diff -urN -X dontdiff stacks-0.2.orig/src/Makefile stacks-0.2/src/Makefile
--- stacks-0.2.orig/src/Makefile	Wed Aug  3 10:15:06 2005
+++ stacks-0.2/src/Makefile	Sat Oct 15 15:14:24 2005
@@ -1,9 +1,9 @@
 .SUFFIXES: .aux .bbl .bib .blg .dvi .html .log .out .pdf .ps .tex .toc
-PDFS = conventions.pdf sites.pdf introduction.pdf categories.pdf hypercovering.pdf desirables.pdf injectives.pdf stacks-groupoids.pdf sets.pdf fdl.pdf
-DVIS = conventions.dvi sites.dvi introduction.dvi categories.dvi hypercovering.dvi desirables.dvi injectives.dvi stacks-groupoids.dvi sets.dvi fdl.dvi
-PSS = conventions.ps sites.ps introduction.ps categories.ps hypercovering.ps desirables.ps injectives.ps stacks-groupoids.ps sets.ps fdl.ps
-AUXS = conventions.aux sites.aux introduction.aux categories.aux hypercovering.aux desirables.aux injectives.aux stacks-groupoids.aux sets.aux
-TOCS = conventions.toc sites.toc introduction.toc categories.toc hypercovering.toc desirables.toc injectives.toc stacks-groupoids.toc sets.toc
+PDFS = conventions.pdf sites.pdf introduction.pdf categories.pdf hypercovering.pdf desirables.pdf injectives.pdf stacks-groupoids.pdf sets.pdf fdl.pdf stacks.pdf
+DVIS = conventions.dvi sites.dvi introduction.dvi categories.dvi hypercovering.dvi desirables.dvi injectives.dvi stacks-groupoids.dvi sets.dvi fdl.dvi stacks.dvi
+PSS = conventions.ps sites.ps introduction.ps categories.ps hypercovering.ps desirables.ps injectives.ps stacks-groupoids.ps sets.ps fdl.ps stacks.ps
+AUXS = conventions.aux sites.aux introduction.aux categories.aux hypercovering.aux desirables.aux injectives.aux stacks-groupoids.aux sets.aux stacks.aux
+TOCS = conventions.toc sites.toc introduction.toc categories.toc hypercovering.toc desirables.toc injectives.toc stacks-groupoids.toc sets.toc stacks.toc
 HTMLS = stacks.html contents.html downloads.html
 
 # Files in INSTALLDIR will be overwritten.
diff -urN -X dontdiff stacks-0.2.orig/src/categories.tex stacks-0.2/src/categories.tex
--- stacks-0.2.orig/src/categories.tex	Tue Sep 27 09:14:24 2005
+++ stacks-0.2/src/categories.tex	Sat Oct 15 15:14:24 2005
@@ -398,7 +398,7 @@
 \end{definition}
 
 \noindent
-Condition (2) frased differently says that 
+Condition (2) phrased differently says that 
 applying the functor $p$ gives a bijection between the sets 
 of dotted arrows in the following commutative diagram below:
 $$
@@ -446,9 +446,21 @@
 \begin{lemma}
 \label{lemma-fibred-groupoids}
 If $p : \mathcal{S} \to \mathcal{C}$ is a category fibred in groupoids then
-all fibre categories are groupoids. Moreover, for a pair of composable
-morphisms $f$, $g$ the functors $f^\ast$ $g^\ast f^\ast$ and 
-$(f \circ g)^\ast $ are isomorphic.
+all fibre categories are groupoids. Choose functors $f^\ast$ as above.
+Then for any pair of composable
+morphisms $f : V \to W$, $g : U \to V$ there is a unique isomorphism of 
+functors $\mathcal{S}_W \to \mathcal{S}_U$ 
+$$
+t : g^\ast f^\ast \to (f \circ g)^\ast 
+$$ 
+such that for every $x\in \text{Ob}(\mathcal{S}_W)$ the following
+diagram commutes
+$$
+\xymatrix{
+g^\ast f^\ast x \ar[r] \ar[d]_{t_x} & f^\ast x \ar[d] \\
+(f\circ g)^\ast x \ar[r] & x
+}
+$$
 \end{lemma}
 
 \begin{proof} FIXME.
@@ -492,19 +504,24 @@
 $\mathcal{S}_F$ as in the example. The notion of equivalence
 depends on the $2$-category we are working with. To make sure
 that everybody knows what we are talking about we define the
-$2$-category of categories over $\mathcal{S}$.
+$2$-category of categories over $\mathcal{C}$.
 
 \begin{definition}
-\label{definition-categories-over-S}
-The $2$-category of categories over $\mathcal{S}$ is defined
+\label{definition-categories-over-C}
+The $2$-category of categories over $\mathcal{C}$ is defined
 as follows. Its objects will be functors 
 $p : \mathcal{S} \to \mathcal{C}$ (belonging to
-some set, see \autoref{sets-section-reflection-principle}). Its 
+some set, see Sets, \autoref{sets-section-reflection-principle}). Its 
 $1$-morphisms will be functors $F : \mathcal{S} \to \mathcal{S}'$
 such that $p' \circ F = p$, and its $2$-morphisms $t : F \to G$
 will be morphisms of functors such that $p'(t_x) = \text{id}_x$
-for all $x \in \text{Ob}(\mathcal{S})$.
+for all $x \in \text{Ob}(\mathcal{C})$.
 \end{definition}
+
+\noindent
+The $2$-category of categories fibred in groupoids over $\mathcal{C}$
+is the full sub-$2$-category of this $2$-category whose objects
+are categories fibred in groupoids.
 
 \begin{lemma} 
 \label{lemma-fibred-strict}
diff -urN -X dontdiff stacks-0.2.orig/src/hypercovering.tex stacks-0.2/src/hypercovering.tex
--- stacks-0.2.orig/src/hypercovering.tex	Tue Sep 27 16:26:23 2005
+++ stacks-0.2/src/hypercovering.tex	Sat Oct 15 15:14:24 2005
@@ -34,7 +34,7 @@
 % \externaldocument[hello-]{hello}
 \externaldocument[conventions-]{conventions}
 \externaldocument[injectives-]{injectives}
-\externaldocument[stacks-groupoids-]{stacks-groupoids}
+\externaldocument[stacks-]{stacks}
 
 % The macro \autoref uses the macros \figurename, etc.
 % We list the default values and we change some of them
@@ -260,8 +260,7 @@
 A doubly simplicial object of $\mathcal{C}$ is a functor
 $U_{\bullet,\bullet} : (\Delta\times\Delta)^\circ \to \mathcal{C}$.
 By subdividing we can make this into a simplicial object 
-$W(U_{\bullet,\bullet})$ with the same cohomology. FIXME: Explain,
-unless this is nonsense.
+$W(U_{\bullet,\bullet})$ with the same cohomology. FIXME: Explain this.
 
 \noindent
 Suppose that $U'_\bullet \to U_\bullet$ is a morphism of simplicial
@@ -273,8 +272,9 @@
 $$
 etc. Compare Example \ref{example-simplicial-products}. 
 Out of this object we can construct a single simplicial object
-$W(U'_{\bullet,\bullet})$ as explained above. Note that there is a 
-natural morphism of simplicial objects $W_\bullet \to U_\bullet$.
+$W(U'_{\bullet,\bullet})$ as explained above. Construct the 
+natural morphism of simplicial objects 
+$W(U'_{\bullet,\bullet}) \to U_\bullet$.
 
 \begin{lemma}
 Suppose that every $\{U'_n \to U_n\}$ is a covering for the topology
@@ -283,24 +283,25 @@
 morphism of complexes
 $$
 R\Gamma(U_\bullet, \mathcal{F}) \to 
-R\Gamma(W_\bullet, \mathcal{F})
+R\Gamma(W(U_{\bullet,\bullet}), \mathcal{F})
 $$
-which is a quasi-isomorphism. FIXME: something like this in any case.
+which is a quasi-isomorphism. FIXME: Something like this in any case.
 \end{lemma}
 
 \section{The general case}
 
 \noindent
 Mention how things work more generally, for example if $\mathcal{C}$
-does not have disjoint unions.
+does not have the property that coverings consisting of a single map
+are cofinal. State the theorem in the correct generality.
 
 
 \smallskip\noindent
-To continue reading, 
+To continue reading,
 \begin{enumerate}
 
-\item visit the next section: Stacks and Groupoids,
-\autoref{stacks-groupoids-section-introduction}, or 
+\item visit the next section: Stacks,
+\autoref{stacks-section-introduction}, or 
 
 \item go back to the
 table of contents: \url{index.html#contents}.
diff -urN -X dontdiff stacks-0.2.orig/src/scripts/contents_html.sh stacks-0.2/src/scripts/contents_html.sh
--- stacks-0.2.orig/src/scripts/contents_html.sh	Tue Aug  2 09:57:31 2005
+++ stacks-0.2/src/scripts/contents_html.sh	Sat Oct 15 15:14:24 2005
@@ -33,7 +33,7 @@
 
 # LIJST is the list of STEMS of .toc files in the order in which you want it
 # to appear on the web-site. Do not include fdl.
-LIJST="introduction conventions sets categories sites injectives hypercovering stacks-groupoids desirables"
+LIJST="introduction conventions sets categories sites injectives hypercovering stacks stacks-groupoids desirables"
 TELLER=0
 
 for STAM in $LIJST; do
diff -urN -X dontdiff stacks-0.2.orig/src/scripts/downloads_html.sh stacks-0.2/src/scripts/downloads_html.sh
--- stacks-0.2.orig/src/scripts/downloads_html.sh	Tue Aug  2 09:49:31 2005
+++ stacks-0.2/src/scripts/downloads_html.sh	Sat Oct 15 15:14:24 2005
@@ -3,7 +3,7 @@
 # Write a downloads section to downloads.html.
 
 # Same list as in contents_html.sh.
-LIJST="introduction conventions sets categories sites injectives hypercovering stacks-groupoids desirables"
+LIJST="introduction conventions sets categories sites injectives hypercovering stacks stacks-groupoids desirables"
 
 cat > downloads.html << "EOF"
 <h3><a name="downloads"></a>Downloads</h3>
diff -urN -X dontdiff stacks-0.2.orig/src/sites.tex stacks-0.2/src/sites.tex
--- stacks-0.2.orig/src/sites.tex	Tue Sep 27 15:01:23 2005
+++ stacks-0.2/src/sites.tex	Sat Oct 15 15:14:24 2005
@@ -168,7 +168,7 @@
 without reference to the topology.
 
 \begin{definition}
-\label{definition-pre-sheaf}
+\label{definition-presheaf}
 A presheaf $\mathcal{F}$ on a category $\mathcal{C}$ with values in a category
 $\mathcal{A}$ is a contravariant functor from $\mathcal{C}$ to $\mathcal{C}$,
 i.e., $\mathcal{F} : \mathcal{C}^\circ \to \mathcal{A}$.
diff -urN -X dontdiff stacks-0.2.orig/src/stacks.tex stacks-0.2/src/stacks.tex
--- stacks-0.2.orig/src/stacks.tex	Wed Dec 31 19:00:00 1969
+++ stacks-0.2/src/stacks.tex	Sat Oct 15 15:14:24 2005
@@ -0,0 +1,214 @@
+\documentclass{amsart}
+
+% The following AMS packages are automatically loaded with amsart 
+% documentclass:
+%\usepackage{amsmath}
+%\usepackage{amssymb}
+%\usepackage{amsthm}
+
+% For commutative diagrams you can use
+% \usepackage{amscd}
+% but Jason prefers xypic
+\usepackage[all]{xy}
+
+% To put source file link in headers.
+% Change "template.tex" to "this_filename.tex"
+\usepackage{fancyhdr}
+\pagestyle{fancy}
+\lhead{}
+\chead{}
+\rhead{Source file: \url{src/stacks.tex}}
+\lfoot{}
+\cfoot{\thepage}
+\rfoot{}
+\renewcommand{\headrulewidth}{0pt}
+\renewcommand{\footrulewidth}{0pt}
+\renewcommand{\headheight}{12pt}
+
+% For cross-file-references
+\usepackage{xr-hyper}
+
+% Package for hypertext links:
+\usepackage[colorlinks=true]{hyperref}
+% For any local file, say "hello.tex" you want to refer to please use
+% \externaldocument[hello-]{hello}
+\externaldocument[conventions-]{conventions}
+\externaldocument[hypercovering-]{hypercovering}
+\externaldocument[sites-]{sites}
+\externaldocument[categories-]{categories}
+\externaldocument[stacks-groupoids-]{stacks-groupoids}
+
+% The macro \autoref uses the macros \figurename, etc.
+% We list the default values and we change some of them
+% to start with a captial.
+% Figure	\figurename
+% Table		\tablename
+% Part		\partname
+% Appendix	\appendixname
+% Equation	\equationname
+% item		\Itemname
+% \renewcommand{\Itemname}{Item}
+\renewcommand{\Itemautorefname}{Item}
+% chapter	\Chaptername
+% \renewcommand{\Chaptername}{Chapter}
+% \renewcommand{\Chapterautorefname}{Chapter}
+% section	\sectionname
+\renewcommand{\sectionname}{Section}
+\renewcommand{\sectionautorefname}{Section}
+% subsection	\subsectionname
+\renewcommand{\subsectionname}{Subsection}
+\renewcommand{\subsectionautorefname}{Subsection}
+% subsubsection	\subsubsectionname
+\renewcommand{\subsubsectionname}{Subsubsection}
+\renewcommand{\subsubsectionautorefname}{Subsubsection}
+% paragraph	\paragraphname
+\renewcommand{\paragraphname}{Paragraph}
+\renewcommand{\paragraphautorefname}{Paragraph}
+% footnote	\Hfootnotename
+% \renewcommand{\Hfootnotename}{Footnote}
+\renewcommand{\Hfootnoteautorefname}{Footnote}
+% Equation	\AMSname
+% Theorem	\theoremname
+
+
+% Theorem environments.
+%
+\newtheorem{theorem}{Theorem}[subsection]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{example}[theorem]{Example}
+\newtheorem{exercise}[theorem]{Exercise}
+\newtheorem{situation}[theorem]{Situation}
+
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newtheorem{remarks}[theorem]{Remarks}
+
+\numberwithin{equation}{subsection}
+
+
+% OK, start here.
+%
+\begin{document}
+
+\title{Stacks}
+
+%\begin{abstract}
+%\end{abstract}
+
+\maketitle
+\thispagestyle{fancy}
+
+\tableofcontents
+
+\section{Introduction}
+\label{section-introduction}
+
+\noindent
+Stacks are defined in this document. See \cite{DM}.
+
+\section{Definition}
+\label{section-definition}
+
+\noindent
+Let $\mathcal{C}$ be a site. The $2$-category of stacks over
+$\mathcal{C}$ will be a full sub-$2$-category of the $2$-category
+of categories over $\mathcal{C}$, see Categories, 
+\autoref{categories-definition-categories-over-C}. Thus a stack will be
+given by a functor of categories $p : \mathcal{S} \to \mathcal{C}$
+which has certain additional properties. Loosely speaking
+the conditions are the following:
+\begin{enumerate}
+\item $p : \mathcal{S} \to \mathcal{C}$ is a category fibred
+in groupoids, see Categories, 
+\autoref{categories-definition-fibred-groupoids},
+\item descend for morphisms holds, and
+\item descent data for objects are effective.
+\end{enumerate}
+To explain this, we choose a collection of pullback functors as in
+Categories, \hyperref[categories-lemma-fibred-groupoids]%
+{Lemma~\ref*{categories-lemma-fibred-groupoids}}.
+
+\smallskip\noindent
+First, suppose that $x,y\in \text{Ob}(\mathcal{S}_U)$ are
+objects in the fibre category over $U$. We are going to define
+a contravariant functor
+$$
+\text{Isom}(x,y) : \mathcal{C}/U \longrightarrow \text{Sets}.
+$$
+In other words this will be a presheaf on $\mathcal{C}/U$, see
+Sites, \hyperref[sites-definition-presheaf]%
+{Definition~\ref*{sites-definition-presheaf}}. Namely, for 
+$f : V \to U$ we set 
+$$
+\text{Isom}(x,y)(f:V\to U) = 
+\text{Mor}_{\mathcal{S}_V}(f^\ast x, f^\ast y).
+$$
+We also have to define the restriction map corresponding to a
+morphism $(g,\text{id}_U) : (f' : V' \to U)  \to (f : V\to U)$ 
+in $\mathcal{C}/U$ (in other words $f' = f \circ g$). This will be a map
+$$
+\text{Mor}_{\mathcal{S}_V}(f^\ast x, f^\ast y) \longrightarrow
+\text{Mor}_{\mathcal{S}_{V'}}({f'}^\ast x, {f'}^\ast y).
+$$
+This map will basically be $g^\ast$, except that this transforms
+an element $\phi$ of the left hand side into an element 
+$g^\ast \phi$
+of $\text{Mor}_{\mathcal{S}_{V'}}(g^\ast f^\ast x, g^\ast f^\ast y)$.
+At this point we use the transformation $t$ of 
+Categories, \hyperref[categories-lemma-fibred-groupoids]%
+{Lemma~\ref*{categories-lemma-fibred-groupoids}}.
+In a formula, the restriction map maps $\phi$ to
+$$
+t_y \circ g^\ast \phi \circ (t_x)^{-1}.
+$$
+
+\begin{lemma}
+\label{lemma-painfull}
+This actually does give a presheaf.
+\end{lemma}
+
+\begin{proof}
+FIXME: A fun way to do this is to use Categories,
+\hyperref[categories-lemma-fibred-strict]%
+{Lemma~\ref*{categories-lemma-fibred-strict}}
+\end{proof}
+
+\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
+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 
+$\text{pr}_{12} : U_{ijk} \to U_{ij}$
+the projection onto the first and second factor, etc.
+
+\smallskip\noindent
+FIXME: To be continued.
+
+\smallskip\noindent
+To continue reading, 
+\begin{enumerate}
+
+\item visit the next section: Stacks and Groupoids,
+\autoref{stacks-groupoids-section-introduction}, or 
+
+\item go back to the
+table of contents: \url{index.html#contents}.
+
+\end{enumerate}
+
+\bibliography{my}
+\bibliographystyle{alpha}
+
+\end{document}