diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/Makefile stacks-0.2/src/Makefile
--- stacks-0.2.orig/src/Makefile	2006-03-05 15:12:12.000000000 +0000
+++ stacks-0.2/src/Makefile	2006-03-20 01:48:12.000000000 +0000
@@ -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 stacks.pdf etale.pdf flat.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 etale.dvi flat.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 etale.ps flat.ps
-AUXS = conventions.aux sites.aux introduction.aux categories.aux hypercovering.aux desirables.aux injectives.aux stacks-groupoids.aux sets.aux stacks.aux etale.aux flat.aux
-TOCS = conventions.toc sites.toc introduction.toc categories.toc hypercovering.toc desirables.toc injectives.toc stacks-groupoids.toc sets.toc stacks.toc etale.toc flat.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 etale.pdf flat.pdf schemes.pdf algebraic.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 etale.dvi flat.dvi schemes.dvi algebraic.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 etale.ps flat.ps schemes.ps algebraic.ps
+AUXS = conventions.aux sites.aux introduction.aux categories.aux hypercovering.aux desirables.aux injectives.aux stacks-groupoids.aux sets.aux stacks.aux etale.aux flat.aux schemes.aux algebraic.aux
+TOCS = conventions.toc sites.toc introduction.toc categories.toc hypercovering.toc desirables.toc injectives.toc stacks-groupoids.toc sets.toc stacks.toc etale.toc flat.toc schemes.toc algebraic.toc
 HTMLS = stacks.html contents.html downloads.html
 
 # Files in INSTALLDIR will be overwritten.
diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/algebraic.tex stacks-0.2/src/algebraic.tex
--- stacks-0.2.orig/src/algebraic.tex	1970-01-01 00:00:00.000000000 +0000
+++ stacks-0.2/src/algebraic.tex	2006-03-20 01:48:12.000000000 +0000
@@ -0,0 +1,226 @@
+\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/algebraic.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[categories-]{categories}
+\externaldocument[schemes-]{schemes}
+
+% 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{Algebraic stacks}
+
+%\begin{abstract}
+%\end{abstract}
+
+\maketitle
+\thispagestyle{fancy}
+
+\tableofcontents
+
+\section{Introduction}
+\label{section-introduction}
+
+\noindent
+This is where we define algebraic stacks and make some very elementary
+observations. The general philosophy will be to have no separation
+conditions whatsoever and add those conditions necessary to make lemmas,
+propositions, theorems true/provable. Thus the notions discussed here 
+differ slightly from those in other places in the literature, e.g.,
+\cite{LM-B}.
+
+\section{Definitions}
+\label{section-definitions}
+
+\subsection{Algebraic spaces}
+\label{subsection-algebraic-spaces}
+
+\noindent
+FIXME.
+
+\begin{definition}
+An algebraic space is a stack $\mathcal{S}$ over $\text{Aff}$ such that
+\begin{enumerate}
+\item every fibre category is setlike, see Categories,
+\autoref{categories-subsection-fibred-in-sets}, 
+\item the diagonal morphism
+$\Delta : \mathcal{S} \to \mathcal{S}\times\mathcal{S}$
+is representable by schemes, see Schemes,
+\autoref{schemes-subsection-definition-representable-by-schemes} and
+\item there exists a stack $\mathcal{X}$ representable by a scheme, see
+Schemes, \autoref{schemes-subsection-stack-representable-by-scheme}
+and an \'etale surjective morphism $\mathcal{X} \to \mathcal{S}$,
+see Schemes,
+\autoref{schemes-definition-property-morphism-representable-by-schemes}.
+\end{enumerate}
+\end{definition}
+
+\begin{remark}
+\label{remark-definition-correct}
+If you try to define some kind of more general algebraic space by requiring
+only that the diagonal is representable by algebraic spaces, and that there is
+a surjective etale morphism of an algebraic space onto $\mathcal{S}$, then 
+you actually end up with the same notion.
+(FIXME: internal references, proofs.)
+\end{remark}
+
+\subsection{Morphisms representable by algebraic spaces}
+\label{subsection-morphism-representable-by-algebraic-spaces}
+
+\noindent
+Here is the formal definition. Please also see the informal discussion below.
+
+\begin{definition}
+\label{definition-representable-by-algebraic-spaces}
+Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of categories
+fibred in groupoids over $\text{Aff}$. We say $f$ is representable by
+algebraic spaces if for every stack $\mathcal{S}$ representable by a scheme
+(see Schemes, Definition \ref{schemes-definition-representable-by-scheme}),
+and every morphism $\mathcal{U} \to \mathcal{Y}$, the 2-fibre product
+$\mathcal{S}\times_\mathcal{Y}\mathcal{X}$ is an algebraic space.
+\end{definition}
+
+\noindent
+Informal discussion. Suppose that, with the notation of the definition,
+$S$ represents $\mathcal{S}$. Suppose that $W$ is a scheme and that
+$\text{Aff}/W \to \mathcal{S}\times_\mathcal{Y}\mathcal{X}$ is 
+etale and surjective. According to
+Schemes, Lemma \ref{schemes-lemma-morphism-stacks-representable-by-schemes}
+we get a morphism of schemes $g : W \to S$ and a 2-commutative diagram
+of stacks
+$$
+\xymatrix{
+\text{Aff}/W \ar[d]^g \ar[r] &
+\mathcal{S}\times_\mathcal{X}\mathcal{Y} \ar[d] \ar[r] &
+\mathcal{Y} \ar[d] \\
+\text{Aff}/S &
+\mathcal{S} \ar[l]^j \ar[r] & \mathcal{X}
+}
+$$
+
+\begin{definition}
+\label{definition-property-morphism-representable-by-algebraic-spaces}
+Let $P$ be a property of morphisms of schemes, that is etale local
+on the source and such that if the morphism $f : X \to Y$ has property $P$,
+then so does every base change of $f$. (FIXME: introduce base change.)
+We say that a morphism of stacks $\mathcal{X}
+\to \mathcal{Y}$ representable by algebraic spaces has property
+$P$ if for every diagram as above the morphism of schemes
+$g : W \to S$ has property $P$.
+\end{definition}
+
+\noindent
+FIXME. Explain rationale behind this definition: what else could it be?
+
+
+\subsubsection{Algebraic stacks}
+\label{subsubsection-algebraic-stacks}
+
+\noindent
+FIXME.
+
+\begin{definition}
+An algebraic stack is a stack $\mathcal{S}$ over $\text{Aff}$ such that
+\begin{enumerate}
+\item the diagonal morphism
+$\Delta : \mathcal{S} \to \mathcal{S}\times\mathcal{S}$
+is representable by algebraic spaces, see Definition,
+\autoref{definition-representable-by-algebraic-spaces} and
+\item there exists a stack $\mathcal{X}$ representable by a scheme, see
+Schemes, \autoref{schemes-subsection-stack-representable-by-scheme}
+and a smooth surjective morphism $\mathcal{X} \to \mathcal{S}$,
+see Definition
+\ref{definition-property-morphism-representable-by-algebraic-spaces}.
+\end{enumerate}
+\end{definition}
+
+
+\bibliography{my}
+\bibliographystyle{alpha}
+
+\end{document}
diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/categories.tex stacks-0.2/src/categories.tex
--- stacks-0.2.orig/src/categories.tex	2006-03-14 02:49:38.000000000 +0000
+++ stacks-0.2/src/categories.tex	2006-03-20 01:48:12.000000000 +0000
@@ -401,25 +401,37 @@
 On the other hand, there are lots of examples where it is quite clear
 how you work with it. Note that we require the $2$-morphisms to be
 isomorphisms. As far as this text is concerned all 2-categories occuring
-is this document are (full) sub 2-categories of the example below.
+in this document are (full) sub 2-categories of the example below.
 FIXME: Remove this definition? Replace by a better one?
 
 \begin{example}
 \label{example-category-of-categories}
-The terminology of $2$-categories applies to categories as follows. 
-Choose an ordinal $\alpha$ (see our discussion in Sets,
-\autoref{sets-section-reflection-principle}). Let 
-$\text{Ob}(\text{Cat}_\alpha)$ be the
-set of all categories which are elements of $\alpha$. This will be the set
-of objects of our $2$-category. A $1$-morphism between $\mathcal{A}, 
-\mathcal{B}\in \text{Ob}(\text{Cat}_\alpha)$ is a functor 
+The category of categories. The terminology of $2$-categories applies to
+categories as follows. Choose an ordinal $\alpha$ (see our discussion in
+Sets, \autoref{sets-section-reflection-principle}). Let
+$\text{Ob}(\text{Cat}_\alpha)$ be the set of all categories which are elements
+of $V_\alpha$. This will be the set of objects of our $2$-category. A
+$1$-morphism between
+$\mathcal{A}, \mathcal{B} \in \text{Ob}(\text{Cat}_\alpha)$ is a functor
 $F : \mathcal{A} \to \mathcal{B}$. A $2$-morphism is an {\it isomorphism}
-of $1$-morphisms.
+of $1$-morphisms, i.e., an invertible natural transformation of functors.
+
+\smallskip\noindent
+Composition of functors and composition of transformations of functors is
+defined above. Datum (6) in Definition \ref{definition-2-category} is 
+given as follows. Suppose that $t : F \to F'$ is a transformation of
+functors $\mathcal{A} \to \mathcal{B}$ and suppose that
+$G : \mathcal{B} \to \mathcal{C}$ is a functor. In this case $G(t)$
+is the transformation of functors $G\circ F \to G \circ F'$ given
+by $G(t_A) : G(F(A)) \to G(F'(A))$. Datum (7) of Definition 
+\ref{definition-2-category} is defined similarly. FIXME. Check the rules
+(1) -- (7) hold in this example (and no more than that in general).
 \end{example}
 
 \noindent
-The notion of equivalence of categories extends to the more general
-setting of $2$-categories.
+The notion of equivalence of categories that we defined in Subsection
+\ref{subsection-categories} extends to the more general setting of
+$2$-categories as follows.
 
 \begin{definition}
 \label{definition-equivalence}
@@ -485,8 +497,8 @@
 a functor. FIXME: Add more as needed.
 \end{remarks}
 
-\subsection{2-fibre products}
-\label{subsection-2-fibre-products}
+\subsubsection{2-fibre products}
+\label{subsubsection-2-fibre-products}
 
 \noindent
 In this subsection we introduce $2$-fibre products. Suppose that $\mathcal{C}$
@@ -626,7 +638,7 @@
 is commutative.
 \end{enumerate}
 The functors $p : \mathcal{A}\times_\mathcal{C}\mathcal{B} \to \mathcal{A}$
-and $q : \mathcal{A}\times_\mathcal{C}\mathcal{B} \to \mathcal{A}$ are the
+and $q : \mathcal{A}\times_\mathcal{C}\mathcal{B} \to \mathcal{B}$ are the
 forgetfull functors in this case. The transformation $\psi : F \circ p \to
 G \circ q$ is given on the object $\xi = (A,B,f)$ by
 $\psi_\xi = f : F(p(\xi)) = F(A) \to G(B) = G(q(\xi))$.
@@ -732,7 +744,8 @@
 $z' \to z$. The uniqueness implies that the morphisms $z' \to z$ and
 $z\to z'$ are mutually inverse, in other words isomorphisms.
 
-\begin{example}\label{example-group-homomorphism-fibreedingroupoids}
+\begin{example}
+\label{example-group-homomorphism-fibreedingroupoids}
 A homomorphism of groups $p : G \to H$ gives rise to a functor 
 $p\colon \mathcal{S}\to\mathcal{C}$ as in Example 
 \ref{example-group-homorphism-functor}. This functor
@@ -742,8 +755,9 @@
 kernel of $p$ as in Example \ref{example-group-groupoid}.
 \end{example}
 
-\smallskip\noindent Suppose that for every $f : V \to U$ and $x\in
-\text{Ob}(\mathcal{S}_U)$ as in the first condition we choose a lift
+\smallskip\noindent
+Suppose that for every $f : V \to U$ and $x\in \text{Ob}(\mathcal{S}_U)$
+as in the first condition we choose a lift
 $f^\ast x \to x$ of $f$; this is possible by the axiom of choice. For
 every morphism $\phi : x \to x'$ in $\mathcal{S}_U$ there is a unique
 morphism $f^\ast \phi : f^\ast x \to f^\ast x'$ in $\mathcal{S}_V$
@@ -930,6 +944,62 @@
 isomorphic to $G(z)$ in $\mathcal{S}'$.
 \end{proof}
 
+\begin{lemma}
+\label{lemma-2-product-categories-over-C} The 2-category of categories
+over $\mathcal{C}$ has 2-fibre products. Suppose that
+$f : \mathcal{X} \to \mathcal{S}$ and
+$g : \mathcal{Y} \to \mathcal{S}$ are morphisms of categories over
+$\mathcal{C}$. An explicit 2-fibre product
+$\mathcal{X} \times_\mathcal{S}\mathcal{Y}$ is given by the following
+description
+\begin{enumerate}
+\item an object of $\mathcal{X}\times_\mathcal{S} \mathcal{Y}$ is a quadruple
+$(U,x,y,f)$, where $U \in \text{Ob}(\mathcal{C})$,
+$x\in \text{Ob}(\mathcal{X}_U)$, $y\in \text{Ob}(\mathcal{Y}_U)$,
+and $f : F(x) \to G(y)$ is an isomorphism in $\mathcal{S}_U$,
+\item a morphism $(U,x,y,f) \to (U',x',y', f')$ is given by a pair $(a,b)$,
+where $a : x \to x'$ is a morphism in $\mathcal{X}$, and $b : y \to y'$ is a
+morphism in $\mathcal{Y}$ such that (1) $a$ and $b$ induced the same
+morphism $U \to U'$, and (2) the diagram 
+$$
+\xymatrix{
+F(A) \ar[r]^f \ar[d]^{F(a)} & G(B) \ar[d]^{G(b)} \\
+F(A') \ar[r]^{f'} & G(B')
+}
+$$
+is commutative.
+\end{enumerate}
+The functors $p : \mathcal{X}\times_\mathcal{S}\mathcal{Y} \to \mathcal{X}$
+and $q : \mathcal{X}\times_\mathcal{S}\mathcal{Y} \to \mathcal{Y}$ are the
+forgetfull functors in this case. The transformation $\psi : F \circ p \to
+G \circ q$ is given on the object $\xi = (U,x,y,f)$ by
+$\psi_\xi = f : F(p(\xi)) = F(x) \to G(y) = G(q(\xi))$.
+\end{lemma}
+
+\begin{proof}
+Let us check the universal property: let $p_W : \mathcal{W}\to \mathcal{C}$
+be a category over $\mathcal{C}$, let $X : \mathcal{W} \to \mathcal{X}$ and
+$Y : \mathcal{W} \to \mathcal{Y}$ be functors over $\mathcal{C}$, and let
+$t : F \circ X \to G \circ Y$ be an isomorphism of functors.
+The desired functor
+$\gamma : \mathcal{W} \to \mathcal{A}\times_\mathcal{C}\mathcal{B}$
+is given by $W \mapsto (p_W(W), X(W), Y(W), t_W)$. What else could it be? 
+(A meta-argument for uniqueness.) FIXME: write this out.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-2-product-fibred-categories}
+In the situation of the lemma above, if $\mathcal{X}$, $\mathcal{Y}$ and 
+$\mathcal{S}$ are fibred in groupoids over $\mathcal{C}$, then so is
+$\mathcal{X}\times_\mathcal{S}\mathcal{Y}$. In particular the 2-category
+of categories fibred in groupoids over $\mathcal{C}$ has 2-fibre products
+(and they are described as above).
+\end{lemma}
+
+\begin{proof} 
+FIXME.
+\end{proof}
+
 \subsection{Categories fibred in sets}
 \label{subsection-fibred-in-sets}
 
@@ -956,8 +1026,8 @@
 
 \begin{definition}
 \label{definition-category-fibred-sets}
-A category fibred in sets $p : \mathcal{S} \to \mathcal{C}$
-is a category fibred in sets if all fibre categories are discrete.
+A category fibred in groupoids $p : \mathcal{S} \to \mathcal{C}$ is said
+to be a category fibred in sets if all fibre categories are discrete.
 \end{definition}
 
 \noindent
@@ -973,20 +1043,55 @@
 as a presheaf on $\mathcal{C}$.
 
 \smallskip\noindent
-Conversely, given a presheaf of sets 
+Conversely, given a presheaf of sets
 $F : \mathcal{C}^{\text{opp}} \to \text{Sets}$
 we can construct a category $\mathcal{S}_F$ fibred in sets
 over $\mathcal{C}$ by taking as fibre category $\mathcal{S}_{F,U}$ 
 the discrete category whose underlying set is $F(U)$. This is explained
 more generally, and in more detail in Example \ref{example-functor-groupoids}
-below. 
+below. Also, here is an important example.
+
+\begin{example}
+\label{example-fibred-category-from-functor-of-points}
+In this example $F = h_X = \text{Mor}(-,X)$ for some
+$X \in \text{Ob}(\mathcal{C})$ (see Example \ref{example-hom-functor}).
+In other words, $F$ is a representable presheaf.
+Since $\mathcal{S}_{F,U}$ is the discrete category whose objects are the
+morphisms from $U$ into $X$ it follows that
+$\mathcal{S}_F\to \mathcal{C}$ is the functor denoted
+$\mathcal{C}/X \to \mathcal{C}$ from
+Example \ref{example-comma-category}.
+FIXME. Improve formulation.
+\end{example}
+
+\smallskip\noindent
+For this reason it is tempting to define a ``representable'' object in the
+2-category of categories fibred in groupoids to be a category fibred in
+sets whose associated presheaf is representable. However, this is would not
+be a good definition since we prefer to have a notion wich is invariant under
+equivalences. Thus we consider first which categories in groupoids are
+equivalent to categories fibred in sets.
 
 \begin{lemma}
+\label{lemma-setlike-fibres}
 Suppose that $p : \mathcal{S} \to \mathcal{C}$ is a category fibred in
 groupoids all of whose fibre categories $\mathcal{S}_U$ are setlike. 
 Then there exists a category fibred in sets $p' : \mathcal{S}' \to
-\mathcal{C}$ and an equivalence $\mathcal{S} \to \mathcal{S}'$
-of categories over $\mathcal{C}$.
+\mathcal{C}$ and an equivalence
+$\text{can}:\mathcal{S} \to \mathcal{S}'$ of categories over $\mathcal{C}$.
+The 1-morphism $\mathcal{S}\to\mathcal{S}'$ is unique up to a unique
+2-morphism. It further has the property that
+$$
+\text{Ob}(\mathcal{S}_U) \longrightarrow \text{Ob}(\mathcal{S}'_U) 
+$$
+(induced by $\text{can}$) identifies the RHS with ismorphism classes of the
+LHS for all $U \in \text{Ob}(\mathcal{C})$. The 1-morphism
+$\mathcal{S}\to\mathcal{S}'$ is unique up to a unique 2-morphism. 
+
+\smallskip\noindent
+Conversely, any category fibred in groupoids over $\mathcal{C}$ which
+is equivalent (as a category over $\mathcal{C}$) to a category fibred 
+in sets, has setlike fibre categories.
 \end{lemma}
 
 \begin{proof}
@@ -1004,8 +1109,44 @@
 FIXME: check this is well-defined. 
 
 \smallskip\noindent
-By construction the rule $(U,\xi) \mapsto U$ is a functor. FIXME: check the
-other properties.
+By construction the rule $(U,\xi) \mapsto U$ is a functor. FIXME: check this
+and the other properties.
+\end{proof}
+
+\noindent
+With this lemma in hand it is easy to recognize those categories over
+$\mathcal{C}$ which are equivalent to a category fibred in sets. Thus we
+now make the following definition.
+
+\begin{definition}
+\label{definition-representable-fibred-category}
+A category fibred in groupoids $p : \mathcal{S} \to \mathcal{C}$ is
+called representable, if the following conditions are satisfied:
+\begin{enumerate}
+\item all fibre categories $\mathcal{S}_U$ are setlike, and
+\item the presheaf $U \mapsto \text{Ob}(\mathcal{S}_U)/\cong$ is 
+representable.
+\end{enumerate}
+\end{definition}
+
+\noindent
+In this case, by Lemma \ref{lemma-setlike-fibres} the category 
+$\mathcal{S}'$ is isomorphic to $\mathcal{C}/X$ over $\mathcal{C}$.
+As usual, by the Yoneda lemma the pair $(X,j)$, where $j$ is the
+equivalence $j : \mathcal{S} \to \mathcal{C}/X$ is uniquely determined
+up to isomorphism.
+
+\begin{lemma}
+\label{lemma-2-product-categories-fibred-sets}
+The 2-category of categories fibred in sets over $\mathcal{C}$
+has 2-fibre products. More precisely, the 2-fibre product described in 
+Lemma \ref{lemma-2-product-categories-over-C} returns a category fibred in
+sets if one starts out with such. A similar result holds for categories
+fibred in groupoids all of whose fibre categories are setlike.
+\end{lemma}
+
+\begin{proof}
+FIXME.
 \end{proof}
 
 \subsection{Presheaves of groupoids}
@@ -1066,15 +1207,6 @@
 is a {\it split} category fibred in groupoids.
 \end{example}
 
-\begin{example}
-\label{example-fibred-category-from-functor-of-points}
-When $F=\text{Mor}(-,X)$ for some $X \in \text{Ob}(\mathcal{C})$,
-$\mathcal{S}_F\to \mathcal{C}$ is the category 
-$\mathcal{C}/X \to \mathcal{C}$ from Example \ref{example-comma-category}.
-\end{example}
-
-
-
 \begin{lemma} 
 \label{lemma-fibred-strict}
 Let $ p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids.
diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/desirables.tex stacks-0.2/src/desirables.tex
--- stacks-0.2.orig/src/desirables.tex	2005-10-17 14:59:55.000000000 +0000
+++ stacks-0.2/src/desirables.tex	2006-03-20 01:48:12.000000000 +0000
@@ -162,8 +162,8 @@
 \noindent
 Do a little bit of theory here. Talk about sheaves, morphisms of sites.
 The category of sheaves on a site now means all sheaves with values in
-$\text{Sets}_\alpha$ where $\alpha$ is suitably large (relative to the
-site).
+$V_\alpha = \text{Sets}_\alpha$ where $\alpha$ is suitably large (relative
+to the site).
 
 Introduce the notion of topos and morphism of topoi. The notion of 
 simplicial and strictly simplicial topos.
diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/injectives.tex stacks-0.2/src/injectives.tex
--- stacks-0.2.orig/src/injectives.tex	2005-09-27 18:03:08.000000000 +0000
+++ stacks-0.2/src/injectives.tex	2006-03-20 01:48:12.000000000 +0000
@@ -116,11 +116,11 @@
 
 \noindent
 Grothendieck proved the existence of injectives in great generality 
-in the paper \cite{Tohoku}. We will prove this is true for the category
-of (pre)sheaves on a site.
+in the paper \cite{Tohoku}. We will prove this is true for abelian
+(pre)sheaves on a site.
 
-\subsection{Algebra}
-\label{subsection-injectives-algebra}
+\subsection{Modules}
+\label{subsection-injectives-modules}
 
 \noindent 
 As an example theorem let us try to prove that there are enough injective
@@ -163,6 +163,43 @@
 $$
 This the kind of construction we would like to have in general.
 
+\subsubsection{Categories of modules}
+\label{subsubsection-category-modules}
+
+\noindent
+As a consequence we obtain a category of modules with a canonical
+resolution. For an ordinal $\alpha$ we denote $\text{Mod}_{R,\alpha}$
+the category of modules contained in $V_\alpha$ (see Sets, 
+\hyperref[sets-subsection-sets-hierarchy]%
+{Subsection~\ref*{sets-subsection-sets-hierarchy}}).
+
+\begin{lemma}
+\label{lemma-injective-module-preserves-category}
+For any given set of $R$-modules $\{M_i\}_{i\in I}$ there exists an ordinal
+$\alpha$ such that $M_i \in \text{Ob}(\text{Mod}_{R,\alpha})$,
+$\forall i\in I$ and such that for any
+$M \in \text{Ob}(\text{Mod}_{R,\alpha})$ we have
+$J(M) \in \text{Ob}(\text{Mod}_{R,\alpha})$.
+\end{lemma}
+
+\begin{proof}
+Consider the formula $\phi(M)$: ``$M$ is an $R$-module and there exists
+an $R$-module $N$ such that $N=J(M)$''. Apply the reflection principle to
+$\phi(M)$, see 
+\hyperref[sets-theorem-reflection-principle]%
+{Theorem~\ref*{sets-theorem-reflection-principle}}. (Use $T = \{M_i\}$.)
+The result follows.
+\end{proof}
+
+\noindent
+Some remarks are in order. First we observe that the modules $J(M)$
+are injective in the absolute sense, and not only injective in the
+category $\text{Mod}_{R,\alpha}$. Second, in exactly the same way we
+can make sure that $\text{Mod}_{R,\alpha}$ has all finite limits,
+finite direct sums, or countable sums and products, etc. Of course
+the category $\text{mod}_{R,\alpha}$ never has arbitrary direct sums,
+which is why working with $\text{mod}_{R,\alpha}$ is somewhat cumbersome.
+
 \subsubsection{Projective resolutions}
 \label{subsubsection-projective-resolution}
 
@@ -187,19 +224,18 @@
 \label{subsection-injectives-presheaves}
 
 \noindent
-Let $\mathcal{C}$ be a category. Consider the category of abelian
-presheaves $\text{Ab}(\mathcal{C})$.
-On the other hand, denote $\text{Ab}(\text{Ob}(\mathcal{C}))$
-the category of families of abelian groups indexed by elements of
-$\text{Ob}(\mathcal{C})$. We will denote a typical object
-of $\text{Ab}(\mathcal{C}))$ by $B$ and a typical object of
-$\text{Ab}(\text{Ob}(\mathcal{C}))$ by $A$. Consider the forgetful 
-functor $v : \text{Ab}(\mathcal{C}) \to 
-\text{Ab}(\text{Ob}(\mathcal{C}))$, denoted $B \mapsto vB$.
+Let $\mathcal{C}$ be a category. On the one hand, consider abelian
+presheaves on $\mathcal{C}$. On the other hand, consider
+families of abelian groups indexed by elements of
+$\text{Ob}(\mathcal{C})$; in other words presheaves on the discrete
+category with underlying set of objects $\text{Ob}(\mathcal{C})$.
+We will denote presheaves on $\mathcal{C}$ by $B$ and presheaves on
+$\text{Ob}(\mathcal{C})$ by $A$. Consider the forgetful functor $v$,
+denoted $B \mapsto vB$.
 
 \smallskip\noindent
-There is a functor $u : \text{Ab}(\text{Ob}(\mathcal{C})) \to
-\text{Ab}(\mathcal{C})$ defined as follows:
+There is a functor $u$ that assigns a presheaf on $\mathcal{C}$
+to a presheaf on $\text{Ob}(\mathcal{C})$. It is defined as follows:
 $$
 \Gamma(U, uA) = \prod\nolimits_{U' \to U} A(U').
 $$
@@ -213,54 +249,78 @@
 There is a canonical surjective map $vuA \to A$ and a canonical map
 injective map $B \to uvB$. We leave it to the reader to show that
 $$
-\text{Mor}_{\text{Ab}(\text{Ob}(\mathcal{C}))}(B, uA) =
-\text{Mor}_{\text{Ab}(\mathcal{C})}(vB, A).
+\text{Mor}_{\text{PAb}(\text{Ob}(\mathcal{C}))}(B, uA) =
+\text{Mor}_{\text{PAb}(\mathcal{C})}(vB, A).
 $$
-Thus the pair $(u,v)$ is an example of a pair of adjoint
-functors. FIXME: Discuss this somewhere.
-It is clear that $u$ and $v$ are exact functors. It is clear that
-$\text{Ab}(\text{Ob}(\mathcal{C}))$ has enough injectives. 
-In fact there is a functor $J$ on this category such that
-$A \to J(A)$ is functorial as in \autoref{subsection-injectives-algebra}.
+(Obvious notation.) Thus the pair $(u,v)$ is an example of a pair of adjoint
+functors. FIXME: Discuss this somewhere. It is clear that $u$ and $v$ are exact functors. It is clear that any presheaf on $\text{Ob}(\mathcal{C})$ has an
+injective hull. In fact there is a functor $J$ such that
+$A \mapsto \big(A \to J(A)\big)$ is functorial as in
+\autoref{subsection-injectives-modules}.
+(Namely, $J(A)$ is the assignment $U\mapsto J(A(U))$, where
+$J(A(U))$ is the functor constructed in
+Subsection \ref{subsection-injectives-modules} for the ring $\mathbf{Z}$
+applied to the $\mathbf{Z}$-module $A(U)$.)
 
 \smallskip\noindent
 Putting all of this together gives us the following procedure
-for embedding objects $B$ of $\text{Ab}(\mathcal{C}))$ into
+for embedding objects $B$ of $\text{PAb}(\mathcal{C}))$ into
 an injective object: $B \to uJ(vB)$.
 
 \begin{proposition}
 \label{proposition-presheaves-injectives}
-The category of abelian presheaves on a category has
-enough injectives (in the strongest possible sense).
+For abelian presheaves on a category there is a functorial injective hull.
 \end{proposition}
 
+\subsubsection{Categories of presheaves of abelian groups}
+\label{subsubsection-category-presheaves}
+
+\noindent
+Arguing as in the proof of
+Lemma \ref{lemma-injective-module-preserves-category} we obtain a category
+with an injective resolution functor. For any ordinal $\alpha$,
+we use the notation $\text{PAb}(\mathcal{C})_\alpha$ to denote the category
+of presheaves $\mathcal{F}$ of abelian groups with $\mathcal{F} \in V_\alpha$.
+See Sets, \hyperref[sets-subsection-sets-hierarchy]%
+{Subsection~\ref*{sets-subsection-sets-hierarchy}}.
+
+\begin{lemma} 
+\label{lemma-injective-presheaf-preserves-category}
+Given any set of abelian presheaves $\mathcal{F}_i$, $i\in I$, there
+exists an ordinal $\alpha$ such that $\text{PAb}(\mathcal{C})_\alpha$
+contains all of the $\mathcal{F}_i$, and such that there is a functor
+$\text{PAb}(\mathcal{C})_\alpha \to
+\text{Arrows}(\text{PAb}(\mathcal{C})_\alpha)$
+of the form $\mathcal{F} \mapsto (\mathcal{F} \to J(\mathcal{F}))$
+with the property that $\mathcal{F} \to J(\mathcal{F})$ is an injective
+hull for all $\mathcal{F} \in \text{PAb}(\mathcal{C})_\alpha$.
+\end{lemma}
+
+\begin{proof}
+FIXME. Very similar to the corresponding lemma for modules.
+\end{proof}
+
 \subsection{Abelian Sheaves}
 \label{subsection-injectives-sheaves}
 
 \noindent
 Let $\mathcal{C}$ be a site. In this section we prove that there are 
-enough injectives in the category $\text{Ab}(\mathcal{C})$ of abelian 
-sheaves on $\mathcal{C}$. 
+enough injectives for abelian sheaves on $\mathcal{C}$. 
 
 \smallskip\noindent
-Let us denote $\text{PAb}(\mathcal{C})$ the category of presheaves of
-abelian groups. Denote $i : \text{Ab}(\mathcal{C}) 
-\to \text{PAb}(\mathcal{C})$ the inclusion functor and let
-$\# : \text{PAb}(\mathcal{C}) \to \text{Ab}(\mathcal{C})$
-denote the sheafification functor, see FIXME. 
-In this subsection we will use that 
-$i(\mathcal{F})^\# = \mathcal{F}$, see FIXME.
-Finally, let
-$ \mathcal{F} \to J(\mathcal{F})$ denote the canonical
+Denote $i$ the forgetfull functor from sheaves to presheaves. Let
+$\#$ denote the sheafification functor, see FIXME. In this subsection we
+will use that $i(\mathcal{F})^\# = \mathcal{F}$, see FIXME.
+Finally, let $\mathcal{F} \to J(\mathcal{F})$ denote the canonical
 embedding into an injective presheaf we found in 
-\autoref{subsection-injectives-sheaves}. 
+\autoref{subsection-injectives-presheaves}. 
 
 \smallskip\noindent
 For any sheaf $\mathcal{F}$ in $\text{Ab}(\mathcal{C})$ and
 any ordinal $\beta$ we define a sheaf
-$J_\beta(\mathcal{F}) \in \text{Ab}(\mathcal{C})$ 
-by transfinite induction. FIXME: explain transfinite induction in 
-\url{src/sets.tex}. First we set $J_0(\mathcal{F})=\mathcal{F}$.
+$J_\beta(\mathcal{F})$ by transfinite induction.
+FIXME: explain transfinite induction in \url{src/sets.tex}.
+First we set $J_0(\mathcal{F})=\mathcal{F}$.
 We define $J_1(\mathcal{F})=J(i(\mathcal{F}))^\#$;
 there is a map $\mathcal{F}=i(\mathcal{F})^\# \to J(i\mathcal{F})^\#$
 by functoriality of $\#$. This map $\mathcal{F} \to J_1(\mathcal{F})$
@@ -314,8 +374,10 @@
 FIXME. Hint: First suppose that $T = \lim_{\alpha < \beta} T_\alpha$
 is a limit of sets and that $\varphi : S \to T$ is a map of sets. 
 Then $\varphi$ lifts to a map into $T_\alpha$ for some $\alpha < \beta$
-provided $S$ is smaller than $\alpha$. Use this and some argument
-for equalizers to get through.
+provided that $\beta$ is not a limit of ordinals indexed by $S$.
+In other words, you pick $\beta$ to be a singular cardinal with
+$cf(\beta)$ bigger than the cardinality of $S$. Reference? Use this and
+some argument for equalizers to get through.
 \end{proof}
 
 \noindent
@@ -346,7 +408,54 @@
 \end{theorem}
 
 \begin{proof}
-FIXME.
+FIXME. Idea: Let $\mathcal{G}_i$, $i\in I$ be a set of abelian
+sheaves such that every subsheaf of every $\mathbf{Z}_X^\#$
+occurs as one of the $\mathcal{G}_i$. Apply
+Lemma \ref{lemma-map-into-smaller} to this collection to
+get an ordinal $\beta$. We claim that for any sheaf of abelian
+groups $\mathcal{F}$ the map $\mathcal{F} \to J_\beta(\mathcal{F})$
+is an injection of $\mathcal{F}$ into an injective.
+Note that by construction the assigment $\mathcal{F} \mapsto
+\big(\mathcal{F} \to J_\beta(\mathcal{F})\big)$ is functorial.
+
+\smallskip\noindent
+The proof of the claim comes from the fact that by
+Lemma \ref{lemma-characterize-injectives} it suffices to extend any
+morphism $\gamma : \mathcal{G} \to J_\beta(\mathcal{F})$ 
+from a subsheaf $\mathcal{G}$ of some $\mathbf{Z}_X^\#$ to all of
+$\mathbf{Z}_X^\#$. Then by Lemma \ref{lemma-map-into-smaller} the
+map $\gamma$ lifts into $J_\alpha(\mathcal{F})$ for some
+$\alpha < \beta$. Finally, we apply Lemma \ref{lemma-map-into-next-one}
+to get the desired extension of $\gamma$ to a morphism
+into $J_{\alpha+1}(\mathcal{F}) \to J_\beta(\mathcal{F})$.
+\end{proof}
+
+\subsubsection{Categories of abelian sheaves}
+\label{subsubsection-abelian-sheaves}
+
+\noindent
+Again we obtain a result concerning the existence of a category 
+preserved by the functorial assigment $\mathcal{F} \mapsto
+\big(\mathcal{F} \to J_\beta(\mathcal{F})\big)$ described in
+Theorem \ref{theorem-sheaves-injectives}. As is usual, for an
+ordinal $\alpha$, we denote $\text{Ab}(\mathcal{C})_\alpha$ the
+category of abelian sheaves on $\mathcal{C}$ which are elements
+of $V_\alpha$.
+
+\begin{lemma}
+\label{lemma-injective-sheaf-preserves-category}
+Given any set of abelian sheaves $\mathcal{F}_i$, $i\in I$, there
+exists an ordinal $\alpha$ such that $\text{Ab}(\mathcal{C})_\alpha$
+contains all of the $\mathcal{F}_i$, and such that there is a functor
+$\text{Ab}(\mathcal{C})_\alpha \to
+\text{Arrows}(\text{Ab}(\mathcal{C})_\alpha)$
+of the form $\mathcal{F} \mapsto (\mathcal{F} \to J(\mathcal{F}))$
+with the property that $\mathcal{F} \to J(\mathcal{F})$ is an injective
+hull for all $\mathcal{F} \in \text{Ab}(\mathcal{C})_\alpha$.
+\end{lemma}
+
+\begin{proof}
+FIXME. Very similar to the corresponding lemma for modules.
 \end{proof}
 
 \section{Grothendieck categories and injectives}
diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/schemes.tex stacks-0.2/src/schemes.tex
--- stacks-0.2.orig/src/schemes.tex	1970-01-01 00:00:00.000000000 +0000
+++ stacks-0.2/src/schemes.tex	2006-03-20 01:48:12.000000000 +0000
@@ -0,0 +1,482 @@
+\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/schemes.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[categories-]{categories}
+\externaldocument[stacks-]{stacks}
+\externaldocument[sets-]{sets}
+
+% 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{Schemes as stacks and representability}
+
+%\begin{abstract}
+%\end{abstract}
+
+\maketitle
+\thispagestyle{fancy}
+
+\tableofcontents
+
+\section{Introduction}
+\label{section-introduction}
+
+\noindent
+In this document we explain how we will think of schemes as stacks over the
+category of affine schemes.
+
+\section{Affine schemes, schemes, stacks representable by a scheme}
+\label{section-schemes}
+
+\noindent
+You can skip the first two subsections for sure.
+
+\subsection{Locally ringed spaces}
+\label{subsection-locally-ringed-sapces}
+
+\noindent
+A locally ringed space $(X,\mathcal{O}_X)$ is a pair consisting of a
+topological space $X$ and a sheaf of rings $\mathcal{O}_X$ all of whose stalks
+are local rings. Morphisms in the category of locally ringed spaces are
+maps of pairs $f : (X, \mathcal{O}_X) \to (Y,\mathcal{O}_Y)$ so that
+all the induced ring maps $\mathcal{O}_{Y,f(x)} \to \mathcal{O}_{X,x}$ are
+local ring maps.
+
+\smallskip\noindent
+A reference for this section is \cite{EGA}, I.
+
+\subsection{Affine schemes}
+\label{subsection-affine-schemes}
+
+\noindent
+An affine scheme is a locally ringed space isomorphic to a locally ringed
+space of the form $\text{Spec}(A)$, for some commutative (unital) ring $A$.
+(Note that $A$ can be the zero ring in which case $\text{Spec}(A)$ is
+the empty space.) As a set $\text{Spec}(A)$ is the set of prime ideals of
+$A$. The topology on $\text{Spec}(A)$ is the unique one that has a basis
+of opens of the form $D(f) = \{ \wp \in\text{Spec}(A) \mid
+f \not\in \wp \}$,
+$f\in A$. The structure sheaf $\mathcal{O} = 
+\mathcal{O}_{\text{Spec}(A)}$ is the unique
+sheaf of rings such that (1) $\Gamma(D(f), \mathcal{O}) = A_f$ and
+(2) the restriction map $\Gamma(D(f), \mathcal{O}) \to \Gamma(D(fg),
+\mathcal{O})$ is the canonical map $A_f \to A_{fg}$.
+
+\smallskip\noindent
+A morphism of affine schemes is a morphism in the category of locally ringed 
+spaces.
+
+\subsection{The category of affine schemes}
+\label{subsection-affine-schemes}
+
+\noindent
+It should be clear what the category of affine schemes is, except for a
+little bit of set-theoretical discussion. We will use the notation
+$\text{Aff}$ to denote this category. Our approach is to use only
+categories which are sets. Thus we will choose a supply of affines and
+work with this. For a precise mathematical discussion, see
+Subsection \ref{subsection-sets-of-affines}.
+
+\smallskip\noindent
+The topology on $\text{Aff}$ will be the fppf topology. A covering is
+given by a finite family of maps $\{U_i \to U\}$, where each $U_i \to U$
+is a finitely presented flat morphism of affines, and $\coprod U_i \to U$
+is surjective. 
+
+\smallskip\noindent
+Sometimes we consider $\text{Aff}$ with other topologies, such as the
+etale, Zariski, or fpqc topologies. Notation $\text{Aff}_{etale}$, etc.
+FIXME. Put in internal reference to topology discussion.
+
+\subsubsection{Sets of affine schemes}
+\label{subsection-sets-of-affines}
+
+\noindent
+Choose an ordinal $\alpha$ and denote $\text{Aff}_\alpha$ the
+category of affine schemes which are elements of $V_\alpha$, see
+Sets,\hyperref[sets-subsection-sets-hierarchy]%
+{Subsection~\ref*{sets-subsection-sets-hierarchy}}. So there is a
+theory of algebraic stacks for any $\alpha$. There are some minimal
+conditions on $\alpha$ needed to imply that $\text{Aff}_\alpha$ is a site. 
+These minimal required properties are expressed in the following lemma.
+
+\begin{lemma}
+\label{lemma-Aff-site}
+For any set $S$ may choose an ordinal $\alpha$ with $S \in V_\alpha$ 
+such that $\text{Aff}_\alpha$ has (finite) fibre products, and finite disjoint
+unions. In addition we may assume that for any finitely presented morphism
+of affines $X \to Y$, such that $Y \in \text{Aff}_\alpha$, there exists
+an affine $X' \in \text{Ob}(\text{Aff}_\alpha)$ such that $X' \cong X$.
+\end{lemma}
+
+\begin{proof}
+Consider the following statement: ``For any finite directed graph $\Gamma$,
+for any assignment $v \mapsto F(v)$, $\forall v\in \text{Vertices}(\Gamma)$,
+where $F(v)$ is an affine scheme, and any assignment
+$\big(e : v_1 \to v_2\big) \mapsto \big(F(e) : F(v_1) \to F(v_2)\big)$,
+$\forall e \in \text{Edges}(\Gamma)$ where $F(e)$ is a morphism of affine
+schemes, there exists an affine scheme $X$ and morphisms $f(v) : X \to F(v)$,
+$\forall v\in \text{Vertices}(\Gamma)$ such that $f(v_2) = F(e) \circ f(v_1)$,
+$\forall \big(e : v_1 \to v_2\big) \in \text{Edges}(\Gamma)$, such that
+$(X, \{f(v)\}_{v\in \text{Vertices}(\Gamma)})$ is universal among all such.''
+This statement says that finite limits exist for affine schemes. It is
+proved in a standard way (for example by turning it into ring theory).
+
+\smallskip\noindent
+On the other hand, upon formalizing the statement we obtain a provable
+formula $\phi(\Gamma, F)$ of ZFC set theory. Hence, according to the reflection
+principle, see Sets, \hyperref[sets-section-reflection-principle]%
+{Lemma~\ref*{sets-section-reflection-principle}}
+there exists an ordinal $\alpha$ such that the formula is true in
+$V_\alpha$: If you take $\Gamma \in V_\alpha$ and the $F(v)$ to be in
+$\text{Aff}_\alpha$, then you can find a solution
+$(X, \{f(v)\}_{v\in \text{Vertices}(\Gamma)})$
+with $X$ in $V_\alpha$. This takes care of the statement about fibre 
+products. (Of course as soon as $\alpha$ is infinite then every
+graph is isomorphic to a graph in $V_\alpha$; we can also simply apriori
+require this for $V_\alpha$.).
+
+\smallskip\noindent
+We can similarly write out the condition of the existence of disjoint unions
+as a set theory formula, and similarly the existence of the affine $X'$
+given $X \to Y$. The reflection principle states we can have $S$ inside of
+$V_\alpha$ as well.
+\end{proof}
+
+\noindent
+Clearly, we may assume that $\text{Aff}_\alpha$ is closed under any reasonable
+operation (see Sets, \autoref{sets-section-reflection-principle}).
+Of course, whenever we require such a condition we will need to write out
+the proof that this is so.
+
+\smallskip\noindent
+So, in the following we will work with stacks (or categories) over
+$\text{Aff}_\alpha$\footnote{As per our general philosophy, if we ever need
+an actual 2-category of stacks, we also choose another cardinal $\gamma$ and
+consider only those categories over $\text{Aff}_\alpha$ contained in
+$V_\gamma$.}. If $\alpha < \beta$, then there is an inclusion
+$\text{Aff}_\alpha \subset \text{Aff}_\beta$, and hence any category
+over $\text{Aff}_\beta$ gives rise to a category over $\text{Aff}_\alpha$.
+But this is not the correct thing to do when studying algebraic stacks.
+Instead we want to show that algebraic stacks over $\text{Aff}_\alpha$
+give rise to algebraic stacks over $\text{Aff}_\beta$. In other words we will
+need a theorem saying that the 2-category of algebraic
+stacks over $\text{Aff}_\alpha$ is equivalent to a full sub-2-category of
+algebraic stacks over $\text{Aff}_\beta$. Here it is.
+
+\smallskip\noindent
+FIXME. Improve the theorem below and move it to a more appropriate spot.
+
+\begin{theorem}
+\label{theorem-change-alpha}
+Suppose that $p : \mathcal{S} \to \text{Aff}_\alpha$ is an algebraic stack.
+Let $\beta > \alpha$. Then there exists an algebraic stack
+$p' : \mathcal{S}' \to \text{Aff}_\beta$ and an equivalence
+$$
+\xymatrix{
+(p')^{-1}(\text{Aff}_\alpha) \ar[rd]_{p'} \ar[rr]^c && \mathcal{S}\ar[ld]^p\\
+&\text{Aff}_\alpha.&}
+$$
+The pair $((\mathcal{S'},p'),c)$ is well determined up to a 1-isomorphism
+(which is itself unique up to unique 2-isomorphism).
+\end{theorem}
+
+\begin{proof}
+FIXME. Hint. Choose a representation (in $\text{Stacks}/\text{Aff}_\alpha$)
+$\mathcal{S} = [ \mathcal{X}/\mathcal{R} ]$, with $\mathcal{X}$ representable
+by a scheme $X$ and $\mathcal{R}$ representable by an algebraic space.
+Choose a presentation $\mathcal{R} = [ \mathcal{U}/\mathcal{R}_\mathcal{U} ]$
+where $\mathcal{U}$ and $\mathcal{R}_\mathcal{U}$ are representable
+by schemes $U$ and $R_U$. Now define (in $\text{Stacks}/\text{Aff}_\beta$)
+$\mathcal{U}'$ to be the stack associated to $U$, $\mathcal{R}'_\mathcal{U}$
+to be the stack associated to $R_U$, $\mathcal{R}'$ the stack
+$\mathcal{R}' = [ \mathcal{U}'/\mathcal{R}'_\mathcal{U} ]$, $\mathcal{X}'$
+the stack associated to $X$, and finally
+$\mathcal{S}' = [ \mathcal{X}'/\mathcal{R}' ]$.
+\end{proof}
+
+\noindent
+From now on $\text{Aff}$ will denote a category of affines $\text{Aff}_\alpha$
+such as in Lemma \ref{lemma-Aff-site}. By the theorem above we may increase
+$\alpha$ whenever this is needed.
+
+\begin{remark}
+\label{remark-other-approach}
+There is another approach. Allow yourself to enlarge $\alpha$ at any moment.
+Think of every statement in the text as being preceded by ``There exist
+arbitrarily large $\alpha$ such that''. 
+\end{remark}
+
+\subsection{Schemes}
+\label{subsection-schemes}
+
+\noindent
+We recall the definition of a scheme.
+
+\smallskip\noindent
+A scheme $(X,\mathcal{O}_X)$ is a locally ringed space
+with the property that every point has a neighbourhood which is an
+affine scheme.
+
+\smallskip\noindent
+A scheme $X$ gives rise to a functor (or presheaf)
+$$
+\xymatrix{
+\text{Aff}^{\text{opp}} \ar[r]^{h_X} & \text{Sets}, &
+U \ar@{|->}[r] & \text{Mor}(U, X).}
+$$
+The usual Yoneda lemma tells us that we can recover the scheme from this
+functor. 
+
+\begin{lemma}
+\label{lemma-yoneda-schemes}
+Suppose that $X$, $Y$ are schemes with that have open coverings
+by affines isomorphic to objects of $\text{Aff}$. Then $\text{Mor}(X,Y)
+= \text{Mor}(h_X, h_Y)$.
+\end{lemma}
+
+\begin{proof}
+FIXME.
+\end{proof}
+
+\subsection{Stacks representable by a scheme}
+\label{subsection-stack-representable-by-scheme}
+
+\noindent
+In Categories, \hyperref[categories-definition-representable-fibred-category]%
+{Definition~\ref*{categories-definition-representable-fibred-category}} we
+defined the notion of a representable category fibred in groupoids. This,
+applied to a stack (or a category) over $\text{Aff}$ will define the notion of
+a stack representable by an affine scheme. 
+
+\smallskip\noindent
+Here is the formal definition of a category over $\text{Aff}$ representable by
+a scheme. Please also see the informal discussion below.
+
+\begin{definition}
+\label{definition-representable-by-scheme}
+A category fibred in groupoids $p : \mathcal{S} \to \text{Aff}$ is
+called representable by a scheme, if the following conditions are satisfied:
+\begin{enumerate}
+\item all fibre categories $\mathcal{S}_U$ are setlike, and
+\item the presheaf $U \mapsto \text{Ob}(\mathcal{S}_U)/\cong$ is 
+is isomorphic to $h_S$ for a scheme $S$ as in
+Lemma \ref{lemma-yoneda-schemes}.
+\end{enumerate}
+\end{definition}
+
+\begin{lemma}
+\label{lemma-representable-by-scheme-implies-stack}
+If $\mathcal{S} \to \text{Aff}$ is representable by a scheme then $\mathcal{S}$
+is a stack over $\text{Aff}$.
+\end{lemma}
+
+\begin{proof}
+FIXME.
+\end{proof}
+
+\begin{example}
+\label{example-standard-representable-scheme}
+Let $X$ be a scheme that has a covering by open affines which are isomorphic
+to objects of $\text{Aff}$. There is a standard stack over $\text{Aff}$
+representable by $X$, namely the stack of affines over $X$. Compare Categories,
+\hyperref[categories-example-comma-category]%
+{Example~\ref*{categories-example-comma-category}}.
+This stack will be denoted $\text{Aff}/X$, and it is described as follows.
+\begin{enumerate}
+\item An object of $\text{Aff}/X$ is a morphism of schemes
+$U \to X$, with $U \in \text{Ob}(\text{Aff})$.
+\item A morphism between $U\to X$ and $V \to X$ is a commutative diagram
+$$
+\xymatrix{
+U \ar[rr] \ar[rd] && V \ar[ld] \\
+&X.&}
+$$
+\item The functor $(\text{Aff}/X) \to \text{Aff}$ maps $U\to X$ to $U$.
+\end{enumerate}
+It is clear from the definition that $\text{Aff}/X$ is representable by
+a scheme. 
+
+\smallskip\noindent
+The construction is clearly functorial in $X$, so that a morphism
+of schemes $f : X \to Y$ induces a morphisms of stacks 
+$\text{Aff}/X \to \text{Aff}/Y$. FIXME: more?
+\end{example}
+
+\begin{situation}
+\label{situation-stack-represented-by-scheme}
+The following situation will appear repeatedly in the text. Suppose that
+$\mathcal{S} \to \text{Aff}$ is a stack representable by a scheme. If we
+say the scheme $S$ represents $\mathcal{S}$, then we mean that besides 
+being given the scheme $S$, we are given an equivalence $j : \mathcal{S}
+\to \text{Aff}/S$ of stacks over $\text{Aff}$.
+\end{situation}
+
+\begin{lemma}
+\label{lemma-morphism-stacks-representable-by-schemes}
+Suppose that the stacks $\mathcal{X}$, $\mathcal{Y}$ are represented
+by the schemes $X$ and $Y$. For any morphism of stacks $F : \mathcal{X}
+\to \mathcal{Y}$ there is a unique morphism of schemes $f : X \to Y$
+such that the diagram
+$$
+\xymatrix{
+\mathcal{X} \ar[r]^F \ar[d]_j & \mathcal{Y} \ar[d]^j \\
+\text{Aff}/X \ar[r]^f & \text{Aff}/Y}
+$$
+2-commutes and then the diagram actually commutes.
+\end{lemma}
+
+\begin{proof}
+FIXME.
+\end{proof}
+
+\section{Morphisms representable by schemes}
+\label{section-morphisms-representable-by-schemes}
+
+\noindent
+In this section we define the notion of moprhisms of stacks over $\text{Aff}$
+representable by schemes.
+
+\subsection{Definition}
+\label{subsection-definition-representable-by-schemes}
+
+\noindent
+Here is the formal definition. Please also see the informal discussion below.
+
+\begin{definition}
+\label{definition-representable-by-schemes}
+Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of categories
+fibred in groupoids over $\text{Aff}$. We say $f$ is representable by
+schemes if for every stack $\mathcal{S}$ representable by a scheme
+(see Definition \ref{definition-representable-by-scheme}), and every morphism
+$\mathcal{U} \to \mathcal{Y}$, the 2-fibre product
+$\mathcal{S}\times_\mathcal{Y}\mathcal{X}$ is representable by a scheme.
+\end{definition}
+
+\noindent
+Informal discussion. In the situation of the definition we sometimes 
+say that $\mathcal{X}$ is relatively representable over $\mathcal{Y}$.
+Suppose that, with the notation of the definition, $S$ represents
+$\mathcal{S}$ and $W$ represents $\mathcal{S}\times_\mathcal{Y}\mathcal{X}$.
+According to Lemma \ref{lemma-morphism-stacks-representable-by-schemes}
+we get a morphism of schemes $g : W \to S$ and a 2-commutative diagram
+of stacks
+$$
+\xymatrix{
+\text{Aff}/W \ar[d]^g &
+\mathcal{S}\times_\mathcal{X}\mathcal{Y} \ar[d] \ar[l]^j \ar[r] &
+\mathcal{Y} \ar[d] \\
+\text{Aff}/S &
+\mathcal{S} \ar[l]^j \ar[r] & \mathcal{X}
+}
+$$
+FIXME: more.
+
+\smallskip\noindent
+FIXME. It seems to me that you can define the notion even if 
+$\mathcal{X}$ and $\mathcal{Y}$ are just categories over $\text{Aff}$. Does
+it make sense in this generality?
+
+\begin{definition}
+\label{definition-property-morphism-representable-by-schemes}
+Let $P$ be a property of morphisms of schemes such that
+if the morphism $f : X \to Y$ has property $P$, then so does
+every base change of $f$. (FIXME: introduce base change.)
+We say that a morphism of stacks $\mathcal{X}
+\to \mathcal{Y}$ representable by schemes has property
+$P$ if for every diagram as above the morphism of schemes
+$g : W \to S$ has property $P$.
+\end{definition}
+
+\noindent
+FIXME. Explain rationale behind this definition: what else could it be?
+
+\bibliography{my}
+\bibliographystyle{alpha}
+
+\end{document}
diff -urN -X stacks-0.2/src/documentation/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	2006-03-05 15:12:12.000000000 +0000
+++ stacks-0.2/src/scripts/contents_html.sh	2006-03-20 01:48:12.000000000 +0000
@@ -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 etale injectives hypercovering stacks stacks-groupoids flat desirables"
+LIJST="introduction conventions sets categories sites etale injectives hypercovering stacks stacks-groupoids schemes algebraic flat desirables"
 TELLER=0
 
 for STAM in $LIJST; do
diff -urN -X stacks-0.2/src/documentation/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	2006-03-05 15:12:12.000000000 +0000
+++ stacks-0.2/src/scripts/downloads_html.sh	2006-03-20 01:48:12.000000000 +0000
@@ -3,7 +3,7 @@
 # Write a downloads section to downloads.html.
 
 # Same list as in contents_html.sh.
-LIJST="introduction conventions sets categories sites etale injectives hypercovering stacks stacks-groupoids flat desirables"
+LIJST="introduction conventions sets categories sites etale injectives hypercovering stacks stacks-groupoids schemes algebraic flat desirables"
 
 cat > downloads.html << "EOF"
 <h3><a name="downloads"></a>Downloads</h3>
diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/sets.tex stacks-0.2/src/sets.tex
--- stacks-0.2.orig/src/sets.tex	2005-09-26 19:26:48.000000000 +0000
+++ stacks-0.2/src/sets.tex	2006-03-20 01:48:12.000000000 +0000
@@ -117,6 +117,19 @@
 \noindent
 Explain how everything is a set.
 
+\subsection{The hierarchy of sets}
+\label{subsection-sets-hierarchy}
+
+\noindent
+A set $T$ is transitive if $x\in T$ implies $x\subset T$.
+A set $\alpha$ is an ordinal if it is transitive and wellordered by $\in$.
+We define, by transfinite induction, $V_0 = \emptyset$,
+$V_{\alpha + 1} = P(V_\alpha)$, and for a limit ordinal $\alpha$,
+$$
+V_\alpha = \bigcup_{\beta < \alpha} V_\beta.
+$$
+Every set is contained in one of the $V_\alpha$.
+
 \subsection{Everything is is contained in some ordinal}
 \label{subsection-ordinal}
 
@@ -127,7 +140,28 @@
 \label{section-reflection-principle}
 
 \noindent
-This explains how we deal with set theoretical difficulties.
+This explains how we deal with set theoretical difficulties. 
+
+\subsection{Statement of the theorem}
+\label{subsection-reflection-theorem}
+
+\noindent
+Let $\phi(x_1,\ldots,x_n)$ be a formula of set theory. Let $V$ be a set.
+The formula $V \models \phi(x_1,\ldots,x_n)$ is the formula obtained 
+from $\phi(x_1,\ldots,x_n)$ replacing all the $\forall x$ and $\exists x$
+by $\forall x\in S$ and $\exists x\in S$. (So the formula
+$\phi(x_1,x_2) = \exists x, (x\in x_1 \wedge x\in x_2)$ is turned 
+into $S \models \phi(x_1,x_2) = \exists x, ((x\in S) \wedge 
+(x\in x_1 \wedge x\in x_2))$.
+
+\begin{theorem}
+\label{theorem-reflection-principle}
+Let $\phi(x_1,\ldots,x_n)$ be a formula of set theory, and let $T$ be a set.
+There exists an $\alpha$ such that $\forall x_1,\ldots,x_n \in V_\alpha$,
+$$
+V_\alpha \models \phi(x_1,\ldots,x_n) \Leftrightarrow \phi(x_1,\ldots,x_n).
+$$
+\end{theorem}
 
 \smallskip\noindent
 To continue reading, 
diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/stacks.tex stacks-0.2/src/stacks.tex
--- stacks-0.2.orig/src/stacks.tex	2006-03-14 02:49:38.000000000 +0000
+++ stacks-0.2/src/stacks.tex	2006-03-20 01:48:12.000000000 +0000
@@ -345,11 +345,11 @@
 \end{lemma}
 
 \begin{proof}
-First, note that $f_i \circ {\widetilde{pr}}_1 = f_j \circ {\widetilde{pr}}_2=
-f_k\circ {\widetilde{pr}}_3$. Then note that ${\text{pr}_{13}^\ast \phi_{ik}}$,
-${\text{pr}_{23}^\ast \phi_{jk}}$ and ${\text{pr}_{12}^\ast \phi_{ij}}$ factor
-uniquely through $(f_i \circ {\widetilde{pr}}_1)^\ast x = 
-(f_j \circ {\widetilde{pr}}_2)^\ast x = (f_k \circ {\widetilde{pr}}_3)^\ast x$
+First, note that $f_i \circ\text{pr}_1 = f_j \circ \text{pr}_2=
+f_k\circ \text{pr}_3$. Then note that $\text{pr}_{13}^\ast \phi_{ik}$,
+$\text{pr}_{23}^\ast \phi_{jk}$ and $\text{pr}_{12}^\ast \phi_{ij}$ factor
+uniquely through $(f_i\circ\text{pr}_1)^\ast x = 
+(f_j \circ\text{pr}_2)^\ast x = (f_k \circ\text{pr}_3)^\ast x$
 by Lemma 3.1.3.
 \end{proof}
 
@@ -384,6 +384,26 @@
 \noindent
 Usually the hardest part to check is the third condition.
 
+\begin{lemma}
+\label{lemma-2-product-stacks}
+Suppose that $f : \mathcal{X} \to \mathcal{S}$ and
+$g : \mathcal{Y} \to \mathcal{S}$ are morphisms of stacks over
+$\mathcal{C}$. Let $\mathcal{X} \times_\mathcal{S}\mathcal{Y}$, $p$, $q$,
+$\psi$ be the explicit 2-fibre product of $f$ and $g$ in the 2-category
+of categories over $\mathcal{C}$ described in
+\hyperref[categories-lemma-2-product-categories-over-C]%
+{Lemma~\ref*{categories-lemma-2-product-categories-over-C}}.
+Then $\mathcal{X} \times_\mathcal{S}\mathcal{Y}$ is a stack. In particular
+the 2-category of stacks over $\mathcal{C}$ has 2-fibre products (and
+they are as described in 
+\hyperref[categories-lemma-2-product-categories-over-C]%
+{Lemma~\ref*{categories-lemma-2-product-categories-over-C}}).
+\end{lemma}
+
+\begin{proof}
+FIXME.
+\end{proof}
+
 \subsection{Examples}
 \label{subsection-examples}