diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/sites.tex stacks-0.2/src/sites.tex
--- stacks-0.2.orig/src/sites.tex	2005-10-31 07:38:06.000000000 -0500
+++ stacks-0.2/src/sites.tex	2006-02-08 00:14:07.000000000 -0500
@@ -147,6 +147,36 @@
 \end{definition}
 
 \begin{example}
+Let $X$ be a topological space.  Let $T_X$ be the category whose
+objects consist of all the open sets $U$ in $X$ and whose morphisms
+are just the inclusion maps.  That is, there is at most one morphism
+between any two objects in $T_X$.  Now define a site on this
+category by defining $\{U_i \to U\}\in \text{Cov} T_X$ if $\bigcup
+U_i = U$. Conditions (1) and (2) above are clear, and (3) is also
+clear once we realize that in $T_X$ we have $U \times V = U \bigcap V$.
+Presheaves and sheaves (as defined below) on the site $T_X$ will
+agree exactly with the usual notion of a (pre)sheaf on a topological
+space.
+\end{example}
+
+\begin{example}
+\label{example-site on group} Every category (with products) has a
+canonical topology associated to it. (Note: this is the finest
+topology where all representable presheaves are sheaves). Here is one
+example. Let $G$ be a group. Consider the category whose objects are 
+sets $X$ with a left $G$-action, with $G$-equivariant maps as the
+morphisms.  We define a topology by declaring $\{U_i \to U\}$ to be 
+a covering if $\bigcup U_i = U$. To verify that fibred products do 
+exist in this category, suppose $f: S \rightarrow U$ and 
+$h: T \rightarrow U$ are morphisms in the category. Then let 
+$W = \{(s, t) \in S \times T \mid f(s) = h(t)\}$. This is a $G$-set. 
+The action is given by $g \bullet(s,t) = (g \bullet s, g \bullet t)$.
+Projections onto $S$ and $T$ are clearly $G$-maps and so 
+$W = S \times_U T$.  Conditions (1), (2), and (3) are now easily 
+verified. This site will be denoted $T_G$.
+\end{example}
+
+\begin{example}
 FIXME. We can have a lot of examples linked from here.
 \end{example}
 
@@ -218,7 +248,61 @@
 }
 $$
 is exact as before. If the products in $(*)$ exist then this condition just
-means that the first arrow is the equalizer of the other two. 
+means that the first arrow is the equalizer of the other two.
+
+FIXME: A little about the canonical topology, and in particular some examples.
+
+\begin{example}
+As an example, suppose $\mathcal{F}$ is a sheaf of sets on $T_G$
+(see Example \ref{example-site on group}).  First we note that $G$ 
+itself is an object in the category (the action given by left 
+multiplication). As a $G$-set, denote it ${}_GG$. Next, remark that
+the map
+$$
+\text{Hom}_G({}_GG,{}_GG) \longrightarrow G^{opp}, 
+\varphi \longmapsto \varphi(1)
+$$
+is an isomorphism of groups. The inverse map sends $g \in G$
+to the map $ s \mapsto s \cdot g$ (i.e.\ right multiplication). Then
+$\mathcal{F}({}_GG)$ is also a $G$-set where the action $g \bullet s$
+for $g \in G$ and $s \in \mathcal{F}({}_GG)$ is given by
+$\mathcal{F}(\cdot g)(s)$. Claim: If $\mathcal{F}$ is a sheaf then we
+can recover $\mathcal{F}$ from the $G$-set $\mathcal{F}({}_GG)$
+and vice versa. That is, there is an equivalence of categories
+between left $G$-sets and sheaves of sets on $T_G$. We will show a
+quasi-inverse of the functor $\mathcal{F} \mapsto \mathcal{F}({}_GG)$ is
+given by $U \mapsto \text{Hom}_G(\cdot, U)$ where $U$ is a $G$-set. 
+Since $T_G$ has the canonical topology, the presheaves  
+$\text{Hom}_G(\cdot, U)$ are sheaves.
+Composing $U \rightarrow \text{Hom}_G(\cdot, U)$ with 
+$\mathcal{F} \rightarrow \mathcal{F}({}_GG)$ we get 
+$U \mapsto \text{Hom}_G({}_GG,U)$ which is
+canonically isomorphic to $U$ (namely, a $G$-equivariant map of ${}_GG$ 
+into $U$ is uniquely determined by the image of $1$ in the exact same way 
+as above). Composing in the reverse direction $\mathcal{F} \mapsto 
+\mathcal{F}({}_GG)$ with $U \mapsto \text{Hom}_G(\cdot, U)$ we have to 
+show that the presheaf $\text{Hom}_G(\cdot, \mathcal{F}({}_GG))$ is naturally
+isomorphic to $\mathcal{F}$, provided that $\mathcal{F}$ is a sheaf.
+Suppose $U$ is another $G$-set. Then
+$\{{}_GG \stackrel{\phi_u}{\rightarrow} U\}_{u \in U}$ (where
+$\phi_u(g) = g \bullet u$) is a covering of $U$. Since $\mathcal{F}$ is a 
+sheaf we have the exact sequence:
+$$
+\xymatrix{ \mathcal{F}(U) \ar[r] & \prod\nolimits_{u \in U}
+\mathcal{F}({}_GG) \ar@/^/[r] \ar@/_/[r] & \prod\nolimits_{u, v \in
+U} \mathcal{F}({}_GG \times_U{}_GG) } \leqno{(*)}
+$$
+Now we note that the middle term is exactly 
+$\text{Mor}(U,\mathcal{F}({}_GG))$ (maps of sets). Since the sequence is 
+exact, we have that $\mathcal{F}(U)$ is the equalizer of the second two
+arrows. This means it is exactly isomorphic to the subset of morphisms in
+$\text{Mor}(U, \mathcal{F}(_G G))$ that commute with the $G$-action (FIXME?),
+i.e., $\mathcal{F}(U) \cong \text{Hom}_G(U, \mathcal{F}({}_GG))$.
+This isomorphisms is clearly functorial in $U$ so we have an
+isomorphism of sheaves, as desired. Note that in the special case that 
+$U$ is a left $G$-module rather than just a set, then this process gives 
+an equivalence between left $G$-modules and sheaves of abelian groups on $T_G$.
+\end{example}
 
 \subsection{More about coverings}
 \label{subsection-coverings}
@@ -295,8 +379,55 @@
 induced by $f$~and~$g$ are equal.
 \end{proof}
 
+\smallskip\noindent
+Suppose that $\mathcal{F}$ is a presheaf of abelian groups on a
+fixed site $T$.  We would like to canonically associate a sheaf
+$\mathcal{F}^\#$ to $\mathcal{F}$ such that there exists a
+functorial morphism $\mathcal{F} \rightarrow \mathcal{F}^\#$ and such that for
+any morphism from $\mathcal{F}$ to an abelian sheaf $\mathcal{G}$
+there is a unique factorization $\mathcal{F} \rightarrow
+\mathcal{F}^\# \rightarrow \mathcal{G}$.  $\mathcal{F}^\#$ will be
+called the sheafification of $\mathcal{F}$.
+
+\begin{definition}
+A directed set is a set $S$ together with a relation $\geq$ which is
+transitive and reflexive such that for $a, b \in S$ there exists another
+element $c \in S$ such that $c \geq a$ and $c \geq b$.
+\end{definition}
+\noindent
+Let $\mathcal{J}_U$ be the set of all coverings of $U$.  It is not hard to check
+that $\mathcal{J}_U$, along with all possible refinements of coverings,
+is a category.  It is also clear that given any two coverings of $U$, $\mathcal{U}_1$
+and $\mathcal{U}_2$, there is another covering refining them both.
+That is, the covering $\{U_{1i} \times_U U_{2j} \rightarrow U\}$ is a cover of $U$ and the
+natural projection maps give the refinements.
+(Note: it is exactly conditions 2 and 3 in \ref{definition-site} that allow us to
+know that this will be a cover).
+Now if $\mathcal{F}$ is a functor from $\mathcal{C} \rightarrow \mathcal{A}b$ then we can
+define the direct limit relative to a directed set $I$ to be $\stackrel{\lim}{\rightarrow}\mathcal{F}(U_i)$ to be
+$\bigoplus \mathcal{F}(U_i) / \Box$ where $\Box$ is generated by the relations $m_i = \sigma_{ij}(m_i)$
+where $m_i \in \mathcal{F}(U_i)$ and $\sigma_{ij}: \mathcal{F}(U_i) \rightarrow \mathcal{F}(U_j)$.
+Now, by the above remarks, we see that $\mathcal{J}_U$ is a directed set, and so we can take
+the direct limit over the set of coverings of $U$.
+We let $\mathcal{F}^\dagger(U) := \stackrel{\lim}{\rightarrow}\mathcal{F}(\mathcal{U})$
+where the limit is over the directed set of coverings $\mathcal{J}_U$.
+\\Note that it is exactly \ref{lemma-indepent-refinement} which makes this definition make sense.
+\\Remark: This is also denoted $\check{H}^o(U, F)$.  It corresponds to taking the directed limit of global sections
+over all coverings of $U$, ie, the 0th \v{C}ech cohomology group.
+\\We say that $\mathcal{F}$ is separated if, for all coverings of $U, \{U_i \rightarrow U$ the map
+$\mathcal(F) \rightarrow \prod \mathcal(F)(U_i)$ is injective.
+\begin{theorem}
+With $\mathcal{F}$ as above
+\begin{enumerate}
+\item $\mathcal{F}^\dagger$ is separated
+\item If $\mathcal{F}$ is separated, then $\mathcal{F}^\dagger$ is a sheaf.
+\item $\mathcal{F}^{\dagger\dagger}$ is always a sheaf.
+\end{enumerate}
+\end{theorem}
 \noindent
-FIXME. Use the lemma to explain about sheafification.
+FIXME. PROOF.
+\\\noindent
+FIXME. Discuss the more general case when $\mathcal{F}$ may not be a sheaf with values in $\mathcal{A}b$.
 
 \section{Representable sheaves}
 \label{section-representable-sheaves}