\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[categories-]{categories}
\externaldocument[hypercovering-]{hypercovering}
\externaldocument[schemes-]{schemes}
\externaldocument[desirables-]{desirables}
\externaldocument[fdl-]{fdl}

% 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}



\smallskip\noindent
To continue reading, 
\begin{enumerate}

\item visit the next section: Algebraic stacks desirables,
\autoref{desirables-section-foundational}, or 

\item go back to the
table of contents: \url{index.html#contents}.

\end{enumerate}

\bibliography{my}
\bibliographystyle{alpha}

\end{document}