diff -urN -X dontdiff stacks-0.2.orig/src/stacks.tex stacks-0.2/src/stacks.tex
--- stacks-0.2.orig/src/stacks.tex	Sat Oct 15 15:14:24 2005
+++ stacks-0.2/src/stacks.tex	Sat Oct 15 18:23:19 2005
@@ -178,22 +178,149 @@
 \end{proof}
 
 \noindent
+This lemma says that there is no harm in thinking of 
+$g^\ast \phi$ as a morphism ${f'}^\ast x \to {f'}^\ast y$.
+
+\smallskip\noindent
 OK, so the second condition listed above is simply the condition that
 $\text{Isom}(x,y)$ is a sheaf on the site $\mathcal{C}/U$! 
 
 \smallskip\noindent
 We still have to explain the meaning of the third condition. 
 For this we have to explain what a descent datum is. We
-introduce some notation. Suppose that $\{f_i : U_i \to U\}$ is
-a convering in the site $\mathcal{C}$. Let 
-$x_i \in \text{Ob}(\mathcal{S}_{U_i})$. We will be looking at
+introduce some notation. Suppose that $\{f_i : U_i \to U\}_{i\in I}$
+is a convering in the site $\mathcal{C}$. We will be looking at
 fibre products $U_{ij} = U_i \times_U U_j$ and fibre products
-$U_{ijk} = U_i \times_U U_j \times_U U_k$. We will denote $\text{pr}_a$
-the projection onto the $a$th factor and we will denote 
+$U_{ijk} = U_i \times_U U_j \times_U U_k$. We will denote
+$\text{pr}_1 : U_{ij} \to U_i$, resp.\ $\text{pr}_1 :
+U_{ijk} \to U_i$ the projection onto the first factor,
 $\text{pr}_{12} : U_{ijk} \to U_{ij}$
-the projection onto the first and second factor, etc.
+the projection onto the first and second factor, etc. Let 
+$x_i \in \text{Ob}(\mathcal{S}_{U_i})$, $i\in I$ be a collection of
+objects, and let
+$$
+\phi_{ij} : \text{pr}_1^\ast x_i \longrightarrow
+\text{pr}_2^\ast x_j ,
+$$
+$i,j\in I$ be a collection of morphisms in the fibre categories
+$\mathcal{S}_{U_{ij}}$. 
+
+\begin{definition}
+\label{definition-descent-data}
+The collection $\{x_i, \phi_{ij}\}$ is a descent datum if
+the following cocycle condition is satisfied: For every
+triple $(i,j,k)\in I^3$ the diagram
+$$
+\xymatrix{
+\text{pr}_1^\ast x_i 
+	\ar[rr]^{\text{pr}_{13}^\ast \phi_{ik}}
+	\ar[rd]_{\text{pr}_{12}^\ast \phi_{ij}}
+& & 
+\text{pr}_3^\ast x_k \\
+& \text{pr}_2^\ast x_j \ar[ru]_{\text{pr}_{23}^\ast \phi_{jk}}
+}
+$$
+is commutative.
+\end{definition}
+
+\noindent
+Let us make a small remark about this definition. There is a
+diagonal morphism $\Delta_i : U_i \to U_{ii}$. We can pull back
+$\phi_{ii}$ via this morphism to get an automorphism 
+$\Delta^\ast \phi_{ii} \in \text{Aut}_{U_i}(x_i)$.
+On pulling back the cocycle condition for the triple $(i,i,i)$ 
+by $\Delta_{123} : U_i \to U_{iii}$ we deduce that
+$\Delta^\ast \phi_{ii} \circ \Delta^\ast \phi_{ii} =
+\Delta^\ast \phi_{ii}$, in other words $\Delta^\ast \phi_{ii} =
+\text{id}_{x_i}$. Furthermore, there is a morphism
+$\Delta_{13}: U_{ij} \to U_{iji}$ and we can pull back the
+cocycle condition for the triple $(i,j,i)$ to get the
+identity $\text{switch}^\ast \phi_{ji} \circ \phi_{ij} = 
+\text{id}_{x_i}$. Here $\text{switch}$ is the obvious
+morphism $U_{ij} \to U_{ji}$. 
 
 \smallskip\noindent
+A morphism of descent data $\{x_i, \phi_{ij}\}$ to
+$\{y_i, \psi_{ij}\}$ is given by a collection of 
+morphisms $\alpha_i : x_i \to y_i$ in the fibre category
+over $U_i$ such that the following diagrams
+$$
+\xymatrix{
+\text{pr}_1^\ast x_i
+	\ar[r]^{\text{pr}_1^\ast \alpha_i}
+	\ar[d]_{\phi_{ij}}
+&
+\text{pr}_1^\ast y_i
+	\ar[d]^{\psi_{ij}}
+\\
+\text{pr}_2^\ast x_j
+	\ar[r]_{\text{pr}_2^\ast \alpha_j}
+&
+\text{pr}_2^\ast y_j
+}
+$$
+commute. Note that every morphism of descent data is an isomorphism.
+
+\smallskip\noindent
+An object $x \in \text{Ob}(\mathcal{S}_U)$ gives rise to a descend
+datum in the following manner. First we set $x_i = f_i^\ast x$.
+Second we let $\phi_{ij}$ be the canonical isomorphism
+$$
+\text{pr}_1^\ast x_i = \text{pr}_1^\ast f_i^\ast x
+= (f_i \circ \text{pr}_1)^\ast x =
+(f_j \circ \text{pr}_2)\ast x = 
+\text{pr}_2^\ast f_j^\ast x = \text{pr}_2^\ast x_j.
+$$
+The lemma below shows this is a descent datum; we will call
+this the canonical descent datum associated to $x$.
+
+\begin{lemma}
+\label{lemma-trivial-cocycle}
+The cocycle condition holds for the datum described above.
+\end{lemma}
+
+\begin{proof}
+FIXME: Use Categories,
+\hyperref[categories-lemma-fibred-strict]%
+{Lemma~\ref*{categories-lemma-fibred-strict}}
+\end{proof}
+
+\begin{definition}
+\label{definition-effective-descent-datum}
+A descent datum $\{x_i,\phi_{ij}\}$ is said to be effective
+when there exists an $x\in \text{Ob}(\mathcal{S}_U)$ 
+such that $\{x_i,\phi_{ij}\}$ is isomorphic to the
+canonical descent datum associated to $x$.
+\end{definition}
+
+\noindent
+At this point we are ready to give the definition of a
+stack. 
+
+\begin{definition}
+\label{defintion-stack}
+A stack (in groupoids) over a site $\mathcal{C}$ is a 
+category $p : \mathcal{S} \to \mathcal{C}$ over $\mathcal{C}$
+such that
+\begin{enumerate}
+\item $p : \mathcal{S} \to \mathcal{C}$ is a category fibred 
+in groupoids over $\mathcal{C}$, 
+\item for all $U \in \text{Ob}(\mathcal{C})$ and all
+$x,y\in \text{Ob}(\mathcal{S}_U)$ the presheaf
+$\text{Isom}(x,y)$ is a sheaf on $\mathcal{C}/U$, and
+\item for all coverings $\mathcal{U}=\{U_i \to U\}$ in $\mathcal{C}$, 
+all descent data $\{x_i,\phi_{ij}\}$ for $\mathcal{U}$ are effective.
+\end{enumerate}
+\end{definition}
+
+\noindent
+Usually the hardest part to check is the third condition.
+
+\begin{example}
+FIXME: Need lots of examples.
+\end{example}
+
+\noindent
 FIXME: To be continued.
 
 \smallskip\noindent