
notation is meant to suggest an open covering as in topology.
\begin{definition}
\label{definition-site}
A {\it site}\footnote{This notation differs from that of \cite{SGA4}, as
explained in the introduction.} is given by a category $\mathcal{C}$ and a set
$\text{Cov}(\mathcal{C})$ of families of morphisms with fixed target
$\{U_i \to U\}_{i \in I}$, called {\it coverings of $\mathcal{C}$},
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 $\{U_i \to U\}_{i\in I} \in \text{Cov}(\mathcal{C})$ and for each
$i$ we have $\{V_{ij} \to U_i\}_{j\in J_i} \in \text{Cov}(\mathcal{C})$, then
$\{V_{ij} \to U\}_{i \in I, j\in J_i} \in \text{Cov}(\mathcal{C})$.
\item If $\{U_i \to U\}_{i\in I}\in \text{Cov}(\mathcal{C})$
and $V \to U$ is a morphism of $\mathcal{C}$ then $U_i \times_U V$
exists for all $i$ and
$\{U_i \times_U V \to V \}_{i\in I} \in \text{Cov}(\mathcal{C})$.
\end{enumerate}
\end{definition}
\begin{remark}
\label{remark-no-big-sites}
(On set theoretic issues -- skip on a first reading.)
The main reason for introducing sites is to study the
category of sheaves on a site, because it is the generalization
of the category of sheaves on a topological space that has
been so important in algebraic geometry. In order to avoid thinking
about things like ``classes of classes'' and so on, we will
not allow sites to be ``big'' categories, in contrast to what
we do for categories and $2$-categories.
\medskip\noindent
Suppose that $\mathcal{C}$ is a category and that
$\text{Cov}(\mathcal{C})$ is a proper class of coverings
satisfying (1), (2) and (3) above. We will not allow this as a
site either, mainly because we are going to take limits over coverings.
However, there are several natural
ways to replace $\text{Cov}(\mathcal{C})$ by a set of coverings
or a slightly different structure
that give rise to the same category of sheaves. For example:
\begin{enumerate}
\item In Sets, Section \ref{sets-section-coverings-site}
we show how to pick a suitable set of
coverings that gives the same category of sheaves.
\item Another thing we can do is to take the associated topology
(see Definition \ref{definition-topology-associated-site}).
The resulting topology on $\mathcal{C}$ has the same category of sheaves.
Two topologies have the same categories of sheaves if and only if
they are equal, see Theorem \ref{theorem-topology-and-topos}.
A topology on a category is given by a choice of sieves on objects.
The collection of all possible sieves and even all possible
topologies on $\mathcal{C}$ is a set.
\item We could also slightly modify the
notion of a site, see Remark \ref{remark-shrink-coverings} below, and
end up with a canonical set of coverings which is contained in the
powerset of the set of arrows of $\mathcal{C}$.
\end{enumerate}
Each of these solutions has some minor drawback. For the first, one has
to check that constructions later on do not depend on the choice
of the set of coverings. For the second, one has to learn about topologies
and redo many of the arguments for sites. For the third, see
the last sentence of Remark \ref{remark-shrink-coverings}.
\medskip\noindent
Our approach will be to work with sites as in Definition \ref{definition-site}
above. Given a category $\mathcal{C}$ with a proper class of coverings
as above, we will replace this by a set of coverings producing a site using
Sets, Lemma \ref{sets-lemma-coverings-site}. It is shown in
Lemma \ref{lemma-choice-set-coverings-immaterial} below that the resulting
category of sheaves (the topos) is independent of this choice. We leave it to
the reader to use one of the other two strategies to deal with these issues if
he/she so desires.
\end{remark}
\begin{example}
\label{example-site-topological}
Let $X$ be a topological space. Let $X_{Zar}$ 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 $X_{Zar}$. Now define
$\{U_i \to U\}_{i \in I}\in \text{Cov}(X_{Zar})$ if
and only if $\bigcup U_i = U$.
Conditions (1) and (2) above are clear, and (3) is also
clear once we realize that in $X_{Zar}$ we have
$U \times V = U \cap V$. Note that in particular the empty
set has to be an element of $X_{Zar}$ since otherwise
this would not work in general. Furthermore, it is equally important,
as we will see later, to allow the {\it empty covering of the empty
set as a covering}!
We turn $X_{Zar}$ into a site
by choosing a suitable set of coverings
$\text{Cov}(X_{Zar})_{\kappa, \alpha}$ as in
Sets, Lemma \ref{sets-lemma-coverings-site}.
Presheaves and sheaves (as defined below)
on the site $X_{Zar}$ agree exactly with the usual notion of
a presheaves and sheaves on a topological space, as defined
in Sheaves, Section \ref{sheaves-section-introduction}.
\end{example}
\begin{example}
\label{example-site-on-group}
Let $G$ be a group. Consider the category $G\textit{-Sets}$
whose objects are sets $X$ with a left $G$-action, with
$G$-equivariant maps as the morphisms. An important example
is ${}_GG$ which is the $G$-set whose underlying set is $G$ and
action given by left multiplication. This category has
fiber products, see Categories, Section
\ref{categories-section-example-fibre-products}.
We declare $\{\varphi_i : U_i \to U\}_{i\in I}$ to be
a covering if $\bigcup_{i\in I} \varphi_i(U_i) = U$.
This gives a class of coverings on $G\textit{-Sets}$
which is easily see to satisfy conditions (1), (2), and (3)
of Definition \ref{definition-site}. The result is not a
site since both the collection of objects of the underlying category and
the collection of coverings form a proper class.
We first replace by $G\textit{-Sets}$ by a
full subcategory $G\textit{-Sets}_\alpha$ as in Sets,
Lemma \ref{sets-lemma-sets-with-group-action}.
After this the site
$(G\textit{-Sets}_\alpha,
\text{Cov}_{\kappa, \alpha'}(G\textit{-Sets}_\alpha))$
gotten by suitably restricting the collection of coverings
as in Sets, Lemma \ref{sets-lemma-coverings-site} will be
denoted $\mathcal{T}_G$.
\medskip\noindent
As a special case, if the group $G$ is countable, then we can let
$\mathcal{T}_G$ be the category of countable $G$-sets and coverings
those jointly surjective families of morphisms
$\{\varphi_i : U_i \to U\}_{i \in I}$ such that $I$ is countable.
\end{example}
\begin{example}
\label{example-indiscrete}
Let $\mathcal{C}$ be a category. There is a canonical way to turn this
into a site where $\{\text{id}_U : U \to U\}$ are the coverings.
Sheaves on this site are the presheaves on $\mathcal{C}$.
This corresponding topology is called the {\it chaotic} or
{\it indiscrete topology}.
\end{example}
\section{Sheaves}
\label{section-sheaves}
\noindent
Let $\mathcal{C}$ be a site. Before we introduce the notion of
a sheaf with values in a category we explain what it means
for a presheaf of sets to be a sheaf. Let $\mathcal{F}$ be
a presheaf of sets on $\mathcal{C}$ and let
$\{U_i \to U\}_{i\in I}$ be an element of $\text{Cov}(\mathcal{C})$.
By assumption all the fibre products $U_i \times_U U_j$ exist
in $\mathcal{C}$. There are two natural maps
$$
\xymatrix{
\prod\nolimits_{i\in I}
\mathcal{F}(U_i)
\ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*}
&
\prod\nolimits_{(i_0, i_1) \in I \times I}
\mathcal{F}(U_{i_0} \times_U U_{i_1})
}
$$
which we will denote $\text{pr}^*_i$, $i = 0, 1$ as indicated
in the displayed equation.
Namely, an element of the left hand side corresponds to a
family $(s_i)_{i\in I}$, where each $s_i$ is a section of
$\mathcal{F}$ over $U_i$. For each pair $(i_0, i_1) \in I \times I$
we have the projection morphisms
$$
\text{pr}^{(i_0, i_1)}_{i_0} :
U_{i_0} \times_U U_{i_1}
\longrightarrow
U_{i_0}
\text{ and }
\text{pr}^{(i_0, i_1)}_{i_1} :
U_{i_0} \times_U U_{i_1}
\longrightarrow
U_{i_1}.
$$
Thus we may pull back either the section $s_{i_0}$ via
the first of these maps or the section $s_{i_1}$ via the
second. Explicitly the maps we referred to above are
$$
\text{pr}_0^* :
(s_i)_{i\in I}
\longmapsto
\Big(
\text{pr}^{(i_0, i_1), *}_{i_0}(s_{i_0})
\Big)_{(i_0, i_1) \in I \times I}
$$
and
$$
\text{pr}_1^* :
(s_i)_{i\in I}
\longmapsto
\Big(
\text{pr}^{(i_0, i_1), *}_{i_1}(s_{i_1})
\Big)_{(i_0, i_1) \in I \times I}.
$$
Finally consider the natural map
$$
\mathcal{F}(U)
\longrightarrow
\prod\nolimits_{i\in I}
\mathcal{F}(U_i), \quad
s
\longmapsto
(s|_{U_i})_{i \in I}
$$
where we have used the notation $s|_{U_i}$ to indicate the
pullback of $s$ via the map $U_i \to U$. It is clear from the
functorial nature of $\mathcal{F}$ and the commutativity
of the fibre product diagrams that
$\text{pr}_0^*( (s|_{U_i})_{i \in I} ) =
\text{pr}_1^*( (s|_{U_i})_{i \in I} )$.
\begin{definition}
\label{definition-sheaf-sets}
Let $\mathcal{C}$ be a site, and let $\mathcal{F}$ be a presheaf of sets
on $\mathcal{C}$. We say $\mathcal{F}$ is a {\it sheaf} if
for every covering $\{U_i \to U\}_{i \in I} \in \text{Cov}(\mathcal{C})$
the diagram
\begin{equation}
\label{equation-sheaf-condition}
\xymatrix{
\mathcal{F}(U) \ar[r]
&
\prod\nolimits_{i\in I}
\mathcal{F}(U_i)
\ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*}
&
\prod\nolimits_{(i_0, i_1) \in I \times I}
\mathcal{F}(U_{i_0} \times_U U_{i_1})
}
\end{equation}
represents the first arrow as the equalizer of $\text{pr}_0^*$
and $\text{pr}_1^*$.
\end{definition}
\noindent
Loosely speaking this means that given sections $s_i \in \mathcal{F}(U_i)$
such that
$$
s_i|_{U_i \times_U U_j} = s_j|_{U_i \times_U U_j}
$$
in $\mathcal{F}(U_i \times_U U_j)$ for all pairs $(i, j) \in I \times I$
then there exists a unique $s \in \mathcal{F}(U)$ such
that $s_i = s|_{U_i}$.
\begin{remark}
\label{remark-sheaf-condition-empty-covering}
If the covering $\{U_i \to U\}_{i \in I}$ is the empty family (this means
that $I = \emptyset$), then the sheaf condition signifies that
$\mathcal{F}(U) = \{*\}$ is a singleton set. This is because
in (\ref{equation-sheaf-condition}) the second and third sets
are empty products in the category of sets, which are final objects
in the category of sets, hence singletons.
\end{remark}
\begin{example}
\label{example-sheaves-topological}
Let $X$ be a topological space. Let $X_{Zar}$ be the
site constructed in Example \ref{example-site-topological}.
The notion of a sheaf on $X_{Zar}$ coincides
with the notion of a sheaf on $X$ introduced in
Sheaves, Definition \ref{sheaves-definition-sheaf}.
\end{example}
\begin{example}
\label{example-topological-wrong}
Let $X$ be a topological space. Let us consider the site $X'_{Zar}$ which is
the same as the site $X_{Zar}$ of
Example \ref{example-site-topological} except that
we disallow the empty covering of the empty set.
In other words, we do allow the covering $\{\emptyset \to \emptyset\}$
but we do not allow the covering whose index set is empty.
It is easy to show that this still defines a site. However,
we claim that the sheaves on $X'_{Zar}$ are different
from the sheaves on $X_{Zar}$. For example, as an extreme
case consider the situation where $X = \{p\}$ is a singleton.
Then the objects of $X'_{Zar}$ are $\emptyset, X$
and every covering if $\emptyset$ can be refined by
$\{\emptyset \to \emptyset\}$ and every covering of $X$ by $\{X \to X\}$.
Clearly, a sheaf on this is given by any choice of
a set $\mathcal{F}(\emptyset)$ and any choice of a
set $\mathcal{F}(X)$, together with any restriction map
$\mathcal{F}(X) \to \mathcal{F}(\emptyset)$. Thus sheaves
on $X'_{Zar}$ are the same as usual sheaves on the two point space
$\{\eta, p\}$ with open sets $\{\emptyset, \{\eta\}, \{p, \eta\}\}$.
In general sheaves on $X'_{Zar}$ are the same as sheaves
on the space $X \amalg \{\eta\}$, with opens given by
the empty set and any set of the form $U \cup \{\eta\}$ for
$U \subset X$ open.
\end{example}
\begin{definition}
\label{definition-category-sheaves-sets}
The category {\it $\Sh(\mathcal{C})$}
of sheaves of sets is the full subcategory of the category
$\textit{PSh}(\mathcal{C})$ whose objects are the sheaves of sets.
\end{definition}
\noindent
Let $\mathcal{A}$ be a category. If products indexed by $I$, and
$I \times I$ exist in $\mathcal{A}$ for any $I$ that occurs as an index
set for covering families then Definition \ref{definition-sheaf-sets}
above makes sense, and defines a notion of a sheaf on $\mathcal{C}$
with values in $\mathcal{A}$. Note that the diagram in $\mathcal{A}$
$$
\xymatrix{
\mathcal{F}(U) \ar[r]
&
\prod\nolimits_{i\in I}
\mathcal{F}(U_i)
\ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*}
&
\prod\nolimits_{(i_0, i_1) \in I \times I}
\mathcal{F}(U_{i_0} \times_U U_{i_1})
}
$$
is an equalizer diagram if and only if for every object $X$ of
$\mathcal{A}$ the diagram of sets
$$
\xymatrix{
\Mor_\mathcal{A}(X, \mathcal{F}(U)) \ar[r]
&
\prod
\Mor_\mathcal{A}(X, \mathcal{F}(U_i))
\ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*}
&
\prod
\Mor_\mathcal{A}(X, \mathcal{F}(U_{i_0} \times_U U_{i_1}))
}
$$
is an equalizer diagram.
\medskip\noindent
Suppose $\mathcal{A}$ is arbitrary.
Let $\mathcal{F}$ be a presheaf with values in $\mathcal{A}$.
Choose any object $X\in \Ob(\mathcal{A})$.
Then we get a presheaf of sets $\mathcal{F}_X$ defined
by the rule
$$
\mathcal{F}_X(U) = \Mor_\mathcal{A}(X, \mathcal{F}(U)).
$$
From the above it follows that a good definition is
obtained by requiring all the presheaves $\mathcal{F}_X$ to be
sheaves of sets.
\begin{definition}
\label{definition-sheaf}
Let $\mathcal{C}$ be a site, let $\mathcal{A}$ be a category
and let $\mathcal{F}$ be a presheaf on $\mathcal{C}$ with values in
$\mathcal{A}$. We say that $\mathcal{F}$ is a {\it sheaf}
if for all objects $X$ of $\mathcal{A}$ the presheaf of sets
$\mathcal{F}_X$ (defined above) is a sheaf.
\end{definition}
\section{Families of morphisms with fixed target}
\label{section-refinements}
\noindent
This section is meant to introduce some notions regarding
families of morphisms with the same target.
\begin{definition}
\label{definition-morphism-coverings}
Let $\mathcal{C}$ be a category.
Let $\mathcal{U} = \{U_i \to U\}_{i\in I}$ be a family
of morphisms of $\mathcal{C}$ with fixed target.
Let $\mathcal{V} = \{V_j \to V\}_{j\in J}$ be another.
\begin{enumerate}
\item
A {\it morphism of families of maps with fixed target
of $\mathcal{C}$ from $\mathcal{U}$ to $\mathcal{V}$},
or simply a {\it morphism from $\mathcal{U}$ to $\mathcal{V}$}
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.
\item In the special case that $U = V$ and $U \to V$ is the identity
we call $\mathcal{U}$ a {\it refinement} of the family $\mathcal{V}$.
\end{enumerate}
\end{definition}
\noindent
A trivial but important remark is that if
$\mathcal{V} = \{V_j \to V\}_{j \in J}$
is the {\it empty family of maps}, i.e., if $J = \emptyset$, then no
family $\mathcal{U} = \{U_i \to V\}_{i \in I}$ with $I \not = \emptyset$
can refine $\mathcal{V}$!
\begin{definition}
\label{definition-combinatorial-tautological}
Let $\mathcal{C}$ be a category.
Let $\mathcal{U} = \{\varphi_i : U_i \to U\}_{i\in I}$, and
$\mathcal{V} = \{\psi_j : V_j \to U\}_{j\in J}$ be two families of morphisms
with fixed target.
\begin{enumerate}
\item We say $\mathcal{U}$ and $\mathcal{V}$ are
{\it combinatorially equivalent}
if there exist maps
$\alpha : I \to J$ and $\beta : J\to I$ such that
$\varphi_i = \psi_{\alpha(i)}$ and $\psi_j = \varphi_{\beta(j)}$.
\item We say $\mathcal{U}$ and $\mathcal{V}$ are
{\it tautologically equivalent} if there exist maps
$\alpha : I \to J$ and $\beta : J\to I$ and
for all $i\in I$ and $j \in J$ commutative diagrams
$$
\xymatrix{
U_i \ar[rd] \ar[rr] & &
V_{\alpha(i)} \ar[ld] & &
V_j \ar[rd] \ar[rr] & &
U_{\beta(j)} \ar[ld] \\
&
U & & & &
U &
}
$$
with isomorphisms as horizontal arrows.
\end{enumerate}
\end{definition}
\begin{lemma}
\label{lemma-tautological-combinatorial}
Let $\mathcal{C}$ be a category.
Let $\mathcal{U} = \{\varphi_i : U_i \to U\}_{i\in I}$, and
$\mathcal{V} = \{\psi_j : V_j \to U\}_{j\in J}$ be two families of morphisms
with the same fixed target.
\begin{enumerate}
\item If $\mathcal{U}$ and $\mathcal{V}$ are combinatorially equivalent
then they are tautologically equivalent.
\item If $\mathcal{U}$ and $\mathcal{V}$ are tautologically equivalent
then $\mathcal{U}$ is a refinement of $\mathcal{V}$ and
$\mathcal{V}$ is a refinement of $\mathcal{U}$.
\item The relation ``being combinatorially equivalent'' is an
equivalence relation on all families of morphisms with fixed target.
\item The relation ``being tautologically equivalent'' is an
equivalence relation on all families of morphisms with fixed target.
\item The relation ``$\mathcal{U}$ refines $\mathcal{V}$ and
$\mathcal{V}$ refines $\mathcal{U}$'' is an equivalence relation on
all families of morphisms with fixed target.
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\noindent
In the following lemma, given a category $\mathcal{C}$, a presheaf
$\mathcal{F}$ on $\mathcal{C}$, a
family $\mathcal{U} = \{U_i \to U\}_{i\in I}$ such that
all fibre products $U_i \times_U U_{i'}$ exist, we say that
{\it the sheaf condition for $\mathcal{F}$ with respect to
$\mathcal{U}$} holds if the diagram (\ref{equation-sheaf-condition})
is an equalizer diagram.
\begin{lemma}
\label{lemma-tautological-same-sheaf}
Let $\mathcal{C}$ be a category. Let
$\mathcal{U} = \{\varphi_i : U_i \to U\}_{i\in I}$, and
$\mathcal{V} = \{\psi_j : V_j \to U\}_{j\in J}$ be two families of morphisms
with the same fixed target. Assume that the fibre products
$U_i \times_U U_{i'}$ and $V_j \times_U V_{j'}$ exist.
If $\mathcal{U}$ and $\mathcal{V}$ are
tautologically equivalent, then for any presheaf $\mathcal{F}$ on
$\mathcal{C}$ the sheaf condition for $\mathcal{F}$ with respect to
$\mathcal{U}$ is equivalent to the sheaf condition for $\mathcal{F}$
with respect to $\mathcal{V}$.
\end{lemma}
\begin{proof}
First, note that if $\varphi : A \to B$ is an isomorphism in the
category $\mathcal{C}$, then $\varphi^* : \mathcal{F}(B) \to \mathcal{F}(A)$
is an isomorphism. Let $\beta : J \to I$ be a map and let
$\psi_j : V_j \to U_{\beta(j)}$ be isomorphisms over $U$ which
are assumed to exist by hypothesis. Let us show that the sheaf
condition for $\mathcal{V}$ implies the sheaf condition for $\mathcal{U}$.
Suppose given sections $s_i \in \mathcal{F}(U_i)$ such that
$$
s_i|_{U_i \times_U U_{i'}} = s_{i'}|_{U_i \times_U U_{i'}}
$$
in $\mathcal{F}(U_i \times_U U_{i'})$ for all pairs $(i, i') \in I \times I$.
Then we can define $s_j = \psi_j^*s_{\beta(j)}$. For any pair
$(j, j') \in J \times J$ the morphism
$\psi_j \times_{\text{id}_U} \psi_{j'} : V_j \times_U V_{j'} \to
U_{\beta(j)} \times_U U_{\beta(j')}$ is an isomorphism as well.
Hence by transport of structure we see that
$$
s_j|_{V_j \times_U V_{j'}} = s_{j'}|_{V_j \times_U V_{j'}}
$$
as well. The sheaf condition w.r.t.\ $\mathcal{V}$ implies there
exists a unique $s$ such that $s|_{V_j} = s_j$ for all $j \in J$.
By the first remark of the proof this implies that $s|_{U_i} = s_i$
for all $i \in \Im(\beta)$ as well. Suppose that $i \in I$,
$i \not \in \Im(\beta)$. For such an $i$ we have isomorphisms
$U_i \to V_{\alpha(i)} \to U_{\beta(\alpha(i))}$ over $U$. This gives a
morphism $U_i \to U_i \times_U U_{\beta(\alpha(i))}$ which is a
section of the projection. Because $s_i$ and $s_{\beta(\alpha(i))}$
restrict to the same element on the fibre product we conclude that
$s_{\beta(\alpha(i))}$ pulls back to $s_i$ via $U_i \to U_{\beta(\alpha(i))}$.
Thus we see that also $s_i = s|_{U_i}$ as desired.
\end{proof}
\begin{lemma}
\label{lemma-refine-same-topology}
Let $\mathcal{C}$ be a category. Let $\text{Cov}_i$, $i = 1, 2$
be two sets of families of morphisms with fixed target which
each define the structure of a site on $\mathcal{C}$.
\begin{enumerate}
\item If every $\mathcal{U} \in \text{Cov}_1$ is tautologically
equivalent to some $\mathcal{V} \in \text{Cov}_2$, then
$\Sh(\mathcal{C}, \text{Cov}_2) \subset
\Sh(\mathcal{C}, \text{Cov}_1)$.
If also, every $\mathcal{U} \in \text{Cov}_2$ is tautologically
equivalent to some $\mathcal{V} \in \text{Cov}_1$ then
the category of sheaves are equal.
\item Suppose
that for each $\mathcal{U} \in \text{Cov}_1$ there exists a
$\mathcal{V} \in \text{Cov}_2$ such that $\mathcal{V}$ refines
$\mathcal{U}$. In this case
$\Sh(\mathcal{C}, \text{Cov}_2) \subset
\Sh(\mathcal{C}, \text{Cov}_1)$.
If also for every $\mathcal{U} \in \text{Cov}_2$
there exists a $\mathcal{V} \in \text{Cov}_1$ such that $\mathcal{V}$
refines $\mathcal{U}$, then the categories of sheaves
are equal.
\end{enumerate}
\end{lemma}
\begin{proof}
Part (1) follows directly from Lemma \ref{lemma-tautological-same-sheaf}
and the definitions.
\medskip\noindent
We advise the reader to {\bf skip the proof of (2)} on a first reading.
Let $\mathcal{F}$ be a sheaf of sets for the site
$(\mathcal{C}, \text{Cov}_2)$. Let $\mathcal{U} \in \text{Cov}_1$,
say $\mathcal{U} = \{U_i \to U\}_{i\in I}$. Choose a
refinement $\mathcal{V} \in \text{Cov}_2$ of $\mathcal{U}$, say
$\mathcal{V} = \{V_j \to U\}_{j\in J}$ and refinement given
by $\alpha : J \to I$ and $f_j : V_j \to U_{\alpha(j)}$.
\medskip\noindent
First let $s, s' \in \mathcal{F}(U)$. If for all $i \in I$ we
have $s|_{U_i} = s'|_{U_i}$, then we also have $s|_{V_j} = s'|_{V_j}$
for all $j \in J$. This implies that $s = s'$ by the sheaf condition
for $\mathcal{F}$ with respect to $\text{Cov}_2$. Hence we see that
the unicity in the sheaf condition for $\mathcal{F}$ and the
site $(\mathcal{C}, \text{Cov}_1)$ holds.
\medskip\noindent
Next, suppose given
$s_i \in \mathcal{F}(U_i)$ such that $s_i|_{U_i \times_U U_{i'}}
= s_{i'}|_{U_i \times_U U_{i'}}$ for all $i, i' \in I$.
Set $s_j = f_j^*(s_{\alpha(j)}) \in \mathcal{F}(V_j)$.
Since the morphisms $f_j$ are morphisms over $U$ we obtain
induced morphisms $f_{jj'} : V_j \times_U V_{j'} \to
U_{\alpha(i)} \times_U U_{\alpha(i')}$ compatible with the
$f_j, f_{j'}$ via the projection maps. It follows that
$$
s_j|_{V_j \times_U V_{j'}}
= f_{jj'}^*(s_{\alpha(j)}|_{U_{\alpha(j)} \times_U U_{\alpha(j')}})
= f_{jj'}^*(s_{\alpha(j')}|_{U_{\alpha(j)} \times_U U_{\alpha(j')}})
= s_{j'}|_{V_j \times_U V_{j'}}
$$
for all $j, j' \in J$. Hence, by the sheaf condition
for $\mathcal{F}$ with respect to $\text{Cov}_2$, we get a section
$s \in \mathcal{F}(U)$ which restricts to $s_j$ on each $V_j$.
We are done if we show $s$ restricts to $s_{i_0}$ on $U_{i_0}$
for any $i_0 \in I$. For each $i_0 \in I$ the family
$\mathcal{U}' = \{U_i \times_U U_{i_0} \to U_{i_0}\}_{i \in I}$
is an element of $\text{Cov}_1$ by the axioms of a site.
Also, the family
$\mathcal{V}' = \{V_j \times_U U_{i_0} \to U_{i_0}\}_{j \in J}$
is an element of $\text{Cov}_2$.
Then $\mathcal{V}'$ refines $\mathcal{U}'$
via $\alpha : J \to I$ and the maps $f'_j = f_j \times \text{id}_{U_{i_0}}$.
The element $s_{i_0}$ restricts to $s_i|_{U_i \times_U U_{i_0}}$
on the members of the covering $\mathcal{U}'$ and hence via
$(f_j')^*$ to the elements $s_j|_{V_j \times_U U_{i_0}}$ on the members
of the covering $\mathcal{V}'$. By construction of $s$ this is the
same as the family of restrictions of $s|_{U_{i_0}}$ to the members
of the covering $\mathcal{V}'$. Hence by the sheaf condition
for $\mathcal{F}$ with respect to $\text{Cov}_2$ we see that
$s|_{U_{i_0}} = s_{i_0}$ as desired.
\end{proof}
\begin{lemma}
\label{lemma-choice-set-coverings-immaterial}
Let $\mathcal{C}$ be a category.
Let $\text{Cov}(\mathcal{C})$ be a proper class of coverings
satisfying conditions (1), (2) and (3) of Definition \ref{definition-site}.
Let $\text{Cov}_1, \text{Cov}_2 \subset \text{Cov}(\mathcal{C})$
be two subsets of $\text{Cov}(\mathcal{C})$ which endow
$\mathcal{C}$ with the structure of a site. If
every covering $\mathcal{U} \in \text{Cov}(\mathcal{C})$
is combinatorially equivalent to a covering in
$\text{Cov}_1$ and combinatorially equivalent to a
covering in $\text{Cov}_2$, then
$\Sh(\mathcal{C}, \text{Cov}_1) =
\Sh(\mathcal{C}, \text{Cov}_2)$.
\end{lemma}
\begin{proof}
This is clear from Lemmas \ref{lemma-refine-same-topology}
and \ref{lemma-tautological-combinatorial} above as the hypothesis
implies that every covering
$\mathcal{U} \in \text{Cov}_1 \subset \text{Cov}(\mathcal{C})$
is combinatorially equivalent to an element of $\text{Cov}_2$,
and similarly with the roles of $\text{Cov}_1$ and $\text{Cov}_2$
reversed.
\end{proof}
\section{The example of G-sets}
\label{section-example-sheaf-G-sets}
\noindent
As an example, consider the site $\mathcal{T}_G$ of
Example \ref{example-site-on-group}. We will describe the
category of sheaves on $\mathcal{T}_G$. The answer will turn
out to be independent of the choices made in defining $\mathcal{T}_G$.
In fact, during the proof we will need only the following
properties of the site $\mathcal{T}_G$:
\begin{enumerate}
\item[(a)] $\mathcal{T}_G$ is a full subcategory of $G\textit{-Sets}$,
\item[(b)] $\mathcal{T}_G$ contains the $G$-set ${}_GG$,
\item[(c)] $\mathcal{T}_G$ has fibre products and they are the same as
in $G\textit{-Sets}$,
\item[(d)] given $U \in \Ob(\mathcal{T}_G)$ and a $G$-invariant
subset $O \subset U$, there exists an object of $\mathcal{T}_G$ isomorphic
to $O$, and
\item[(e)] any surjective family of maps $\{U_i \to U\}_{i \in I}$, with
$U, U_i \in \Ob(\mathcal{T}_G)$ is combinatorially equivalent to a
covering of $\mathcal{T}_G$.
\end{enumerate}
These properties hold by Sets, Lemmas \ref{sets-lemma-what-is-in-it-G-sets}
and \ref{sets-lemma-coverings-site}.
\medskip\noindent
Remark that the map
$$
\Hom_G({}_GG, {}_GG)
\longrightarrow
G^{opp},
\varphi
\longmapsto
\varphi(1)
$$
is an isomorphism of groups. The inverse map sends $g \in G$
to the map $R_g : s \mapsto sg$ (i.e.\ right multiplication).
Note that $R_{g_1g_2} = R_{g_2} \circ R_{g_1}$ so the opposite
is necessary.
\medskip\noindent
This implies that for every presheaf $\mathcal{F}$ on $\mathcal{T}_G$
the value $\mathcal{F}({}_GG)$ inherits the structure of a $G$-set
as follows: $g \cdot s$ for $g \in G$ and $s \in \mathcal{F}({}_GG)$
defined by $\mathcal{F}(R_g)(s)$. This is a left action
because
$$
(g_1g_2) \cdot s = \mathcal{F}(R_{g_1g_2})(s) =
\mathcal{F}(R_{g_2}\circ R_{g_1})(s) =
\mathcal{F}(R_{g_1})( \mathcal{F}(R_{g_2})(s)) =
g_1 \cdot (g_2 \cdot s).
$$
Here we've used that $\mathcal{F}$
is contravariant. Note that if $\mathcal{F} \to \mathcal{G}$
is a morphism of presheaves of sets on $\mathcal{T}_G$
then we get a map $\mathcal{F}({}_GG) \to \mathcal{G}({}_GG)$
which is compatible with the $G$-actions we have just defined.
All in all we have constructed a functor
$$
\textit{PSh}(\mathcal{T}_G)
\longrightarrow
G\textit{-Sets}, \quad
\mathcal{F}
\longmapsto
\mathcal{F}({}_GG).
$$
We leave it to the reader to verify that this construction
has the pleasing property that the representable presheaf
$h_U$ is mapped to something canonically isomorphic to $U$.
In a formula $h_U({}_GG) = \Hom_G({}_GG, U) \cong U$.
\medskip\noindent
Suppose that $S$ is a $G$-set. We define a presheaf
$\mathcal{F}_S$ by the formula\footnote{It may appear this is the
representable presheaf defined by $S$. This may not be the case
because $S$ may not be an object of $\mathcal{T}_G$ which was chosen
to be a sufficiently large set of $G$-sets.}
$$
\mathcal{F}_S(U)
=
\Mor_{G\textit{-Sets}}(U, S).
$$
This is clearly a presheaf. On the other hand, suppose that
$\{U_i \to U\}_{i\in I}$ is a covering in $\mathcal{T}_G$.
This implies that $\coprod_i U_i \to U$ is surjective. Thus it is
clear that the map
$$
\mathcal{F}_S(U)
=
\Mor_{G\textit{-Sets}}(U, S)
\longrightarrow
\prod \mathcal{F}_S(U_i)
=
\prod \Mor_{G\textit{-Sets}}(U_i, S)
$$
is injective. And, given a family of $G$-equivariant
maps $s_i : U_i \to S$, such that all the diagrams
$$
\xymatrix{
U_i \times_U U_j \ar[d] \ar[r]
&
U_j \ar[d]^{s_j}
\\
U_i \ar[r]^{s_i}
&
S
}
$$
commute, there is a unique $G$-equivariant map
$s : U \to S$ such that $s_i$ is the composition
$U_i \to U \to S$. Namely, we just define $s(u) = s_i(u_i)$
where $i\in I$ is any index such that there exists some
$u_i \in U_i$ mapping to $u$ under the map $U_i \to U$.
The commutativity of the diagrams above implies exactly
that this construction is well defined. All in all we have
constructed a functor
$$
G\textit{-Sets}
\longrightarrow
\Sh(\mathcal{T}_G), \quad
S
\longmapsto
\mathcal{F}_S
.
$$
\medskip\noindent
We now have the following diagram of categories and functors
$$
\xymatrix{
\textit{PSh}(\mathcal{T}_G) \ar[rr]^{\mathcal{F} \mapsto \mathcal{F}({}_GG)}
&
&
G\textit{-Sets} \ar[ld]_{S \mapsto \mathcal{F}_S}
\\
&
\Sh(\mathcal{T}_G) \ar[lu]
&
}
$$
It is immediate from the definitions that $\mathcal{F}_S({}_GG)
= \Mor_G({}_GG, S) = S$, the last equality by evaluation at $1$.
This almost proves the following.
\begin{proposition}
\label{proposition-sheaves-on-group}
The functors $\mathcal{F} \mapsto \mathcal{F}({}_GG)$
and $S \mapsto \mathcal{F}_S$ define quasi-inverse
equivalences between $\Sh(\mathcal{T}_G)$
and $G\textit{-Sets}$.
\end{proposition}
\begin{proof}
We have already seen that composing the functors one way around
is isomorphic to the identity functor.
In the other direction, for any sheaf $\mathcal{H}$ there is a natural
map of sheaves
$$
can :
\mathcal{H}
\longrightarrow
\mathcal{F}_{\mathcal{H}({}_GG)}.
$$
Namely, for any object $U$ of $\mathcal{T}_G$ we let $can_U$
be the map
$$
\begin{matrix}
\mathcal{H}(U)
&
\longrightarrow
&
\mathcal{F}_{\mathcal{H}({}_GG)}(U)
=
\Mor_G(U, \mathcal{H}({}_GG))
\\
s
&
\longmapsto
&
(u \mapsto \alpha_u^*s).
\end{matrix}
$$
Here $\alpha_u : {}_GG \to U$ is the map
$\alpha_u(g) = gu$ and $\alpha_u^* : \mathcal{H}(U)
\to \mathcal{H}({}_GG)$ is the pullback map. A trivial
but confusing verification shows that this is indeed a map
of presheaves. We have to show that $can$ is an isomorphism.
We do this by showing $can_U$ is an isomorphism for all $U
\in \text{ob}(\mathcal{T}_G)$. We leave the (important but
easy) case that $U = {}_GG$ to the reader.
A general object $U$ of $\mathcal{T}_G$ is a disjoint union of
$G$-orbits: $U = \coprod_{i\in I} O_i$. The family of maps
$\{O_i \to U\}_{i \in I}$ is tautologically equivalent
to a covering in $\mathcal{T}_G$ (by the properties of $\mathcal{T}_G$
listed at the beginning of this section). Hence by Lemma
\ref{lemma-tautological-same-sheaf} the sheaf $\mathcal{H}$
satisfies the sheaf property with respect to
$\{O_i \to U\}_{i \in I}$. The sheaf property for this covering
implies $\mathcal{H}(U) = \prod_i \mathcal{H}(O_i)$.
Hence it suffices to show that $can_U$ is an
isomorphism when $U$ consists of a single $G$-orbit. Let $u \in U$
and let $H \subset G$ be its stabilizer. Clearly,
$\Mor_G(U, \mathcal{H}({}_GG)) = \mathcal{H}({}_GG)^H$
equals the subset of $H$-invariant elements. On the other hand
consider the covering $\{{}_GG \to U\}$ given by $g \mapsto
gu$ (again it is just combinatorially equivalent to some covering
of $\mathcal{T}_G$, and again this doesn't matter).
Note that the fibre product $({}_GG)\times_U ({}_GG)$
is equal to $\{(g, gh), g\in G, h\in H\} \cong \coprod_{h\in H} {}_GG$.
Hence the sheaf property for this covering reads as
$$
\xymatrix{
\mathcal{H}(U) \ar[r]
&
\mathcal{H}({}_GG)
\ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*}
&
\prod_{h \in H}
\mathcal{H}({}_GG).
}
$$
The two maps $\text{pr}_i^*$ into the factor
$\mathcal{H}({}_GG)$ differ by multiplication by $h$.
Now the result follows from this and the fact that $can$
is an isomorphism for $U = {}_GG$.
\end{proof}
\section{Sheafification}
\label{section-sheafification}
\noindent
In order to define the sheafification we study the zeroth
{\v C}ech cohomology group of a covering and its functoriality
properties.
\medskip\noindent
Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$, and let
$\mathcal{U} = \{U_i \to U\}_{i \in I}$ be a covering of $\mathcal{C}$.
Let us use the notation $\mathcal{F}(\mathcal{U})$ to indicate the equalizer
$$
H^0(\mathcal{U}, \mathcal{F})
=
\{
(s_i)_{i\in I} \in \prod\nolimits_i \mathcal{F}(U_i)
\mid
s_i|_{U_i \times_U U_j} = s_j|_{U_i \times_U U_j}
\ \forall i, j \in I
\}.
$$
As we will see later, this is the zeroth {\v C}ech cohomology
of $\mathcal{F}$ over $U$ with respect to the covering $\mathcal{U}$.
A small remark is that we can define $H^0(\mathcal{U}, \mathcal{F})$
as soon as all the morphisms $U_i \to U$ are representable, i.e.,
$\mathcal{U}$ need not be a covering of the site.
There is a canonical map $\mathcal{F}(U) \to H^0(\mathcal{U}, \mathcal{F})$.
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 $H^0(\mathcal{V}, \mathcal{F}) \to
H^0(\mathcal{U}, \mathcal{F})$, compatible with the map $\mathcal{F}(V)
\to \mathcal{F}(U)$.
\medskip\noindent
By construction, a presheaf $\mathcal{F}$ is a sheaf if and only if for
every covering $\mathcal{U}$ of $\mathcal{C}$ the natural map
$\mathcal{F}(U) \to H^0(\mathcal{U}, \mathcal{F})$ is bijective.
We will use this notion to prove the following
simple lemma about limits of sheaves.
\begin{lemma}
\label{lemma-limit-sheaf}
Let $\mathcal{F} : \mathcal{I} \to \Sh(\mathcal{C})$
be a diagram. Then $\lim_\mathcal{I} \mathcal{F}$ exists
and is equal to the limit in the category of presheaves.
\end{lemma}
\begin{proof}
Let $\lim_i \mathcal{F}_i$ be the limit as a presheaf.
We will show that this is a sheaf and then it will trivially follow
that it is a limit in the category of sheaves. To prove the sheaf
property, let $\mathcal{V} = \{V_j \to V\}_{j\in J}$ be a covering.
Let $(s_j)_{j\in J}$ be an element of $H^0(\mathcal{V}, \lim_i \mathcal{F}_i)$.
Using the projection maps we get elements $(s_{j, i})_{j\in J}$
in $H^0(\mathcal{V}, \mathcal{F}_i)$. By the sheaf property for
$\mathcal{F}_i$ we see that there is a unique $s_i \in \mathcal{F}_i(V)$
such that $s_{j, i} = s_i|_{V_j}$. Let $\phi : i \to i'$ be a morphism
of the index category. We would like to show that
$\mathcal{F}(\phi) : \mathcal{F}_i \to \mathcal{F}_{i'}$
maps $s_i$ to $s_{i'}$. We know this is true for the sections
$s_{i, j}$ and $s_{i', j}$ for all $j$ and hence by the sheaf property
for $\mathcal{F}_{i'}$ this is true. At this point we have an
element $s = (s_i)_{i \in \Ob(\mathcal{I})}$ of
$(\lim_i \mathcal{F}_i)(V)$. We leave it to the reader to see
this element has the required property that $s_j = s|_{V_j}$.
\end{proof}
\begin{example}
\label{example-singleton-sheaf}
A particular example is the limit over the empty diagram.
This gives the final object in the category of (pre)sheaves.
It is the sheaf that associates to each object $U$ of $\mathcal{C}$
a singleton set, with unique restriction mappings. We often denote
this sheaf by $*$.
\end{example}
\noindent
Let $\mathcal{J}_U$ be the category of all coverings of $U$.
In other words, the objects of $\mathcal{J}_U$ are the coverings
of $U$ in $\mathcal{C}$, and the morphisms are the refinements.
By our conventions on sites this is indeed a category, i.e.,
the collection of objects and morphisms forms a set.
Note that $\Ob(\mathcal{J}_U)$ is not empty since
$\{\text{id}_U\}$ is an object of it. According to the remarks
above the construction $\mathcal{U} \mapsto H^0(\mathcal{U}, \mathcal{F})$
is a contravariant functor on $\mathcal{J}_U$.
We define
$$
\mathcal{F}^{+}(U)
=
\colim_{\mathcal{J}_U^{opp}}
H^0(\mathcal{U}, \mathcal{F})
$$
See Categories, Section \ref{categories-section-limits} for
a discussion of limits and colimits. We point out that later
we will see that $\mathcal{F}^{+}(U)$ is the zeroth {\v C}ech
cohomology of $\mathcal{F}$ over $U$.
\medskip\noindent
Before we say more about the structure of the colimit, we turn
the collection of sets
$\mathcal{F}^{+}(U)$, $U \in \Ob(\mathcal{C})$
into a presheaf. Namely, let $V \to U$ be a morphism of $\mathcal{C}$.
By the axioms of a site there is a functor\footnote{This construction
actually involves a choice of the fibre products $U_i \times_U V$
and hence the axiom of choice. The resulting map does not depend on
the choices made, see below.}
$$
\mathcal{J}_U
\longrightarrow
\mathcal{J}_V, \quad
\{U_i \to U\}
\longmapsto
\{U_i \times_U V \to V\}.
$$
Note that the projection maps furnish a functorial
morphism of coverings $\{U_i \times_U V \to V\} \to \{U_i \to U\}$
and hence, by the construction above, a functorial map of sets
$H^0(\{U_i \to U\}, \mathcal{F}) \to
H^0(\{U_i \times_U V \to V\}, \mathcal{F})$.
In other words, there is a transformation of functors
from $H^0(-, \mathcal{F}) : \mathcal{J}_U \to \textit{Sets}$
to the composition $\mathcal{J}_U \to \mathcal{J}_V
\xrightarrow{H^0(-, \mathcal{F})} \textit{Sets}$. Hence by
generalities of colimits we obtain a canonical map
$\mathcal{F}^+(U) \to \mathcal{F}^+(V)$. In terms of the description
of the set $\mathcal{F}^+(U)$ above, it just takes the element
associated with $s = (s_i) \in H^0(\{U_i \to U\}, \mathcal{F})$ to the
element associated with $(s_i|_{V \times_U U_i})
\in H^0(\{U_i \times_U V \to V\}, \mathcal{F})$.
\begin{lemma}
\label{lemma-plus-presheaf}
The constructions above define a presheaf
$\mathcal{F}^+$ together with a canonical
map of presheaves $\mathcal{F} \to \mathcal{F}^+$.
\end{lemma}
\begin{proof}
All we have to do is to show that given morphisms
$W \to V \to U$ the composition $\mathcal{F}^+(U)
\to \mathcal{F}^+(V) \to \mathcal{F}^+(W)$
equals the map $\mathcal{F}^+(U) \to \mathcal{F}^+(W)$.
This can be shown directly by verifying that, given
a covering $\{U_i \to U\}$ and
$s = (s_i) \in H^0(\{U_i \to U\}, \mathcal{F})$,
we have canonically
$W \times_U U_i \cong W \times_V (V \times_U U_i)$,
and
$s_i|_{W \times_U U_i}$
corresponds to
$(s_i|_{V \times_U U_i})|_{W \times_V (V \times_U U_i)}$
via this isomorphism.
\end{proof}
\noindent
More indirectly, the result of
Lemma \ref{lemma-independent-refinement} shows that
we may pullback an element $s$ as above via any morphism
from any covering of $W$ to $\{U_i \to U\}$
and we will always end up with the same element in
$\mathcal{F}^+(W)$.
\begin{lemma}
\label{lemma-plus-functorial}
The association $\mathcal{F} \mapsto
(\mathcal{F} \to \mathcal{F}^+)$
is a functor.
\end{lemma}
\begin{proof}
Instead of proving this we state exactly what needs to be proven.
Let $\mathcal{F} \to \mathcal{G}$ be a map of presheaves. Prove
the commutativity of:
$$
\xymatrix{
\mathcal{F} \ar[r] \ar[d]
&
\mathcal{F}^{+} \ar[d]
\\
\mathcal{G} \ar[r]
&
\mathcal{G}^{+}
}
$$
\end{proof}
\noindent
The next two lemmas imply that the colimits above are colimits
over a directed set.
\begin{lemma}
\label{lemma-common-refinement}
Given a pair of coverings $\{U_i \to U\}$
and $\{V_j \to U\}$ of a given object $U$ of the site
$\mathcal{C}$, there exists a covering which is a
common refinement.
\end{lemma}
\begin{proof}
Since $\mathcal{C}$ is a site we have that for every $i$ the
family $\{V_j \times_U U_i \to U_i\}_j$ is a covering.
And, then another axiom implies that $\{V_j \times_U U_i \to U\}_{i, j}$
is a covering of $U$. Clearly this covering refines both given
coverings.
\end{proof}
\begin{lemma}
\label{lemma-independent-refinement}
Any two morphisms $f, g: \mathcal{U} \to \mathcal{V}$ of coverings
inducing the same morphism $U \to V$ induce the same
map $H^0(\mathcal{V}, \mathcal{F}) \to H^0(\mathcal{U}, \mathcal{F})$.
\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 $\alpha : I \to J$ and
maps $f_i : U_i \to V_{\alpha(i)}$.
Likewise, $g$~determines a map $\beta : I \to J$ and maps
$g_i : U_i \to V_{\beta(i)}$.
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
$$
\xymatrix{
&
V_{\alpha(i)}
\\
U_i \ar[r]^-\varphi \ar[ur]^{f_i} \ar[dr]_{g_i}
&
V_{\alpha(i)} \times_V V_{\beta(i)} \ar[u]_{\text{pr}_1} \ar[d]^{\text{pr}_2}
\\
&
V_{\beta(i)}.
}
$$
Now let $s = (s_j)_j \in H^0(\mathcal{V}, \mathcal{F})$.
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 $H^0(\mathcal{V}, \mathcal{F})$.
This shows that the maps
$H^0(\mathcal{V}, \mathcal{F}) \to H^0(\mathcal{U}, \mathcal{F})$
induced by $f$ and $g$ are equal.
\end{proof}
\begin{remark}
\label{remark-both-refine-same-H0}
In particular this lemma shows that if $\mathcal{U}$ is
a refinement of $\mathcal{V}$, and if $\mathcal{V}$ is a
refinement of $\mathcal{U}$, then there is a canonical
identification $H^0(\mathcal{U}, \mathcal{F}) =
H^0(\mathcal{V}, \mathcal{F})$.
\end{remark}
\noindent
From these two lemmas, and the fact that $\mathcal{J}_U$ is nonempty,
it follows that the diagram $H^0(-, \mathcal{F}) : \mathcal{J}_U^{opp}
\to \textit{Sets}$ is filtered, see
Categories, Definition \ref{categories-definition-directed}.
Hence, by Categories,
Section \ref{categories-section-directed-colimits}
the colimit $\mathcal{F}^{+}(U)$ may be described
in the following straightforward manner. Namely, every element in the set
$\mathcal{F}^{+}(U)$ arises from an element
$s \in H^0(\mathcal{U}, \mathcal{F})$ for some covering
$\mathcal{U}$ of $U$. Given a second element $s' \in
H^0(\mathcal{U}', \mathcal{F})$ then $s$ and $s'$ determine
the same element of the colimit if and only if there exists a covering
$\mathcal{V}$ of $U$ and refinements $f : \mathcal{V} \to \mathcal{U}$ and
$f' : \mathcal{V} \to \mathcal{U}'$ such that $f^*s = (f')^*s'$
in $H^0(\mathcal{V}, \mathcal{F})$. Since the trivial covering
$\{\text{id}_U\}$ is an object of $\mathcal{J}_U$ we get
a canonical map $\mathcal{F}(U) \to \mathcal{F}^+(U)$.
\begin{lemma}
\label{lemma-plus-surjective}
The map $\theta : \mathcal{F} \to \mathcal{F}^+$ has the following
property: For every object $U$ of $\mathcal{C}$ and every section
$s \in \mathcal{F}^+(U)$ there exists a covering $\{U_i \to U\}$
such that $s|_{U_i}$ is in the image of $\theta : \mathcal{F}(U_i)
\to \mathcal{F}^{+}(U_i)$.
\end{lemma}
\begin{proof}
Namely, let $\{U_i \to U\}$ be a covering such that $s$ arises
from the element $(s_i) \in H^0(\{U_i \to U\}, \mathcal{F})$.
According to Lemma \ref{lemma-independent-refinement} we may
consider the covering $\{U_i \to U_i\}$ and the (obvious) morphism
of coverings $\{U_i \to U_i\} \to \{U_i \to U\}$ to compute the
pullback of $s$ to an element of $\mathcal{F}^+(U_i)$. And indeed,
using this covering we get exactly $\theta(s_i)$ for the restriction
of $s$ to $U_i$.
\end{proof}
\begin{definition}
\label{definition-separated}
We say that a presheaf of sets $\mathcal{F}$ on a site
$\mathcal{C}$ is {\it separated} if, for all coverings
of $\{U_i \rightarrow U\}$, the map
$\mathcal{F}(U) \to \prod \mathcal{F}(U_i)$ is injective.
\end{definition}
\begin{theorem}
\label{theorem-plus}
With $\mathcal{F}$ as above
\begin{enumerate}
\item
\label{item-sep}
The presheaf $\mathcal{F}^+$ is separated.
\item
\label{item-sheaf}
If $\mathcal{F}$ is separated, then $\mathcal{F}^+$ is a sheaf
and the map of presheaves $\mathcal{F} \to \mathcal{F}^+$ is injective.
\item
\label{item-plus-iso}
If $\mathcal{F}$ is a sheaf, then $\mathcal{F} \to \mathcal{F}^+$
is an isomorphism.
\item
\label{item-plusplus}
The presheaf $\mathcal{F}^{++}$ is always a sheaf.
\end{enumerate}
\end{theorem}
\begin{proof}
Proof of (\ref{item-sep}).
Suppose that $s, s' \in \mathcal{F}^+(U)$ and suppose that
there exists some covering $\{U_i \to U\}$ such that
$s|_{U_i} = s'|_{U_i}$ for all $i$. We now have three coverings
of $U$: the covering $\{U_i \to U\}$ above, a covering $\mathcal{U}$
for $s$ as in Lemma \ref{lemma-plus-surjective},
and a similar covering $\mathcal{U}'$ for $s'$. By Lemma
\ref{lemma-common-refinement}, we can find a common refinement,
say $\{W_j \to U\}$. This means we have $s_j, s'_j \in \mathcal{F}(W_j)$
such that $s|_{W_j} = \theta(s_j)$, similarly for $s'|_{W_j}$, and
such that $\theta(s_j) = \theta(s'_j)$. This last equality means
that there exists some covering $\{W_{jk} \to W_j\}$ such that
$s_j|_{W_{jk}} = s'_j|_{W_{jk}}$. Then since $\{W_{jk} \to U\}$
is a covering we see that $s, s'$ map to the same element of
$H^0(\{W_{jk} \to U\}, \mathcal{F})$ as desired.
\medskip\noindent
Proof of (\ref{item-sheaf}). It is clear that $\mathcal{F} \to
\mathcal{F}^+$ is injective because all the maps
$\mathcal{F}(U) \to H^0(\mathcal{U}, \mathcal{F})$
are injective. It is also clear that, if $\mathcal{U} \to
\mathcal{U}'$ is a refinement, then $H^0(\mathcal{U}', \mathcal{F})
\to H^0(\mathcal{U}, \mathcal{F})$ is injective. Now,
suppose that $\{U_i \to U\}$ is a covering, and let
$(s_i)$ be a family of elements of $\mathcal{F}^+(U_i)$
satisfying the sheaf condition
$s_i|_{U_i \times_U U_j} = s_j|_{U_i \times_U U_j}$
for all $i, j \in I$. Choose coverings (as in
Lemma \ref{lemma-plus-surjective}) $\{U_{ij} \to U_i\}$
such that $s_i|_{U_{ij}}$ is the image of the (unique)
element $s_{ij} \in \mathcal{F}(U_{ij})$. The sheaf condition
implies that $s_{ij}$ and $s_{i'j'}$ agree over
$U_{ij} \times_U U_{i'j'}$ because it maps to
$U_i \times_U U_{i'}$ and we have the equality there.
Hence $(s_{ij}) \in H^0(\{U_{ij} \to U\}, \mathcal{F})$
gives rise to an element $s \in \mathcal{F}^+(U)$. We leave
it to the reader to verify that $s|_{U_i} = s_i$.
\medskip\noindent
Proof of (\ref{item-plus-iso}). This is immediate from the definitions
because the sheaf property says exactly that every map
$\mathcal{F} \to H^0(\mathcal{U}, \mathcal{F})$ is bijective
(for every covering $\mathcal{U}$ of $U$).
\medskip\noindent
Statement (\ref{item-plusplus}) is now obvious.
\end{proof}
\begin{definition}
\label{definition-associated-sheaf}
Let $\mathcal{C}$ be a site and let $\mathcal{F}$ be a presheaf
of sets on $\mathcal{C}$. The sheaf $\mathcal{F}^\# := \mathcal{F}^{++}$
together with the canonical map $\mathcal{F} \to \mathcal{F}^\#$
is called the {\it sheaf associated to $\mathcal{F}$}.
\end{definition}
\begin{proposition}
\label{proposition-sheafification-adjoint}
The canonical map $\mathcal{F} \to \mathcal{F}^\#$ has the
following universal property: For any map $\mathcal{F} \to \mathcal{G}$,
where $\mathcal{G}$ is a sheaf of sets, there is a unique map
$\mathcal{F}^\# \to \mathcal{G}$ such that $\mathcal{F} \to \mathcal{F}^\#
\to \mathcal{G}$ equals the given map.
\end{proposition}
\begin{proof}
By Lemma \ref{lemma-plus-functorial} we get a commutative diagram
$$
\xymatrix{
\mathcal{F} \ar[r] \ar[d]
&
\mathcal{F}^{+} \ar[r] \ar[d]
&
\mathcal{F}^{++} \ar[d]
\\
\mathcal{G} \ar[r]
&
\mathcal{G}^{+} \ar[r]
&
\mathcal{G}^{++}
}
$$
and by Theorem \ref{theorem-plus} the lower horizontal maps
are isomorphisms. The uniqueness follows from Lemma
\ref{lemma-plus-surjective} which says that every section of
$\mathcal{F}^\#$ locally comes from sections of $\mathcal{F}$.
\end{proof}
\noindent
It is clear from this result that the functor $\mathcal{F}
\mapsto (\mathcal{F} \to \mathcal{F}^\#)$ is unique
up to unique isomorphism of functors. Actually, let us temporarily
denote $i : \Sh(\mathcal{C}) \to \textit{PSh}(\mathcal{C})$
the functor of inclusion. The result above actually says that
$$
\Mor_{\textit{PSh}(\mathcal{C})}(\mathcal{F}, i(\mathcal{G}))
=
\Mor_{\Sh(\mathcal{C})}(\mathcal{F}^\#, \mathcal{G}).
$$
In other words, the functor of sheafification is the left adjoint
to the inclusion functor $i$. We finish this section with a couple
of lemmas.
\begin{lemma}
\label{lemma-colimit-sheaves}
Let $\mathcal{F} : \mathcal{I} \to \Sh(\mathcal{C})$
be a diagram. Then $\colim_\mathcal{I} \mathcal{F}$ exists
and is the sheafification of the colimit in the category of presheaves.
\end{lemma}
\begin{proof}
Since the sheafification functor is a left adjoint it commutes
with all colimits, see Categories,
Lemma \ref{categories-lemma-adjoint-exact}.
Hence, since $\textit{PSh}(\mathcal{C})$ has colimits, we deduce
that $\Sh(\mathcal{C})$ has colimits (which are the
sheafifications of the colimits in presheaves).
\end{proof}
\begin{lemma}
\label{lemma-sheafification-exact}
The functor $\textit{PSh}(\mathcal{C}) \to \Sh(\mathcal{C})$,
$\mathcal{F} \mapsto \mathcal{F}^\#$ is exact.
\end{lemma}
\begin{proof}
Since it is a left adjoint it is right exact, see
Categories, Lemma \ref{categories-lemma-exact-adjoint}.
On the other hand, by Lemmas \ref{lemma-common-refinement}
and Lemma \ref{lemma-independent-refinement} the colimits
in the construction of $\mathcal{F}^+$ are really over the
directed set $\Ob(\mathcal{J}_U)$ where
$\mathcal{U} \geq \mathcal{U}'$ if and only if
$\mathcal{U}$ is a refinement of $\mathcal{U}'$. Hence by
Categories, Lemma \ref{categories-lemma-directed-commutes}
we see that $\mathcal{F} \to \mathcal{F}^+$ commutes
with finite limits (as a functor from presheaves to
presheaves). Then we conclude using
Lemma \ref{lemma-limit-sheaf}.
\end{proof}
\begin{lemma}
\label{lemma-sections-sheafification}
Let $\mathcal{C}$ be a site.
Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$.
Denote $\theta^2 : \mathcal{F} \to \mathcal{F}^\#$ the canonical
map of $\mathcal{F}$ into its sheafification.
Let $U$ be an object of $\mathcal{C}$.
Let $s \in \mathcal{F}^\#(U)$. There exists
a covering $\{U_i \to U\}$ and sections
$s_i \in \mathcal{F}(U_i)$ such that
\begin{enumerate}
\item $s|_{U_i} = \theta^2(s_i)$, and
\item for every $i, j$ there exists a covering
$\{U_{ijk} \to U_i \times_U U_j\}$ of $\mathcal{C}$ such that
the pullback of $s_i$ and $s_j$ to each $U_{ijk}$ agree.
\end{enumerate}
Conversely, given any covering $\{U_i \to U\}$, elements
$s_i \in \mathcal{F}(U_i)$ such that (2) holds, then there
exists a unique section $s \in \mathcal{F}^\#(U)$ such
that (1) holds.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\section{Injective and surjective maps of sheaves}
\label{section-sheaves-injective}
\begin{definition}
\label{definition-sheaves-injective-surjective}
Let $\mathcal{C}$ be a site, and let $\varphi : \mathcal{F}
\to \mathcal{G}$ be a map of sheaves of sets.
\begin{enumerate}
\item We say that $\varphi$ is {\it injective} if for every object
$U$ of $\mathcal{C}$ the map $\varphi : \mathcal{F}(U)
\to \mathcal{G}(U)$ is injective.
\item We say that $\varphi$ is {\it surjective} if for every object
$U$ of $\mathcal{C}$ and every section $s\in \mathcal{G}(U)$
there exists a covering $\{U_i \to U\}$ such that for
all $i$ the restriction $s|_{U_i}$ is in the image of
$\varphi : \mathcal{F}(U_i) \to \mathcal{G}(U_i)$.
\end{enumerate}
\end{definition}
\begin{lemma}
\label{lemma-mono-epi-sheaves}
The injective (resp.\ surjective) maps defined above
are exactly the monomorphisms (resp.\ epimorphisms) of
the category $\Sh(\mathcal{C})$. A map of sheaves
is an isomorphism if and only if it is both injective
and surjective.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{lemma}
\label{lemma-coequalizer-surjection}
Let $\mathcal{C}$ be a site. Let $\mathcal{F} \to \mathcal{G}$
be a surjection of sheaves of sets. Then the diagram
$$
\xymatrix{
\mathcal{F} \times_\mathcal{G} \mathcal{F}
\ar@<1ex>[r] \ar@<-1ex>[r]
&
\mathcal{F} \ar[r]
&
\mathcal{G}}
$$
represents $\mathcal{G}$ as a coequalizer.
\end{lemma}
\begin{proof}
Let $\mathcal{H}$ be a sheaf of sets and let
$\varphi : \mathcal{F} \to \mathcal{H}$ be a map of sheaves equalizing
the two maps $\mathcal{F} \times_\mathcal{G} \mathcal{F} \to \mathcal{F}$.
Let $\mathcal{G}' \subset \mathcal{G}$ be the presheaf image of
the map $\mathcal{F} \to \mathcal{G}$. As the product
$\mathcal{F} \times_\mathcal{G} \mathcal{F}$ may be computed in the
category of presheaves we see that it is equal to the presheaf product
$\mathcal{F} \times_{\mathcal{G}'} \mathcal{F}$. Hence $\varphi$
induces a unique map of presheaves $\psi' : \mathcal{G}' \to \mathcal{H}$.
Since $\mathcal{G}$ is the sheafification of $\mathcal{G}'$ by
Lemma \ref{lemma-mono-epi-sheaves}
we conclude that $\psi'$ extends uniquely to a map of sheaves
$\psi : \mathcal{G} \to \mathcal{H}$. We omit the verification that
$\varphi$ is equal to the composition of $\psi$ and the given map.
\end{proof}
\section{Representable sheaves}
\label{section-representable-sheaves}
\noindent
Let $\mathcal{C}$ be a category. The canonical topology is
the finest topology such that all representable presheaves
are sheaves (it is formally defined in
Definition \ref{definition-canonical-topology} but we will not
need this).
This topology is not always the topology associated to the
structure of a site on $\mathcal{C}$.
We will give a collection of coverings that generates this topology
in case $\mathcal{C}$ has fibered products. First we give
the following general definition.
\begin{definition}
\label{definition-universal-effective-epimorphisms}
Let $\mathcal{C}$ be a category. We say that a family $\{U_i \to U\}_{i \in I}$
is an {\it effective epimorphism} if all the morphisms $U_i \to U$ are
representable (see
Categories, Definition \ref{categories-definition-representable-morphism}),
and for any $X\in \Ob(\mathcal{C})$ the sequence
$$
\xymatrix{
\Mor_\mathcal{C}(U, X) \ar[r]
&
\prod\nolimits_{i \in I} \Mor_\mathcal{C}(U_i, X)
\ar@<1ex>[r] \ar@<-1ex>[r]
&
\prod\nolimits_{(i, j) \in I^2} \Mor_\mathcal{C}(U_i \times_U U_j, X)
}
$$
is an equalizer diagram. We say that a family $\{U_i \to U\}$ is a
{\it universal effective epimorphism} if for any morphism $V \to U$
the base change $\{U_i \times_U V \to V\}$ is an effective epimorphism.
\end{definition}
\noindent
The class of families which are universal effective epimorphisms
satisfies the axioms of Definition \ref{definition-site}.
If $\mathcal{C}$ has fibre products, then the associated topology is
the canonical topology. (In this case, to get a site argue as in Sets,
Lemma \ref{sets-lemma-coverings-site}.)
\medskip\noindent
Conversely, suppose that $\mathcal{C}$ is a site such that
all representable presheaves are sheaves. Then clearly, all
coverings are universal effective epimorphisms.
Thus the following definition is the ``correct'' one in the
setting of sites.
\begin{definition}
\label{definition-weaker-than-canonical}
We say that the topology on a site $\mathcal{C}$ is
{\it weaker than the canonical topology}, or that the topology is
{\it subcanonical} if all the coverings
of $\mathcal{C}$ are universal effective epimorphisms.
\end{definition}
\noindent
A representable sheaf is a representable presheaf which is also a
sheaf. Since it is perhaps better to avoid this terminology when the
topology is not subcanonical, we only define it formally in that case.
\begin{definition}
\label{definition-representable-sheaf}
Let $\mathcal{C}$ be a site whose topology is subcanonical.
The Yoneda embedding $h$ (see
Categories, Section \ref{categories-section-opposite})
presents $\mathcal{C}$ as a full subcategory of the
category of sheaves of $\mathcal{C}$. In this case
we call sheaves of the form $h_U$ with $U \in \Ob(\mathcal{C})$
{\it representable sheaves} on $\mathcal{C}$.
Notation: Sometimes, the representable sheaf $h_U$ associated to $U$ is
denoted {\it $\underline{U}$}.
\end{definition}
\noindent
Note that we have in the situation of the definition
$$
\Mor_{\Sh(\mathcal{C})}(h_U, \mathcal{F}) = \mathcal{F}(U)
$$
for every sheaf $\mathcal{F}$, since it holds for presheaves, see
(\ref{equation-map-representable-into-presheaf}). In general the
presheaves $h_U$ are not sheaves and to get a sheaf you have to
sheafify them. In this case we still have
\begin{equation}
\label{equation-map-representable-into-sheaf}
\Mor_{\Sh(\mathcal{C})}(h_U^\#, \mathcal{F}) =
\Mor_{\textit{PSh}(\mathcal{C})}(h_U, \mathcal{F}) =
\mathcal{F}(U)
\end{equation}
for every sheaf $\mathcal{F}$. Namely, the first equality holds
by the adjointness property of $\#$ and the second is
(\ref{equation-map-representable-into-presheaf}).
\begin{lemma}
\label{lemma-covering-surjective-after-sheafification}
\begin{slogan}
Coverings become surjective after sheafification.
\end{slogan}
Let $\mathcal{C}$ be a site. If
$\{U_i \to U\}_{i \in I}$ is a covering of the site
$\mathcal{C}$, then the morphism of presheaves of sets
$$
\coprod\nolimits_{i \in I} h_{U_i} \to h_U
$$
becomes surjective after sheafification.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-mono-epi-sheaves} above we have to show that
$\coprod\nolimits_{i \in I} h_{U_i}^\# \to h_U^\#$
is an epimorphism. Let $\mathcal{F}$ be a sheaf of sets.
A morphism $h_U^\# \to \mathcal{F}$
corresponds to a section $s \in \mathcal{F}(U)$.
Hence the injectivity of $\Mor(h_U^\#, \mathcal{F})
\to \prod_i \Mor(h_{U_i}^\#, \mathcal{F})$ follows
directly from the sheaf property of $\mathcal{F}$.
\end{proof}
\noindent
The next lemma says, in the case the topology is weaker than the
canonical topology, that every sheaf is made up out of
representable sheaves in a way.
\begin{lemma}
\label{lemma-sheaf-coequalizer-representable}
Let $\mathcal{C}$ be a site. Let $E \subset \Ob(\mathcal{C})$ be a
subset such that every object of $\mathcal{C}$ has a covering by
elements of $E$. Let $\mathcal{F}$ be a sheaf of sets. There exists a
diagram of sheaves of sets
$$
\xymatrix{
\mathcal{F}_1 \ar@<1ex>[r] \ar@<-1ex>[r] &
\mathcal{F}_0 \ar[r] &
\mathcal{F}
}
$$
which represents $\mathcal{F}$ as a coequalizer,
such that $\mathcal{F}_i$, $i = 0, 1$ are coproducts
of sheaves of the form $h_U^\#$ with $U \in E$.
\end{lemma}
\begin{proof}
First we show there is an epimorphism $\mathcal{F}_0 \to \mathcal{F}$
of the desired type. Namely, just take
$$
\mathcal{F}_0 =
\coprod\nolimits_{U \in E, s \in \mathcal{F}(U)}
(h_U)^\# \longrightarrow \mathcal{F}
$$
Here the arrow restricted to the component corresponding to $(U, s)$ maps
the element $\text{id}_U \in h_U^\#(U)$ to the section $s \in \mathcal{F}(U)$.
This is an epimorphism according to Lemma \ref{lemma-mono-epi-sheaves} and
our condition on $E$. To construct $\mathcal{F}_1$ first set
$\mathcal{G} = \mathcal{F}_0 \times_\mathcal{F} \mathcal{F}_0$ and
then construct an epimorphism $\mathcal{F}_1 \to \mathcal{G}$
as above. See Lemma \ref{lemma-coequalizer-surjection}.
\end{proof}
\section{Continuous functors}
\label{section-continuous-functors}
\begin{definition}
\label{definition-continuous}
Let $\mathcal{C}$ and $\mathcal{D}$ be sites.
A functor $u : \mathcal{C} \to \mathcal{D}$ is called
{\it continuous} if for every
$\{V_i \to V\}_{i\in I} \in \text{Cov}(\mathcal{C})$
we have the following
\begin{enumerate}
\item $\{u(V_i) \to u(V)\}_{i\in I}$ is in $\text{Cov}(\mathcal{D})$, and
\item for any morphism $T \to V$ in $\mathcal{C}$ the morphism
$u(T \times_V V_i) \to u(T) \times_{u(V)} u(V_i)$ is an isomorphism.
\end{enumerate}
\end{definition}
\noindent
Recall that given a functor $u$ as above, and a presheaf of sets
$\mathcal{F}$ on $\mathcal{D}$ we have defined
$u^p\mathcal{F}$ to be simply the presheaf
$\mathcal{F} \circ u$, in other words
$$
u^p\mathcal{F} (V) = \mathcal{F}(u(V))
$$
for every object $V$ of $\mathcal{C}$.
\begin{lemma}
\label{lemma-pushforward-sheaf}
Let $\mathcal{C}$ and $\mathcal{D}$ be sites.
Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous functor.
If $\mathcal{F}$ is a sheaf on $\mathcal{D}$ then
$u^p\mathcal{F}$ is a sheaf as well.
\end{lemma}
\begin{proof}
Let $\{V_i \to V\}$ be a covering.
By assumption $\{u(V_i) \to u(V)\}$ is a covering
in $\mathcal{D}$ and $u(V_i \times_V V_j) =
u(V_i)\times_{u(V)}u(V_j)$. Hence the sheaf condition for
$u^p\mathcal{F}$ and the covering $\{V_i \to V\}$
is precisely the same as the sheaf condition for $\mathcal{F}$
and the covering $\{u(V_i) \to u(V)\}$.
\end{proof}
\noindent
In order to avoid confusion we sometimes denote
$$
u^s :
\Sh(\mathcal{D})
\longrightarrow
\Sh(\mathcal{C})
$$
the functor $u^p$ restricted to the subcategory of sheaves of sets.
\begin{lemma}
\label{lemma-adjoint-sheaves}
In the situation of Lemma \ref{lemma-pushforward-sheaf}.
The functor $u_s : \mathcal{G} \mapsto (u_p \mathcal{G})^\#$
is a left adjoint to $u^s$.
\end{lemma}
\begin{proof}
Follows directly from Lemma \ref{lemma-adjoints-u} and
Proposition \ref{proposition-sheafification-adjoint}.
\end{proof}
\noindent
Here is a technical lemma.
\begin{lemma}
\label{lemma-technical-up}
In the situation of Lemma \ref{lemma-pushforward-sheaf}.
For any presheaf $\mathcal{G}$ on $\mathcal{C}$
we have $(u_p\mathcal{G})^\# = (u_p(\mathcal{G}^\#))^\#$.
\end{lemma}
\begin{proof}
For any sheaf $\mathcal{F}$ on $\mathcal{D}$ we have
\begin{eqnarray*}
\Mor_{\Sh(\mathcal{D})}(u_s(\mathcal{G}^\#), \mathcal{F})
& = &
\Mor_{\Sh(\mathcal{C})}(\mathcal{G}^\#, u^s\mathcal{F}) \\
& = &
\Mor_{\textit{PSh}(\mathcal{C})}(\mathcal{G}^\#, u^p\mathcal{F}) \\
& = &
\Mor_{\textit{PSh}(\mathcal{C})}(\mathcal{G}, u^p\mathcal{F}) \\
& = &
\Mor_{\textit{PSh}(\mathcal{D})}(u_p\mathcal{G}, \mathcal{F}) \\
& = &
\Mor_{\Sh(\mathcal{D})}((u_p\mathcal{G})^\#, \mathcal{F})
\end{eqnarray*}
and the result follows from the Yoneda lemma.
\end{proof}
\begin{lemma}
\label{lemma-pullback-representable-sheaf}
Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous functor
between sites.
For any object $U$ of $\mathcal{C}$ we have $u_sh_U^\# = h_{u(U)}^\#$.
\end{lemma}
\begin{proof}
Follows from
Lemmas \ref{lemma-pullback-representable-presheaf}
and \ref{lemma-technical-up}.
\end{proof}
\begin{remark}
\label{remark-quasi-continuous}
(Skip on first reading.)
Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let us
use the definition of tautologically equivalent families of maps,
see Definition \ref{definition-combinatorial-tautological}
to (slightly) weaken the conditions defining continuity.
Let $u : \mathcal{C} \to \mathcal{D}$ be a functor.
Let us call $u$ {\it quasi-continuous} if for every
$\mathcal{V} = \{V_i \to V\}_{i\in I} \in \text{Cov}(\mathcal{C})$
we have the following
\begin{enumerate}
\item[(1')] the family of maps
$\{u(V_i) \to u(V)\}_{i\in I}$ is tautologically equivalent
to an element of $\text{Cov}(\mathcal{D})$, and
\item[(2)] for any morphism $T \to V$ in $\mathcal{C}$ the morphism
$u(T \times_V V_i) \to u(T) \times_{u(V)} u(V_i)$ is an isomorphism.
\end{enumerate}
We are going to see that Lemmas \ref{lemma-pushforward-sheaf}
and \ref{lemma-adjoint-sheaves} hold in case
$u$ is quasi-continuous as well.
\medskip\noindent
We first remark that the morphisms $u(V_i) \to u(V)$ are representable, since
they are isomorphic to representable morphisms (by the first condition).
In particular, the family $u(\mathcal{V}) = \{u(V_i) \to u(V)\}_{i\in I}$
gives rise to a zeroth {\v C}ech cohomology group
$H^0(u(\mathcal{V}), \mathcal{F})$ for any presheaf $\mathcal{F}$ on
$\mathcal{D}$.
Let $\mathcal{U} = \{U_j \to u(V)\}_{j \in J}$ be an element
of $\text{Cov}(\mathcal{D})$ tautologically
equivalent to $\{u(V_i) \to u(V)\}_{i \in I}$. Note that $u(\mathcal{V})$
is a refinement of $\mathcal{U}$ and vice versa. Hence by Remark
\ref{remark-both-refine-same-H0} we see that
$H^0(u(\mathcal{V}), \mathcal{F}) = H^0(\mathcal{U}, \mathcal{F})$.
In particular, if $\mathcal{F}$ is a sheaf, then
$\mathcal{F}(u(V)) = H^0(u(\mathcal{V}), \mathcal{F})$ because
of the sheaf property expressed in terms of zeroth {\v C}ech cohomology
groups. We conclude that $u^p\mathcal{F}$ is a sheaf if $\mathcal{F}$
is a sheaf, since $H^0(\mathcal{V}, u^p\mathcal{F}) =
H^0(u(\mathcal{V}), \mathcal{F})$ which we just observed is
equal to $\mathcal{F}(u(V)) = u^p\mathcal{F}(V)$. Thus Lemma
\ref{lemma-pushforward-sheaf} holds. Lemma \ref{lemma-adjoint-sheaves}
follows immediately.
\end{remark}
\section{Morphisms of sites}
\label{section-morphism-sites}
\begin{definition}
\label{definition-morphism-sites}
Let $\mathcal{C}$ and $\mathcal{D}$ be sites.
A {\it morphism of sites} $f : \mathcal{D} \to \mathcal{C}$
is given by a continuous functor $u : \mathcal{C} \to \mathcal{D}$
such that the functor $u_s$ is exact.
\end{definition}
\noindent
Notice how the functor $u$ goes in the direction {\it opposite}
the morphism $f$. If $f \leftrightarrow u$ is a morphism of sites
then we use the notation $f^{-1} = u_s$ and $f_* = u^s$.
The functor $f^{-1}$ is called the {\it pullback functor} and
the functor $f_*$ is called the {\it pushforward functor}.
As in topology we have the following adjointness property
$$
\Mor_{\Sh(\mathcal{D})}(f^{-1}\mathcal{G}, \mathcal{F})
=
\Mor_{\Sh(\mathcal{C})}(\mathcal{G}, f_*\mathcal{F})
$$
The motivation for this definition comes from the following
example.
\begin{example}
\label{example-continuous-map}
Let $f : X \to Y$ be a continuous map of topological spaces.
Recall that we have sites $X_{Zar}$ and $Y_{Zar}$,
see Example \ref{example-site-topological}. Consider the functor
$u : Y_{Zar} \to X_{Zar}$, $V \mapsto f^{-1}(V)$.
This functor is clearly continuous because inverse images of
open coverings are open coverings. (Actually, this depends on how
you chose sets of coverings for $X_{Zar}$ and $Y_{Zar}$.
But in any case the functor is quasi-continuous, see Remark
\ref{remark-quasi-continuous}.)
It is easy to verify that
the functor $u^s$ equals the usual pushforward functor $f_*$
from topology. Hence, since $u_s$ is an adjoint and since
the usual topological pullback functor $f^{-1}$ is an adjoint as well,
we get a canonical isomorphism $f^{-1} \cong u_s$. Since $f^{-1}$
is exact we deduce that $u_s$ is exact. Hence $u$ defines a morphism
of sites $f : X_{Zar} \to Y_{Zar}$, which we may denote
$f$ as well since we've already seen the functors $u_s, u^s$ agree
with their usual notions anyway.
\end{example}
\begin{lemma}
\label{lemma-composition-morphisms-sites}
Let $\mathcal{C}_i$, $i = 1, 2, 3$ be sites. Let
$u : \mathcal{C}_2 \to \mathcal{C}_1$ and
$v : \mathcal{C}_3 \to \mathcal{C}_2$ be continuous functors
which induce morphisms of sites. Then the functor
$u \circ v : \mathcal{C}_3 \to \mathcal{C}_1$ is continuous and
defines a morphism of sites $\mathcal{C}_1 \to \mathcal{C}_3$.
\end{lemma}
\begin{proof}
It is immediate from the definitions that $u \circ v$ is a continuous functor.
In addition, we clearly have $(u \circ v)^p = v^p \circ u^p$, and hence
$(u \circ v)^s = v^s \circ u^s$. Hence functors $(u \circ v)_s$ and
$u_s \circ v_s$ are both left adjoints of $(u \circ v)^s$. Therefore
$(u \circ v)_s \cong u_s \circ v_s$ and we conclude that $(u \circ v)_s$
is exact as a composition of exact functors.
\end{proof}
\begin{definition}
\label{definition-composition-morphisms-sites}
Let $\mathcal{C}_i$, $i = 1, 2, 3$ be sites. Let
$f : \mathcal{C}_1 \to \mathcal{C}_2$ and
$g : \mathcal{C}_2 \to \mathcal{C}_3$ be morphisms of sites
given by continuous functors $u : \mathcal{C}_2 \to \mathcal{C}_1$
and $v : \mathcal{C}_3 \to \mathcal{C}_2$. The {\it composition}
$g \circ f$ is the morphism of sites corresponding to the
functor $u \circ v$.
\end{definition}
\noindent
In this situation we have $(g \circ f)_* = g_* \circ f_*$ and
$(g \circ f)^{-1} = f^{-1} \circ g^{-1}$ (see proof of
Lemma \ref{lemma-composition-morphisms-sites}).
\begin{lemma}
\label{lemma-directed-morphism}
Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let
$u : \mathcal{C} \to \mathcal{D}$ be continuous.
Assume all the categories $(\mathcal{I}_V^u)^{opp}$ of
Section \ref{section-functoriality-PSh}
are filtered. Then $u$ defines a morphism of sites $\mathcal{D} \to
\mathcal{C}$, in other words $u_s$ is exact.
\end{lemma}
\begin{proof}
Since $u_s$ is the left adjoint of $u^s$ we see that $u_s$ is right
exact, see Categories, Lemma \ref{categories-lemma-exact-adjoint}.
Hence it suffices to show that $u_s$ is left exact. In other words
we have to show that $u_s$ commutes with finite limits.
Because the categories $\mathcal{I}_Y^{opp}$ are filtered
we see that $u_p$ commutes with finite limits, see
Categories, Lemma \ref{categories-lemma-directed-commutes}
(this also uses the description of limits in $\textit{PSh}$,
see Section \ref{section-limits-colimits-PSh}).
And since sheafification commutes with finite limits as well
(Lemma \ref{lemma-sheafification-exact}) we conclude because
$u_s = \# \circ u_p$.
\end{proof}
\begin{proposition}
\label{proposition-get-morphism}
Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let
$u : \mathcal{C} \to \mathcal{D}$ be continuous.
Assume furthermore the following:
\begin{enumerate}
\item the category $\mathcal{C}$ has a final object $X$ and
$u(X)$ is a final object of $\mathcal{D}$ , and
\item the category $\mathcal{C}$ has fibre products and
$u$ commutes with them.
\end{enumerate}
Then $u$ defines a morphism of sites $\mathcal{D} \to
\mathcal{C}$, in other words $u_s$ is exact.
\end{proposition}
\begin{proof}
This follows from Lemmas \ref{lemma-directed} and
\ref{lemma-directed-morphism}.
\end{proof}
\begin{remark}
\label{remark-explain-left-exact}
The conditions of Proposition \ref{proposition-get-morphism} above
are equivalent to saying that $u$ is left exact, i.e., commutes
with finite limits. See
Categories, Lemmas
\ref{categories-lemma-finite-limits-exist} and
\ref{categories-lemma-characterize-left-exact}.
It seems more natural to phrase it in terms of final objects
and fibre products since this seems to have more geometric meaning
in the examples.
\end{remark}
\noindent
Lemma \ref{lemma-continuous-with-continuous-left-adjoint} will
provide another way to prove a continuous functor
gives rise to a morphism of sites.
\begin{remark}
\label{remark-quasi-continuous-morphism-sites}
(Skip on first reading.)
Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Analogously to
Definition \ref{definition-morphism-sites} we say that
a {\it quasi-morphism of sites $f : \mathcal{D} \to \mathcal{C}$}
is given by a quasi-continuous functor $u : \mathcal{C} \to \mathcal{D}$
(see Remark \ref{remark-quasi-continuous}) such that $u_s$ is exact.
The analogue of Proposition \ref{proposition-get-morphism} in this
setting is obtained by replacing the word ``continuous''
by the word ``quasi-continuous'', and replacing the word
``morphism'' by ``quasi-morphism''. The proof is literally the
same.
\end{remark}
\noindent
In Definition \ref{definition-morphism-sites} the condition that $u_s$
be exact cannot be omitted. For example, the conclusion of the following
lemma need not hold if one only assumes that $u$ is continuous.
\begin{lemma}
\label{lemma-morphism-of-sites-covering}
Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites given by the
functor $u : \mathcal{C} \to \mathcal{D}$. Given any object $V$ of
$\mathcal{D}$ there exists a covering $\{V_j \to V\}$ such that for every
$j$ there exists a morphism $V_j \to u(U_j)$ for some object $U_j$
of $\mathcal{C}$.
\end{lemma}
\begin{proof}
Since $f^{-1} = u_s$ is exact we have $f^{-1}* = *$ where $*$ denotes the
final object of the category of sheaves
(Example \ref{example-singleton-sheaf}).
Since $f^{-1}* = u_s*$ is the sheafification of $u_p*$ we see
there exists a covering $\{V_j \to V\}$ such that $(u_p*)(V_j)$
is nonempty. Since $(u_p*)(V_j)$ is a colimit over the category
$\mathcal{I}^u_{V_j}$ whose objects are morphisms $V_j \to u(U)$
the lemma follows.
\end{proof}
\section{Topoi}
\label{section-topoi}
\noindent
Here is a definition of a topos which is suitable for our purposes.
Namely, a topos is the category of sheaves on a site. In order to specify
a topos you just specify the site. The real difference between a topos
and a site lies in the definition of morphisms. Namely, it turns out that
there are lots of morphisms of topoi which do not come from morphisms
of the underlying sites.
\begin{definition}[Topoi]
\label{definition-topos}
A {\it topos} is the category $\Sh(\mathcal{C})$ of sheaves
on a site $\mathcal{C}$.
\begin{enumerate}
\item Let $\mathcal{C}$, $\mathcal{D}$ be sites.
A {\it morphism of topoi} $f$ from $\Sh(\mathcal{D})$
to $\Sh(\mathcal{C})$ is given by a pair of functors
$f_* : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$
and
$f^{-1} : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$
such that
\begin{enumerate}
\item we have
$$
\Mor_{\Sh(\mathcal{D})}(f^{-1}\mathcal{G}, \mathcal{F})
=
\Mor_{\Sh(\mathcal{C})}(\mathcal{G}, f_*\mathcal{F})
$$
bifunctorially, and
\item the functor $f^{-1}$ commutes with finite limits, i.e.,
is left exact.
\end{enumerate}
\item Let $\mathcal{C}$, $\mathcal{D}$, $\mathcal{E}$ be sites.
Given morphisms of topoi
$f :\Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ and
$g :\Sh(\mathcal{E}) \to \Sh(\mathcal{D})$ the
{\it composition $f\circ g$} is the morphism of topoi defined
by the functors
$(f \circ g)_* = f_* \circ g_*$ and
$(f \circ g)^{-1} = g^{-1} \circ f^{-1}$.
\end{enumerate}
\end{definition}
\noindent
Suppose that $\alpha : \mathcal{S}_1 \to \mathcal{S}_2$ is an equivalence of
(possibly ``big'') categories. If $\mathcal{S}_1$, $\mathcal{S}_2$
are topoi, then setting $f_* = \alpha$ and $f^{-1}$ equal to a quasi-inverse
of $\alpha$ gives a morphism $f : \mathcal{S}_1 \to \mathcal{S}_2$ of topoi.
Moreover this morphism is an equivalence in the $2$-category of topoi (see
Section \ref{section-2-category}).
Thus it makes sense to say ``$\mathcal{S}$ is a topos'' if $\mathcal{S}$
is equivalent to the category of sheaves on a site (and not necessarily
equal to the category of sheaves on a site). We will occasionally
use this abuse of notation.
\medskip\noindent
Two examples of topoi. The {\it empty topos} is topos
of sheaves on the site $\mathcal{C}$, where $\mathcal{C}$ has
a single object $\emptyset$ and a single morphism $\text{id}_\emptyset$
and a single covering, namely the empty covering of $\emptyset$.
We will sometimes write $\emptyset$ for this site.
This is a site and every sheaf on $\mathcal{C}$ assigns a singleton to
$\emptyset$. Thus $\Sh(\emptyset)$ is equivalent to the category
having a single object and a single morphism.
The {\it punctual topos} is the topos of sheaves on the site
$\mathcal{C}$ which has a single object $pt$ and one morphism
$\text{id}_{pt}$ and whose only covering is the covering
$\{\text{id}_{pt}\}$. We will simply write $pt$ for this site.
It is clear that the category of sheaves $ = $ the category of
presheaves $ = $ the category of sets.
In a formula $\Sh(pt) = \textit{Sets}$.
\medskip\noindent
Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let
$f : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ be a morphism of topoi.
Note that $f_*$ commutes with all limits and that
$f^{-1}$ commutes with all colimits, see Categories,
Lemma \ref{categories-lemma-adjoint-exact}.
In particular, the condition on $f^{-1}$ in the definition above
guarantees that $f^{-1}$ is exact. Morphisms of topoi are often constructed
using either Lemma \ref{lemma-cocontinuous-morphism-topoi}
or the following lemma.
\begin{lemma}
\label{lemma-morphism-sites-topoi}
Given a morphism of sites $f : \mathcal{D} \to \mathcal{C}$
corresponding to the functor $u : \mathcal{C} \to \mathcal{D}$
the pair of functors $(f^{-1} = u_s, f_* = u^s)$ is a morphism of topoi.
\end{lemma}
\begin{proof}
This is obvious from Definition \ref{definition-morphism-sites}.
\end{proof}
\begin{remark}
\label{remark-pt-topos}
There are many sites that give rise to the topos $\Sh(pt)$.
A useful example is the following. Suppose that $S$ is a set (of sets)
which contains at least one nonempty element. Let $\mathcal{S}$ be the
category whose objects are elements of $S$ and whose morphisms are
arbitrary set maps. Assume that $\mathcal{S}$ has fibre products.
For example this will be the case if $S = \mathcal{P}(\text{infinite set})$
is the power set of any infinite set (exercise in set theory).
Make $\mathcal{S}$ into a site by declaring
surjective families of maps to be coverings (and choose
a suitable sufficiently large set of covering families as in
Sets, Section \ref{sets-section-coverings-site}).
We claim that $\Sh(\mathcal{S})$ is equivalent to the category of
sets.
\medskip\noindent
We first prove this in case $S$ contains $e \in S$ which is a singleton.
In this case, there is an equivalence of topoi
$i : \Sh(pt) \to \Sh(\mathcal{S})$ given by
the functors
\begin{equation}
\label{equation-sheaves-pt-sets}
i^{-1}\mathcal{F} = \mathcal{F}(e), \quad
i_*E = (U \mapsto \Mor_{\textit{Sets}}(U, E))
\end{equation}
Namely, suppose that $\mathcal{F}$ is a sheaf on $\mathcal{S}$.
For any $U \in \Ob(\mathcal{S}) = S$ we can find a covering
$\{\varphi_u : e \to U\}_{u \in U}$, where $\varphi_u$
maps the unique element of $e$ to $u \in U$. The sheaf condition
implies in this case that
$\mathcal{F}(U) = \prod_{u \in U} \mathcal{F}(e)$.
In other words
$\mathcal{F}(U) = \Mor_{\textit{Sets}}(U, \mathcal{F}(e))$.
Moreover, this rule is compatible with restriction mappings. Hence
the functor
$$
i_* :
\textit{Sets} = \Sh(pt)
\longrightarrow
\Sh(\mathcal{S}), \quad
E \longmapsto (U \mapsto \Mor_{\textit{Sets}}(U, E))
$$
is an equivalence of categories, and its inverse is the functor
$i^{-1}$ given above.
\medskip\noindent
If $\mathcal{S}$ does not contain a singleton, then the functor
$i_*$ as defined above still makes sense. To show that it is still
an equivalence in this case, choose any nonempty $\tilde e \in S$
and a map $\varphi : \tilde e \to \tilde e$ whose image is a singleton.
For any sheaf $\mathcal{F}$ set
$$
\mathcal{F}(e) :=
\Im(
\mathcal{F}(\varphi) :
\mathcal{F}(\tilde e)
\longrightarrow
\mathcal{F}(\tilde e)
)
$$
and show that this is a quasi-inverse to $i_*$. Details omitted.
\end{remark}
\begin{remark}
\label{remark-morphism-topoi-big}
(Set theoretical issues related to morphisms of topoi. Skip
on a first reading.)
A morphism of topoi as defined above is not a set but a class.
In other words it is given by a mathematical formula rather
than a mathematical object. Although we may contemplate
the collection of all morphisms between two given topoi,
it is not a good idea to introduce it as a mathematical object.
On the other hand, suppose $\mathcal{C}$ and $\mathcal{D}$ are
given sites. Consider a functor
$\Phi : \mathcal{C} \to \Sh(\mathcal{D})$.
Such a thing is a set, in other words, it is a mathematical object.
We may, in succession, ask the following questions on $\Phi$.
\begin{enumerate}
\item Is it true, given a sheaf $\mathcal{F}$ on $\mathcal{D}$,
that the rule
$U \mapsto \Mor_{\Sh(\mathcal{D})}(\Phi(U), \mathcal{F})$
defines a sheaf on $\mathcal{C}$? If so, this defines a functor
$\Phi_* : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$.
\item Is it true that $\Phi_*$ has a left adjoint? If so,
write $\Phi^{-1}$ for this left adjoint.
\item Is it true that $\Phi^{-1}$ is exact?
\end{enumerate}
If the last question still has the answer ``yes'', then we obtain
a morphism of topoi $(\Phi_*, \Phi^{-1})$. Moreover, given any
morphism of topoi $(f_*, f^{-1})$ we may set
$\Phi(U) = f^{-1}(h_U^\#)$ and obtain a functor $\Phi$ as above
with $f_* \cong \Phi_*$ and $f^{-1} \cong \Phi^{-1}$ (compatible
with adjoint property).
The upshot is that by working with the collection of $\Phi$
instead of morphisms of topoi, we (a) replaced the notion of
a morphism of topoi by a mathematical object, and (b)
the collection of $\Phi$ forms a class (and not a collection
of classes). Of course, more can be said, for example one can work
out more precisely the significance of conditions (2) and (3) above;
we do this in the case of points of topoi in Section \ref{section-points}.
\end{remark}
\begin{remark}
\label{remark-quasi-continuous-morphism-topoi}
(Skip on first reading.)
Let $\mathcal{C}$ and $\mathcal{D}$ be sites.
A quasi-morphism of sites $f : \mathcal{D} \to \mathcal{C}$
(see Remark \ref{remark-quasi-continuous-morphism-sites})
gives rise to a morphism of topoi $f$ from
$\Sh(\mathcal{D})$ to $\Sh(\mathcal{C})$
exactly as in Lemma \ref{lemma-morphism-sites-topoi}.
\end{remark}
\section{G-sets and morphisms}
\label{section-G-sets-morphisms}
\noindent
Let $\varphi : G \to H$ be a homomorphism of groups.
Choose (suitable) sites $\mathcal{T}_G$ and $\mathcal{T}_H$ as in
Example \ref{example-site-on-group} and
Section \ref{section-example-sheaf-G-sets}.
Let $u : \mathcal{T}_H \to \mathcal{T}_G$ be the functor which assigns
to a $H$-set $U$ the $G$-set $U_\varphi$ which has the same underlying
set but $G$ action defined by $g \cdot u = \varphi(g)u$.
It is clear that $u$ commutes with finite limits and is
continuous\footnote{Set theoretical remark: First choose $\mathcal{T}_H$.
Then choose $\mathcal{T}_G$ to contain $u(\mathcal{T}_H)$ and such that
every covering in $\mathcal{T}_H$ corresponds to a covering in
$\mathcal{T}_G$. This is possible by
Sets, Lemmas
\ref{sets-lemma-sets-with-group-action},
\ref{sets-lemma-what-is-in-it-G-sets} and
\ref{sets-lemma-coverings-site}.}.
Applying
Proposition \ref{proposition-get-morphism}
and
Lemma \ref{lemma-morphism-sites-topoi}
we obtain a morphism of topoi
$$
f : \Sh(\mathcal{T}_G) \longrightarrow \Sh(\mathcal{T}_H)
$$
associated with $\varphi$. Using
Proposition \ref{proposition-sheaves-on-group}
we see that we get a pair of adjoint functors
$$
f_* : G\textit{-Sets} \longrightarrow H\textit{-Sets}, \quad
f^{-1} : H\textit{-Sets} \longrightarrow G\textit{-Sets}.
$$
Let's work out what are these functors in this case.
\medskip\noindent
We first work out a formula for $f_*$.
Recall that given a $G$-set $S$ the corresponding sheaf
$\mathcal{F}_S$ on $\mathcal{T}_G$ is given by the rule
$\mathcal{F}_S(U) = \Mor_G(U, S)$. And on the other hand, given
a sheaf $\mathcal{G}$ on $\mathcal{T}_H$ the corresponding $H$-set
is given by the rule $\mathcal{G}({}_HH)$. Hence we see that
$$
f_*S = \Mor_{G\textit{-Sets}}(({}_HH)_\varphi, S).
$$
If we work this out a little bit more then we get
$$
f_*S = \{ a : H \to S \mid a(gh) = ga(h) \}
$$
with left $H$-action given by
$(h \cdot a)(h') = a(h'h)$
for any element $a \in f_*S$.
\medskip\noindent
Next, we explicitly compute $f^{-1}$. Note that since the topology
on $\mathcal{T}_G$ and $\mathcal{T}_H$ is subcanonical, all representable
presheaves are sheaves. Moreover, given an object $V$ of $\mathcal{T}_H$
we see that $f^{-1}h_V$ is equal to $h_{u(V)}$ (see
Lemma \ref{lemma-pullback-representable-sheaf}). Hence we see that
$f^{-1}S = S_\varphi$ for representable sheaves. Since every sheaf on
$\mathcal{T}_H$ is a coproduct of representable sheaves we conclude that
this is true in general. Hence we see that
for any $H$-set $T$ we have
$$
f^{-1}T = T_\varphi.
$$
The adjunction between $f^{-1}$ and $f_*$ is evidenced by the formula
$$
\Mor_{G\textit{-Sets}}(T_\varphi, S) =
\Mor_{H\textit{-Sets}}(T, f_*S)
$$
with $f_*S$ as above. This can be proved directly. Moreover, it is then clear
that $(f^{-1}, f_*)$ form an adjoint pair and that $f^{-1}$ is exact.
So alternatively to the above the morphism of topoi
$f : G\textit{-Sets} \to H\textit{-Sets}$ can be defined directly in this
manner.
\section{Quasi-compact objects and colimits}
\label{section-quasi-compact}
\noindent
To be able to use the same language as in the case of topological spaces
we introduce the following terminology.
\begin{definition}
\label{definition-quasi-compact}
Let $\mathcal{C}$ be a site. An object $U$ of $\mathcal{C}$ is
{\it quasi-compact} if given a covering $\mathcal{U} = \{U_i \to U\}_{i \in I}$
in $\mathcal{C}$ there exists another covering
$\mathcal{V} = \{V_j \to U\}_{j \in J}$ and a morphism
$\mathcal{V} \to \mathcal{U}$ of families of maps with fixed target
given by $\alpha : J \to I$ and $V_j \to U_{\alpha(j)}$
(see Definition \ref{definition-morphism-coverings})
such that the image of $\alpha$ is a finite subset of $I$.
\end{definition}
\noindent
Of course the usual notion is sufficient to conclude that
$U$ is quasi-compact.
\begin{lemma}
\label{lemma-conclude-quasi-compact}
Let $\mathcal{C}$ be a site. Let $U$ be an object of $\mathcal{C}$.
Consider the following conditions
\begin{enumerate}
\item $U$ is quasi-compact,
\item for every covering $\{U_i \to U\}_{i \in I}$ in $\mathcal{C}$
there exists a finite covering $\{V_j \to U\}_{j = 1, \ldots, m}$
of $\mathcal{C}$ refining $\mathcal{U}$, and
\item for every covering $\{U_i \to U\}_{i \in I}$ in $\mathcal{C}$
there exists a finite subset $I' \subset I$ such that
$\{U_i \to U\}_{i \in I'}$ is a covering in $\mathcal{C}$.
\end{enumerate}
Then we always have (3) $\Rightarrow$ (2) $\Rightarrow$ (1)
but the reverse implications do not hold in general.
\end{lemma}
\begin{proof}
The implications are immediate from the definitions.
Let $X = [0, 1] \subset \mathbf{R}$
as a topological space (with the usual $\epsilon$-$\delta$ topology).
Let $\mathcal{C}$ be the category of open subspaces of $X$ with
inclusions as morphisms and usual open coverings (compare with
Example \ref{example-site-topological}). However, then we change the notion
of covering in $\mathcal{C}$ to exclude all finite coverings, except
for the coverings of the form $\{U \to U\}$. It is easy to see that this
will be a site as in Definition \ref{definition-site}.
In this site the object $X = U = [0, 1]$ is quasi-compact in the sense of
Definition \ref{definition-quasi-compact} but $U$ does not satisfy (2).
We leave it to the reader to make an example where (2) holds but not (3).
\end{proof}
\noindent
Here is the topos theoretic meaning of a quasi-compact object.
\begin{lemma}
\label{lemma-quasi-compact}
Let $\mathcal{C}$ be a site. Let $U$ be an object of $\mathcal{C}$.
The following are equivalent
\begin{enumerate}
\item $U$ is quasi-compact, and
\item for every surjection of sheaves
$\coprod_{i \in I} \mathcal{F}_i \to h_U^\#$
there is a finite subset $J \subset I$ such that
$\coprod_{i \in J} \mathcal{F}_i \to h_U^\#$ is surjective.
\end{enumerate}
\end{lemma}
\begin{proof}
Assume (1) and let $\coprod_{i \in I} \mathcal{F}_i \to h_U^\#$
be a surjection. Then $\text{id}_U$ is a section of
$h_U^\#$ over $U$. Hence there exists a covering
$\{U_a \to U\}_{a \in A}$ and for each $a \in A$
a section $s_a$ of $\coprod_{i \in I} \mathcal{F}_i$
over $U_a$ mapping to $\text{id}_U$. By the construction of coproducts as
sheafification of coproducts of presheaves
(Lemma \ref{lemma-colimit-sheaves}), for each $a$
there exists a covering $\{U_{ab} \to U_a\}_{b \in B_a}$ and
for all $b \in B_a$ an $\iota(b) \in I$ and a section
$s_{b}$ of $\mathcal{F}_{\iota(b)}$ over $U_{ab}$
mapping to $\text{id}_U|_{U_{ab}}$. Thus after replacing
the covering $\{U_a \to U\}_{a \in A}$ by
$\{U_{ab} \to U\}_{a \in A, b \in B_a}$
we may assume we have a map $\iota : A \to I$
and for each $a \in A$ a section $s_a$ of $\mathcal{F}_{\iota(a)}$
over $U_a$ mapping to $\text{id}_U$.
Since $U$ is quasi-compact, there is a covering
$\{V_c \to U\}_{c \in C}$, a map $\alpha : C \to A$
with finite image, and $V_c \to U_{\alpha(c)}$ over $U$.
Then we see that $J = \Im(\iota \circ \alpha) \subset I$ works
because $\coprod_{c \in C} h_{V_c}^\# \to h_U^\#$ is surjective
(Lemma \ref{lemma-covering-surjective-after-sheafification})
and factors through $\coprod_{i \in J} \mathcal{F}_i \to h_U^\#$.
(Here we use that the composition
$h_{V_c}^\# \to h_{U_{\alpha(c)}}
\xrightarrow{s_{\alpha(c)}} \mathcal{F}_{\iota(\alpha(c))} \to h_U^\#$
is the map $h_{V_c}^\# \to h_U^\#$ coming from the morphism
$V_c \to U$ because $s_{\alpha(c)}$ maps to $\text{id}_U|_{U_{\alpha(c)}}$.)
\medskip\noindent
Assume (2). Let $\{U_i \to U\}_{i \in I}$ be a covering.
By Lemma \ref{lemma-covering-surjective-after-sheafification}
we see that $\coprod_{i \in I} h_{U_i}^\# \to h_U^\#$ is surjective.
Thus we find a finite subset $J \subset I$ such that
$\coprod_{j \in J} h_{U_j}^\# \to h_U^\#$ is surjective.
Then arguing as above we find a covering
$\{V_c \to U\}_{c \in C}$ of $U$ in $\mathcal{C}$
and a map $\iota : C \to J$
such that $\text{id}_U$ lifts to a section
of $s_c$ of $h_{U_{\iota(c)}}^\#$ over $V_c$.
Refining the covering even further we may assume
$s_c \in h_{U_{\iota(c)}}(V_c)$ mapping to $\text{id}_U$.
Then $s_c : V_c \to U_{\iota(c)}$ is a morphism over $U$
and we conclude.
\end{proof}
\noindent
The lemma above motivates the following definition.
\begin{definition}
\label{definition-quasi-compact-topos}
An object $\mathcal{F}$ of a topos $\Sh(\mathcal{C})$ is {\it quasi-compact}
if for any surjective map $\coprod_{i \in I} \mathcal{F}_i \to \mathcal{F}$
of $\Sh(\mathcal{C})$ there exists a finite subset $J \subset I$ such
that $\coprod_{i \in J} \mathcal{F}_i \to \mathcal{F}$ is surjective.
A topos $\Sh(\mathcal{C})$ is said to be {\it quasi-compact}
if its final object $*$ is a quasi-compact object.
\end{definition}
\noindent
By Lemma \ref{lemma-quasi-compact}
if the site $\mathcal{C}$ has a final object $X$, then
$\Sh(\mathcal{C})$ is quasi-compact if and only if $X$ is quasi-compact.
The following lemma is the analogue of
Sheaves, Lemma \ref{sheaves-lemma-directed-colimits-sections}
for sites.
\begin{lemma}
\label{lemma-directed-colimits-sections}
Let $\mathcal{C}$ be a site. Let
$\mathcal{I} \to \Sh(\mathcal{C})$, $i \mapsto \mathcal{F}_i$
be a filtered diagram of sheaves of sets.
Let $U \in \Ob(\mathcal{C})$.
Consider the canonical map
$$
\Psi :
\colim_i \mathcal{F}_i(U)
\longrightarrow
\left(\colim_i \mathcal{F}_i\right)(U)
$$
With the terminology introduced above:
\begin{enumerate}
\item If all the transition maps are injective then
$\Psi$ is injective for any $U$.
\item If $U$ is quasi-compact, then $\Psi$ is injective.
\item If $U$ is quasi-compact and all the transition maps are injective
then $\Psi$ is an isomorphism.
\item If $U$ has a cofinal system of coverings
$\{U_j \to U\}_{j \in J}$ with
$J$ finite and $U_j \times_U U_{j'}$ quasi-compact
for all $j, j' \in J$, then $\Psi$ is bijective.
\end{enumerate}
\end{lemma}
\begin{proof}
Assume all the transition maps are injective. In this case the presheaf
$\mathcal{F}' : V \mapsto \colim_i \mathcal{F}_i(V)$ is
separated (see Definition \ref{definition-separated}).
By Lemma \ref{lemma-colimit-sheaves}
we have
$(\mathcal{F}')^\# = \colim_i \mathcal{F}_i$.
By Theorem \ref{theorem-plus}
we see that $\mathcal{F}' \to (\mathcal{F}')^\#$ is injective.
This proves (1).
\medskip\noindent
Assume $U$ is quasi-compact. Suppose that $s \in \mathcal{F}_i(U)$ and
$s' \in \mathcal{F}_{i'}(U)$ give rise to elements on
the left hand side which have the same image under $\Psi$.
This means we can choose a covering $\{U_a \to U\}_{a \in A}$
and for each $a \in A$ an index $i_a \in I$, $i_a \geq i$, $i_a \geq i'$
such that $\varphi_{ii_a}(s) = \varphi_{i'i_a}(s')$.
Because $U$ is quasi-compact we can choose a covering
$\{V_b \to U\}_{b \in B}$, a map $\alpha : B \to A$ with finite image,
and morphisms $V_b \to U_{\alpha(b)}$ over $U$.
Pick $i''\in I$ to be $\geq$ than all of the $i_{\alpha(b)}$
which is possible because the image of $\alpha$ is finite.
We conclude that $\varphi_{ii''}(s)$ and $\varphi_{i'i''}(s)$
agree on $V_b$ for all $b \in B$ and hence that
$\varphi_{ii''}(s) = \varphi_{i'i''}(s)$. This proves (2).
\medskip\noindent
Assume $U$ is quasi-compact and all transition maps injective.
Let $s$ be an element of the target of $\Psi$. There exists a covering
$\{U_a \to U\}_{a \in A}$ and for each $a \in A$ an index $i_a \in I$
and a section $s_a \in \mathcal{F}_{i_a}(U_a)$
such that $s|_{U_a}$ comes from $s_a$ for all $a \in A$.
Because $U$ is quasi-compact we can choose a covering
$\{V_b \to U\}_{b \in B}$, a map $\alpha : B \to A$ with finite image,
and morphisms $V_b \to U_{\alpha(b)}$ over $U$.
Pick $i \in I$ to be $\geq$ than all of the $i_{\alpha(b)}$
which is possible because the image of $\alpha$ is finite.
By (1) the sections
$s_b = \varphi_{i_{\alpha(b)} i}(s_{\alpha(b)})|_{V_b}$
agree over $V_b \times_U V_{b'}$.
Hence they glue to a section
$s' \in \mathcal{F}_i(U)$ which maps to $s$ under $\Psi$.
This proves (3).
\medskip\noindent
Assume the hypothesis of (4).
Let $s$ be an element of the target of $\Psi$.
By assumption there exists a finite covering
$\{U_j \to U\}_{j = 1, \ldots, m} U_j$, with $U_j \times_U U_{j'}$
quasi-compact for all $j, j' \in J$ and
for each $j$ an index $i_j \in I$ and $s_j \in \mathcal{F}_{i_j}(U_j)$
such that $s|_{U_j}$ is the image of $s_j$ for all $j$.
Since $U_j \times_U U_{j'}$ is quasi-compact we can apply (2)
and we see that there exists an $i_{jj'} \in I$,
$i_{jj'} \geq i_j$, $i_{jj'} \geq i_{j'}$ such that
$\varphi_{i_ji_{jj'}}(s_j)$ and $\varphi_{i_{j'}i_{jj'}}(s_{j'})$
agree over $U_j \times_U U_{j'}$. Choose an index $i \in I$
wich is bigger or equal than all the $i_{jj'}$. Then we see that
the sections $\varphi_{i_ji}(s_j)$ of $\mathcal{F}_i$ glue
to a section of $\mathcal{F}_i$ over $U$. This section is mapped
to the element $s$ as desired.
\end{proof}
\noindent
We need an analogue of the above result in the case that the site
is the limit of an inverse system of sites. For simplicity we only
explain the construction in case the index sets of coverings are finite.
\begin{situation}
\label{situation-inverse-limit-sites}
Here we are given
\begin{enumerate}
\item a cofiltered index category $\mathcal{I}$,
\item for $i \in \Ob(\mathcal{I})$ a site $\mathcal{C}_i$ such that every
covering in $\mathcal{C}_i$ has a finite index set,
\item for a morphism $a : i \to j$ in $\mathcal{I}$ a morphism of sites
$f_a : \mathcal{C}_i \to \mathcal{C}_j$ given by a continuous functor
$u_a : \mathcal{C}_j \to \mathcal{C}_i$,
\end{enumerate}
such that $f_a \circ f_b = f_c$ whenever $c = a \circ b$ in $\mathcal{I}$.
\end{situation}
\begin{lemma}
\label{lemma-colimit-sites}
In Situation \ref{situation-inverse-limit-sites} we can construct
a site $(\mathcal{C}, \text{Cov}(\mathcal{C}))$ as follows
\begin{enumerate}
\item as a category $\mathcal{C} = \colim \mathcal{C}_i$, and
\item $\text{Cov}(\mathcal{C})$ is the union of the images
of $\text{Cov}(\mathcal{C}_i)$ by $u_i : \mathcal{C}_i \to \mathcal{C}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Our definition of composition of morphisms of sites implies that
$u_b \circ u_a = u_c$ whenever $c = a \circ b$ in $\mathcal{I}$.
The formula $\mathcal{C} = \colim \mathcal{C}_i$ means that
$\Ob(\mathcal{C}) = \colim \Ob(\mathcal{C}_i)$ and
$\text{Arrows}(\mathcal{C}) = \colim \text{Arrows}(\mathcal{C}_i)$.
Then source, target, and composition are inherited from the
source, target, and composition on $\text{Arrows}(\mathcal{C}_i)$.
In this way we obtain a category.
Denote $u_i : \mathcal{C}_i \to \mathcal{C}$ the obvious functor.
Remark that given any finite diagram in $\mathcal{C}$
there exists an $i$ such that this diagram is
the image of a diagram in $\mathcal{C}_i$.
\medskip\noindent
Let $\{U^t \to U\}$ be a covering of $\mathcal{C}$. We first prove that if
$V \to U$ is a morphism of $\mathcal{C}$, then $U^t \times_U V$ exists.
By our remark above and our definition of coverings, we can find an
$i$, a covering $\{U_i^t \to U_i\}$ of $\mathcal{C}_i$ and a morphism
$V_i \to U_i$ whose image by $u_i$ is the given data. We claim that
$U^t \times_U V$ is the image of $U^t_i \times_{U_i} V_i$ by $u_i$.
Namely, for every $a : j \to i$ in $\mathcal{I}$ the functor $u_a$
is continuous, hence
$u_a(U^t_i \times_{U_i} V_i) = u_a(U^t_i) \times_{u_a(U_i)} u_a(V_i)$.
In particular we can replace $i$ by $j$, if we so desire.
Thus, if $W$ is another object of $\mathcal{C}$, then we may
assume $W = u_i(W_i)$ and we see that
\begin{align*}
& \Mor_\mathcal{C}(W, u_i(U^t_i \times_{U_i} V_i)) \\
& =
\colim_{a : j \to i}
\Mor_{\mathcal{C}_j}(u_a(W_i), u_a(U^t_i \times_{U_i} V_i)) \\
& =
\colim_{a : j \to i}
\Mor_{\mathcal{C}_j}(u_a(W_i), u_a(U^t_i))
\times_{\Mor_{\mathcal{C}_j}(u_a(W_i), u_a(U_i))}
\Mor_{\mathcal{C}_j}(u_a(W_i), u_a(V_i)) \\
& =
\Mor_\mathcal{C}(W, U^t)
\times_{\Mor_\mathcal{C}(W, U)}
\Mor_\mathcal{C}(W, V)
\end{align*}
as filtered colimits commute with finite limits
(Categories, Lemma \ref{categories-lemma-directed-commutes}).
It also follows that
$\{U^t \times_U V \to V\}$ is a covering in $\mathcal{C}$.
In this way we see that axiom (3) of Definition \ref{definition-site} holds.
\medskip\noindent
To verify axiom (2) of Definition \ref{definition-site}
let $\{U^t \to U\}_{t \in T}$ be a covering of $\mathcal{C}$
and for each $t$ let $\{U^{ts} \to U^t\}$ be a covering of
$\mathcal{C}$. Then we can find an $i$ and a covering
$\{U^t_i \to U_i\}_{t \in T}$ of $\mathcal{C}_i$ whose image by $u_i$ is
$\{U^t \to U\}$. Since $T$ is {\bf finite} we may choose an $a : j \to i$
in $\mathcal{I}$ and coverings $\{U^{ts}_j \to u_a(U^t_i)\}$ of
$\mathcal{C}_j$ whose image by $u_j$ gives $\{U^{ts} \to U^t\}$.
Then we conclude that $\{U^{ts} \to U\}$ is a covering of $\mathcal{C}$
by an application of axiom (2) to the site $\mathcal{C}_j$.
\medskip\noindent
We omit the proof of axiom (1) of Definition \ref{definition-site}.
\end{proof}
\begin{lemma}
\label{lemma-compute-pullback-to-limit}
In Situation \ref{situation-inverse-limit-sites} let
$u_i : \mathcal{C}_i \to \mathcal{C}$ be as constructed in
Lemma \ref{lemma-colimit-sites}. Then $u_i$ defines a morphism
of sites $f_i : \mathcal{C} \to \mathcal{C}_i$. For
$U_i \in \Ob(\mathcal{C}_i)$ and sheaf $\mathcal{F}$ on $\mathcal{C}_i$ we have
\begin{equation}
\label{equation-compute-pullback-to-limit}
f_i^{-1}\mathcal{F}(u_i(U_i)) =
\colim_{a : j \to i} f_a^{-1}\mathcal{F}(u_a(U_i))
\end{equation}
\end{lemma}
\begin{proof}
It is immediate from the arguments in the proof of
Lemma \ref{lemma-colimit-sites} that the functors $u_i$ are continuous.
To finish the proof we have to show that $f_i^{-1} := u_{i, s}$
is an exact functor $\Sh(\mathcal{C}_i) \to \Sh(\mathcal{C})$.
In fact it suffices to show that $f_i^{-1}$ is left exact, because
it is right exact as a left adjoint
(Categories, Lemma \ref{categories-lemma-exact-adjoint}).
We first prove (\ref{equation-compute-pullback-to-limit})
and then we deduce exactness.
\medskip\noindent
For an arbitrary object $V$ of $\mathcal{C}$ we can pick a $a : j \to i$
and an object $V_j \in \Ob(\mathcal{C})$ with $V = u_j(V_j)$. Then we
can set
$$
\mathcal{G}(V) = \colim_{b : k \to j} f_{a \circ b}^{-1}\mathcal{F}(u_b(V_j))
$$
The value $\mathcal{G}(V)$ of the colimit is independent of the choice
of $b : j \to i$ and of the object $V_j$ with $u_j(V_j) = V$; we omit
the verification. Moreover, if $\alpha : V \to V'$ is a morphism of
$\mathcal{C}$, then we can choose $b : j \to i$ and a morphism
$\alpha_j : V_j \to V'_j$ with $u_j(\alpha_j) = \alpha$. This induces