%%% amshelp.tex
%%% This is an AMS-LaTeX file

%%% Copyright (c) 1992, 2000, 2008, 2009, 2011, 2013 Philip S. Hirschhorn
%
% This work may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either version 1.3
% of this license or (at your option) any later version.
% The latest version of this license is in
%   http://www.latex-project.org/lppl.txt
% and version 1.3 or later is part of all distributions of LaTeX
% version 2003/12/01 or later.


%%% Philip Hirschhorn
%%% Department of Mathematics
%%% Wellesley College
%%% Wellesley, MA 02481
%%% psh@math.mit.edu
%%% psh@poincare.wellesley.edu


%%% This is an attempt to explain how to get up and running with
%%% AmS-LaTeX for someone having some familiarity with TeX,
%%% AMS-TeX, or LaTeX.




\newcommand{\filedate}{January 28, 2013}
%\newcommand{\filedate}{\today}
\newcommand{\fileversion}{Version 2.3}

%---------------------------------------------------------------------
%\documentclass[12pt]{amsart}
\documentclass{amsart}

\usepackage{url}

%\usepackage[pdfborderstyle={/S/U/W 1},hyperfootnotes=false]{hyperref}
%\usepackage[colorlinks=false,pdfborder={0 0 0}]{hyperref}
%\usepackage[colorlinks,pdfborder={0 0 0}]{hyperref}
\usepackage[colorlinks]{hyperref}
%\usepackage{hyperref}

% In case we're not using hyperref.sty:
\providecommand{\texorpdfstring}[2]{#1}
% The following can safely be used in \section commands
% without causing pdf warnings:
\newcommand{\bs}{\texorpdfstring{\char`\\}{}}

% We have to load amsrefs *after* hyperref.  (Most packages must
% be loaded before hyperref; amsrefs is an exception to that.)

% We have to load amsrefs *before* loading Xy-pic, or else the
% \newcommand{\cir}{\textasciicircum} in textcmds.sty will complain.
% We can also use the optional argument ``lite'', as in
% \usepackage[lite]{amsrefs}, to avoid the problem by suppressing
% the reading of textcmds.sty
\usepackage[lite]{amsrefs}

\usepackage[all,cmtip]{xy}
\let\objectstyle=\displaystyle



%---------------------------------------------------------------------
%---------------------------------------------------------------------

\numberwithin{equation}{section}


%       Theorem environments

\theoremstyle{plain} %% This is the default, anyway
\newtheorem{thm}[equation]{Theorem}
\newtheorem{cor}[equation]{Corollary}
\newtheorem{lem}[equation]{Lemma}
\newtheorem{prop}[equation]{Proposition}


\theoremstyle{definition}
\newtheorem{defn}[equation]{Definition}

\theoremstyle{remark}
\newtheorem{rem}[equation]{Remark}
\newtheorem{ex}[equation]{Example}
\newtheorem{notation}[equation]{Notation}
\newtheorem{terminology}[equation]{Terminology}


%---------------------------------------------------------------------
%---------------------------------------------------------------------
%---------------------------------------------------------------------
%---------------------------------------------------------------------

\begin{document}


\title[Running \AmS-\LaTeX]{Getting up and running\\
                            with \AmS-\LaTeX}

\author{Philip S. Hirschhorn}

\address{Department of Mathematics\\
         Wellesley College\\
         Wellesley, Massachusetts 02481}

\email{psh@math.mit.edu}

\date{\filedate, \fileversion}

\begin{abstract}
  Together with the template file \texttt{template.tex}, these notes
  are an attempt to tell you enough about \LaTeX{} and \AmS-\LaTeX{}
  so that you can get started without having to read the book.
\end{abstract}

\maketitle

\tableofcontents
%---------------------------------------------------------------------
%---------------------------------------------------------------------
\section{Introduction}

This is an attempt to get you up and running with \AmS-\LaTeX{} as
quickly as possible.  These instructions (along with the template file
\texttt{template.tex}) won't be a substitute for the full
documentation, but they'll give you enough to get started quickly and
only occasionally have to refer to the main documentation.

The current version of \AmS-\LaTeX{} (version 2.2) is a collection of
document classes and optional packages for the current version of
standard \LaTeX.  \AmS-\LaTeX{} provides the document classes
\texttt{amsart}, \texttt{amsproc}, and \texttt{amsbook} (see
section~\ref{sec:DocClsCom}) to replace the standard document classes
\texttt{article}, \texttt{proc}, and \texttt{book}, and several
optional packages (mainly \texttt{amsmath}) that can be used with the
standard \LaTeX{} document classes.  Thus, using \AmS-\LaTeX{} is
really using a variety of \LaTeX.  If you're new to \LaTeX{}, and
these last few sentences made no sense to you at all, don't worry
about it.  You don't have to know what the standard \LaTeX{} document
classes are in order to use the \AmS-\LaTeX{} replacements for them.

I'll be assuming that you have at least some experience with either
plain \TeX, \AmS-\TeX{} or \LaTeX, and I'll try to tell you what you
need to know so that you can get started with \AmS-\LaTeX{}
\emph{without} actually reading the \LaTeX{} user's
guide~\cite{latex}, or even taking much of a look at the \AmS-\LaTeX{}
user's guide~\cite{amslatexusersguide} or the short math guide for
\LaTeX~\cite{mathguide}.

If you've never used \emph{any} version of \TeX{} or \LaTeX, then I
recommend ``The not so short introduction to \LaTeXe{}'' by Tobias
Oetiker, Hubert Partl, Irene Hyna, and Elisabeth Schlegl
\cite{NotShort}.  This is intended for those with no knowledge of
\TeX{} or \LaTeX, and concisely gives a description of what a \LaTeX{}
document looks like and how you type text and simple mathematics in a
\LaTeX{} document.

These instructions come with a template file \verb"template.tex",
which is an attempt to give you enough to fake your way through an
\AmS-\LaTeX{} file \emph{almost} without even reading these
instructions.  I've included the text of that file in these
instructions as section~\ref{sec:template}, so you might want to take
a look at that now, and then just use the table of contents of these
instructions to find more information on whatever in that file
confuses you.

In case you haven't guessed, these instructions were printed using the
\texttt{amsart} document class of \AmS-\LaTeX, so you can get some
idea what it all looks like.



%---------------------------------------------------------------------
%---------------------------------------------------------------------
\section{Basic \LaTeX{} stuff}
\label{sec:basicstuff}

In this section, we'll describe the three commands that must appear in
every \LaTeX{} document: \verb"\documentclass",
\verb"\begin{document}", and \verb"\end{document}".  The complete
explanation of these can be found in the \LaTeX{} User's
Guide~\cite{latex} or in \emph{The not so short introduction to
  \LaTeXe}~\cite{NotShort}.  We'll also explain how to begin a new
section or subsection of the paper, and how \LaTeX{} manages to get
the cross-references right (which is also the explanation of why you
need to run a file through \LaTeX{} \emph{twice} to be sure that all
the cross-references are correct).

%--------------------------------------------------------------------
\subsection{The \texttt{\bs documentclass} command}
\label{sec:DocClsCom}

Before you type anything that actually appears in the paper, you must
include a \verb"\documentclass" command.  It's easiest to just put the
\verb"\documentclass" command at the very beginning of the file,
possibly with a few lines of comments before it.

It's the choice of document class that determines whether you're using
\AmS-\LaTeX{} or just plain old \LaTeX.  \AmS-\LaTeX{} provides the
document classes \texttt{amsart}, \texttt{amsproc}, and
\texttt{amsbook} as replacements for the standard \LaTeX{} document
classes \texttt{article}, \texttt{proc}, and \texttt{book}.  If for
some reason you prefer to use the standard \LaTeX{} classes
\texttt{article}, \texttt{proc}, or \texttt{book}, you can still get
many of the features of \AmS-\LaTeX{} by including the command
\verb"\usepackage{amsmath}" after your \verb"\documentclass" command.
I'll only be discussing the \texttt{amsart} document class here.  For
the others, see the \AmS-\LaTeX{} User's
Guide~\cite{amslatexusersguide}.

The simplest version of the \texttt{\bs documentclass} command is
\begin{center}
  \verb"\documentclass{amsart}"
\end{center}
This will give you the default type size, which is 10~point type.  If
you'd like to use 12~point type, then you should include the optional
argument \verb"[12pt]"; this makes the command
\begin{center}
  \verb"\documentclass[12pt]{amsart}"
\end{center}

%--------------------------------------------------------------------
\subsection{Loading optional packages}
\label{sec:optpack}

There are at least three optional packages that are of interest.  The
first is the \texttt{amsrefs} package, which makes it much easier to
create a bibliography (see section~\ref{sec:amsrefs}).  To load the
\texttt{amsrefs} package, you put the line
\begin{center}
  \verb"\usepackage[lite]{amsrefs}"
\end{center}
after the \verb"\documentclass" command.  (For an explanation of why
we recommend using the optional argument \texttt{lite}, see
section~\ref{sec:refoptions}.)

Another important package is for when you want to use some of the
special symbols contained in the \AmS-Fonts package.  These are
listed, along with all of the standard \LaTeX{} symbols, in
\texttt{symbols.pdf}, available at
\begin{center}
  \url{http://www.ctan.org/tex-archive/info/symbols/math/symbols.pdf}
\end{center}
If you want the standard names for these symbols to be defined for
your use, then you need to use the optional package \texttt{amssymb}.
Thus, to use the default 10~point type, use \texttt{amsrefs} to create
a bibliography, and have the special symbols defined, use the commands
\begin{center}
  \begin{tabular}{l}
    \verb"\documentclass{amsart}"\\
    \verb"\usepackage[lite]{amsrefs}"\\
    \verb"\usepackage{amssymb}"
  \end{tabular}
\end{center}

Another widely used optional package is \Xy-pic, which enables you to
draw commutative diagrams as part of your \LaTeX{} file rather than
creating them with a graphics package and importing the graphics.
(For commutative diagrams, see section~\ref{sec:xypic}).  To use
\Xy-pic, you should include the commands
\begin{center}
  \begin{tabular}{l}
    \verb"\usepackage[all,cmtip]{xy}"\\
    \verb"\let\objectstyle=\displaystyle"
  \end{tabular}
\end{center}
That loads the \Xy-pic package and sets it so that the arrowheads used
are the same ones used in the rest of the document and the nodes in
the diagram are, by default, in \verb"\displaystyle".  If you'd like
the default style for the nodes to be \verb"\textstyle", you should
omit the second of those two lines.

This document uses all of those packages, and so we used the commands
\begin{center}
  \begin{tabular}{l}
    \verb"\documentclass{amsart}"\\
    \verb"\usepackage[lite]{amsrefs}"\\
    \verb"\usepackage{amssymb}"\\
    \verb"\usepackage[all,cmtip]{xy}"\\
    \verb"\let\objectstyle=\displaystyle"
  \end{tabular}
\end{center}

%---------------------------------------------------------------------

\subsection{\texttt{\bs begin\{document\}} and  \texttt{\bs end\{document\}}}

Everything that is to appear in the document must appear in between
the \verb"\begin{document}" and \verb"\end{document}" commands. There
are no optional arguments for these commands, so they always look the
same.  Anything following the \verb"\end{document}" command is
ignored.  In addition to \verb"\usepackage" commands (see
section~\ref{sec:optpack}), you are allowed to have macro definitions
(i.e., newcommands; see section~\ref{sec:definitions}) before the
\verb"\begin{document}", and that's actually a good place for them,
  but that's about all.

%---------------------------------------------------------------------
\subsection{Sections and subsections}
\label{sec:sections}

To begin a new section, you give the command
\begin{center}
  \verb"\section{Section name}"
\end{center}
To begin the present section, I gave the command
\begin{center}
  \verb"\section{Basic \LaTeX{} stuff}"
\end{center}
A section number is supplied automatically.  If you want to be able to
make reference to that section, then you need to \emph{label} it.
Since I wanted to be able to demonstrate the cross-reference commands,
I actually began this section with the lines
\begin{center}
  \begin{tabular}{l}
    \verb"\section{Basic \LaTeX{} stuff}"\\
    \verb"\label{sec:basicstuff}"
  \end{tabular}
\end{center}
This allows me to type ``\verb"section~\ref{sec:basicstuff}"'' and
have it printed as ``section~\ref{sec:basicstuff}''.

To begin a new subsection, you give the command
\begin{center}
  \verb"\subsection{Subsection name}"
\end{center}
To begin the present subsection, I gave the command
\begin{center}
  \verb"\subsection{Sections and subsections}"
\end{center}
A subsection number is supplied automatically.  If you want to be able
to make reference to that subsection, then you need to \emph{label}
it.  This subsection was begun with the lines
\begin{center}
  \begin{tabular}{l}
    \verb"\subsection{Sections and subsections}"\\
    \verb"\label{sec:sections}"
  \end{tabular}
\end{center}
so if we type ``\verb"section~\ref{sec:sections}",'' it is printed as
``section~\ref{sec:sections}''.

Labels always take the number of the smallest enclosing structure.
Thus, a \verb"\label" command that's inside a section but \emph{not}
inside a subsection or Theorem or anything else will take the value of
the section counter, while a \verb"\label" command that's inside the
statement of a Theorem will take the value of that Theorem number.
For more information on this, see section~\ref{sec:xreferences}.

%--------------------------------------------------------------------
\subsubsection{Yes, there are subsubsections too}

I began this subsubsection with the command
\begin{center}
  \verb"\subsubsection{Yes, there are subsubsections too}"
\end{center}

%--------------------------------------------------------------------
\subsubsection*{Sections without numbers}

I began this subsubsection with the command
\begin{center}
  \verb"\subsubsection*{Sections without numbers}"
\end{center}
and got a subsubsection that wasn't numbered.  If you give the command
\begin{center}
  \verb"\section*{A Section Title}"
\end{center}
then you'll begin a new section that will not have a number.


%---------------------------------------------------------------------
\subsection{Italics \emph{for emphasis}}

If you want to use italics to emphasize a word or two, the \LaTeX{}
convention is not to explicitly switch to italics, but rather to use
the command \verb"\emph" (which means \emph{emphasize}).  This command
works just like a font change command, except that it switches you
\emph{into} italics if the current font is upright and switches you
\emph{out of} italics if the current font is italics.

For example, if you type
\begin{center}
  \verb"The whole is \emph{more} than the sum of its parts."
\end{center}
you'll get
\begin{center}
  The whole is \emph{more} than the sum of its parts.
\end{center}
but if you type
\begin{verbatim}
\begin{thm}
  The whole is \emph{more} than the sum of its parts.
\end{thm}
\end{verbatim}
you'll get
\begin{thm}
  The whole is \emph{more} than the sum of its parts.
\end{thm}

\subsubsection*{Note}

The \verb"\emph" command is a recent addition to \LaTeX, and it has
the feature that it automatically inserts an italic correction where
needed.  If you don't know what an italic correction is, you can
safely ignore this paragraph, but I will at least mention that all
those ``\verb"\/"'' commands frequently seen in \TeX{} (and older
\LaTeX) documents are all inserting italic corrections; the point of
this paragraph is that, with the current version of \LaTeX, you don't
have to do that anymore.

%---------------------------------------------------------------------
\subsection{Cross references and the table of contents}

This is an explanation of how \LaTeX{} manages to fill in
cross-references (see section~\ref{sec:xreferences}) to parts of the
file it hasn't processed yet, and what those \verb".aux" and
\verb".toc" files are.

%--------------------------------------------------------------------
\subsubsection*{Cross-References}

Every time \LaTeX{} processes your file, it writes an \emph{auxiliary}
file.  Since the file containing these instructions is called
\verb"amshelp.tex", the auxiliary file is called \verb"amshelp.aux".
The auxiliary file contains the definitions of all the keys used for
cross-references.  When \LaTeX{} begins to process your file, it first
looks for an \verb".aux" file, and reads it in if it exists.  Of
course, this is the \verb".aux" file that was produced the \emph{last}
time that your file was processed, so the Theorem numbers, Section
numbers, etc., are all the ones from the last time the file was
processed.

The very first time that \LaTeX{} processes your file, there is no
\verb".aux" file, and so \LaTeX{} gives \emph{lots} of warning
messages about undefined labels, or whatever.  Ignore all of this.
The \emph{next} time that you run \LaTeX, there \emph{will} be an
\verb".aux" file, and all the references will be filled in.  (Yes, it
is possible, at least in theory, for some page number to change every
time you run \LaTeX{} on your file, even without any changes in the
source file, but this isn't very likely.)

\subsubsection*{The Table of Contents}

If you give the command \verb"\tableofcontents", then \LaTeX{} will
try to write a table of contents at that point, including the page
numbers of the sections.  Obviously, \LaTeX{} can't know those page
numbers or section titles yet, so as \LaTeX{} processes your file, it
writes a \verb".toc" file containing the information it needs.  (The
\verb".toc" file for these instructions is \verb"amshelp.toc".)  Once
again, \LaTeX{} is always using the information from the \emph{last}
time that it processed your file.

If you \emph{do} include a table of contents in your document, and if
the table of contents takes up at least a page or so of space, then
you might have to run \LaTeX{} \emph{three} times in order to get all
of the cross-references right.  The reason for this is that the first
time you run \LaTeX{} there isn't any \verb".toc" file listing the
section titles, and so the table of contents has nothing in it.  The
second time you run \LaTeX{} you'll get a table of contents that lists
the page numbers for the sections from the last time you ran \LaTeX,
when the table of contents took up no space at all.  Unfortunately,
during this second run, the table of contents will be created, and
will take up enough space to change the page numbers of the sections
from what they were during the first run.  Only during the
\emph{third} run will the correct page numbers be written into the
table of contents.  Since this doesn't change the amount of space that
the table of contents occupies, this version will be correct.

\subsubsection*{How do I know when everything is correct?}

After processing your file, \LaTeX{} checks whether all the
cross-reference numbers that it read from the \verb".aux" file are
correct.  If any of them are incorrect, it prints a warning on the
screen at the very end of the run advising you that labels may have
changed and that you should run \LaTeX{} again to get the
cross-references right.  Unfortunately, \LaTeX{} doesn't seem to check
that the table of contents entries are correct, so if you change the
name of a section in a way that doesn't make any page references
incorrect, you won't be warned to run \LaTeX{} again.


%---------------------------------------------------------------------
%---------------------------------------------------------------------
\section{Title, Author, and the \texttt{\bs maketitle} command}

This stuff should go right after the \verb"\begin{document}" command.
I'll give a quick sketch here, which is probably all you'll ever
need, but the full explanation is given in \emph{Instructions for
  preparation of papers and monographs: \AmS-\LaTeX} \cite{instr-l}.
If you are already familiar with \LaTeX, then you should be warned
that this part is slightly different from what you do when using the
standard \LaTeX{} \verb"article" document class.

%---------------------------------------------------------------------
\subsection{The title}
You specify the title with the command
\begin{center}
  \verb"\title[Optional running title]{Actual title}"
\end{center}
These instructions used the title command
\begin{verbatim}
\title[Running \AmS-\LaTeX]{Getting up and running\\
                            with \AmS-\LaTeX}
\end{verbatim}
Notice that you indicate line breaks in the title with a double
backslash.  If I had decided to omit the line break and also to have
the full title printed in the head of the odd numbered pages, I would
have used the command
\begin{center}
  \verb"\title{Getting up and running with \AmS-\LaTeX}"
\end{center}

%---------------------------------------------------------------------
\subsection{The author, and the author's address}

The author is specified with an \verb"author" command:
\begin{center}
\verb"\author{Author's name}"
\end{center}
These directions used the command
\verb"\author{Philip S. Hirschhorn}".
The author's address is given in an address command, with double
backslashes to indicate line breaks.  These instructions used the
command
\begin{center}
  \begin{tabular}{l}
    \verb"\address{Department of Mathematics\\"\\
    \verb"Wellesley College\\"\\
    \verb"Wellesley, Massachusetts 02481}"
  \end{tabular}
\end{center}
If the author's current address is different from the address at which
the research was carried out, then you can specify the current address
with the command \verb"\curraddr".  For example, you might type
\begin{center}
  \begin{tabular}{l}
    \verb"\curraddr{Department of Mechanics\\"\\
    \verb"Brake and Wheel Bearing Division\\"\\
    \verb"Serene Service Center\\"\\
    \verb"Salem, Massachusetts 02139}"
  \end{tabular}
\end{center}
You can also include an email address, with the \verb"\email"
command.  These instructions used the command
\begin{center}
  \verb"\email{psh@math.mit.edu}"
\end{center}
To acknowledge support, use the command \verb"\thanks", e.g.,
\begin{center}
  \verb"\thanks{Supported in part by NSF grant 3.14159}"
\end{center}
This will be printed as a footnote on the first page.

%--------------------------------------------------------------------
\subsubsection*{Multiple authors}

If there are several authors, then each one should have a separate
\verb"\author" command, with each individual's address, current
address, email address, and thanks following that individual's
\verb"\author" command, in its own \verb"\address" command (and
\verb"\curraddr" command, and \verb"\thanks" command, and
\verb"\email" command).  If there \emph{are} several authors, and
their combined names are too long for the running head on the even
numbered pages, you can give an optional argument to each
\verb"\author" command to supply a shortened form to use in the
running head, as in
\begin{center}
  \verb"\author[P.S. Hirschhorn]{Philip S. Hirschhorn}".
\end{center}
(It's apparently a convention that the running head in a multiple
author paper should have only initials for the first and middle names,
but I don't think that I was invited to that convention.)


%---------------------------------------------------------------------
\subsection{The date}

This is pretty straightforward:
\begin{center}
\verb"\date{Whatever date you please}"
\end{center}
To have the date of processing used, use the command
\verb"\date{\today}".

%---------------------------------------------------------------------
\subsection{\texttt{\bs maketitle}}

After you've given all of the commands mentioned in this section, you
can give the command \verb"\maketitle".  If you \emph{don't} give the
command \verb"\maketitle", a title won't be made.  The exact
arrangement of all this information is determined by the document
class.  In particular, the \verb"amsart" document class puts the
author's address at the \emph{end} of the paper.

%---------------------------------------------------------------------
%---------------------------------------------------------------------
\section{Theorems, Propositions, Lemmas, etc.}

The instructions in this section assume that you're using the
\verb"\newtheorem" commands that I put in the file \verb"template.tex"
(see section~\ref{sec:template}).

%--------------------------------------------------------------------
\subsection{Stating theorems, propositions, etc.}
\label{sec:theorems}

To state a theorem, you do the following:
\begin{verbatim}
\begin{thm}
  The square of the hypotenuse of a right triangle is equal to the
  sum of the squares of the two adjacent sides.
\end{thm}
\end{verbatim}
If you do that, you'll get the following:
\begin{thm}
  \label{pythagthm}
  The square of the hypotenuse of a right triangle is equal to the sum
  of the squares of the two adjacent sides.
\end{thm}
If you thought that it was only a proposition, you'd use
\begin{verbatim}
\begin{prop}
  The square of the hypotenuse of a right triangle is equal to the
  sum of the squares of the two adjacent sides.
\end{prop}
\end{verbatim}
and you'd get
\begin{prop}
  The square of the hypotenuse of a right triangle is equal to the sum
  of the squares of the two adjacent sides.
\end{prop}

If you think it's a theorem again, but you'd like to make reference to
it in some other part of the paper, you have to choose a \emph{key}
with which you'll refer to it, and then \emph{label} the theorem.  If
you want to use the key \emph{pythagthm}, then it would look like the
following:
\begin{verbatim}
\begin{thm}
  \label{pythagthm}
  The square of the hypotenuse of a right triangle is equal to the
  sum of the squares of the two adjacent sides.
\end{thm}
\end{verbatim}
If you later give the command \verb"\ref{pythagthm}", then that
command will expand to the \emph{number} that was assigned to that
theorem (in this case, \ref{pythagthm}).  For more explanation of
cross-references, see section~\ref{sec:xreferences}.

If you'd like to state a theorem and give a \emph{name} to it, then
you can add an optional argument to the \verb"\begin{thm}" command. 
If you type
\begin{verbatim}
\begin{thm}[Pythagoras]
  The square of the hypotenuse of a right triangle is equal to the
  sum of the squares of the two adjacent sides.
\end{thm}
\end{verbatim}
you'll get
\begin{thm}[Pythagoras]
  The square of the hypotenuse of a right triangle is equal to the sum
  of the squares of the two adjacent sides.
\end{thm}

%--------------------------------------------------------------------
\subsubsection*{Summary of environments provided in the template}

All of the following structures are numbered in the same sequence,
in the form SectionNumber.Number.  Equations (i.e., displayed
formulas, whether they are equations or not) will be numbered in the
same sequence.
\begin{displaymath}
  \begin{tabular}{c@{\hspace{4em}}l@{\hspace{4em}}c}
    \multicolumn{3}{c}{Theorem Environments}\\*[8pt]
    \hspace{1em}Name&  Printed Form& Body font\\*[6pt]
    \texttt{thm}&           \textbf{Theorem}&      Italic\\
    \texttt{cor}&           \textbf{Corollary}&    Italic\\
    \texttt{lem}&           \textbf{Lemma}&        Italic\\
    \texttt{prop}&          \textbf{Proposition}&  Italic\\
    \texttt{defn}&          \textbf{Definition}&   Normal\\
    \texttt{rem}&           \textit{Remark}&       Normal\\
    \texttt{ex}&            \textit{Example}&      Normal\\
    \texttt{notation}&      \textit{Notation}&     Normal\\
    \texttt{terminology}&   \textit{Terminology}&  Normal\\*[3pt]
  \end{tabular}
\end{displaymath}
For full details, see the beginning of the template file (reproduced
here in section~\ref{sec:template}), after the comment ``The Theorem
Environments.''

%---------------------------------------------------------------------
\subsection{Proofs}

To give a proof, you do the following:
\begin{verbatim}
\begin{proof}
  As any fool can plainly see, it's true!
\end{proof}
\end{verbatim}
and you'll get the following:
\begin{proof}
  As any fool can plainly see, it's true!
\end{proof}
If the theorem said that a condition was both necessary and sufficient
for something, and you want to prove each part separately,
you can do the following:
\begin{verbatim}
\begin{proof}[Proof (sufficiency)]
  Well, it's \emph{obviously} sufficient!
\end{proof}
\end{verbatim}
and you'll get
\begin{proof}[Proof (sufficiency)]
  Well, it's \emph{obviously} sufficient!
\end{proof}
That is, the \verb"proof" environment allows you to use an optional
second argument that will appear in place of the word \verb"Proof".

If the proof of Theorem~\ref{pythagthm} does not appear immediately
after its statement, you might use the following:
\begin{verbatim}
\begin{proof}[Proof of Theorem~\ref{pythagthm}]
  As any fool can plainly see, it's true!
\end{proof}
\end{verbatim}
and you'd get
\begin{proof}[Proof of Theorem~\ref{pythagthm}]
  As any fool can plainly see, it's true!
\end{proof}

%---------------------------------------------------------------------
%---------------------------------------------------------------------
\section{Cross-References}
\label{sec:xreferences}

This section explains how to make reference to numbered sections,
theorems, equations, and bibliography items, with the correct
reference numbers filled in automatically by \LaTeX.

%---------------------------------------------------------------------
\subsection{References to sections, theorems and equations}
\label{sec:thmrefs}

For each structure in the manuscript to which you'll be making
reference, you must assign a \emph{key} that you'll use to refer to
that structure.  For sections, theorems, numbered equations, and items
in an enumerated list (see section~\ref{sec:enumref}), you assign the
key using the \verb"\label" command and refer to it using either the
\verb"\ref" command or the \verb"\eqref" command.  Each of these
commands takes one argument, which is the \emph{key} you're assigning
to the object.  The command \verb"\ref{key}" produces the number that
was assigned to that structure and the command \verb"\eqref{key}"
produces that number enclosed in parentheses.  (The \verb"\eqref"
command also ensures that the number and parentheses are always in an
upright font; see section~\ref{sec:EqRef}.)  The convention is to use
\verb"\eqref" to refer to equation numbers, \verb"\cite" to refer to
bibliography entries (see section~\ref{sec:bibreferences}), and
\verb"\ref" to refer to everything else.

Consider the following example.
\begin{thm}
  \label{homotopy}
  If the maps $f\colon X \to Y$ and $g\colon X \to Y$ are homotopic,
  then the induced homomorphisms $f_{*} \colon \mathrm{H}_{*}X \to
  \mathrm{H}_{*}Y$ and $g_{*} \colon \mathrm{H}_{*}X \to
  \mathrm{H}_{*}Y$ are equal.
\end{thm}

We typed that theorem as follows.
\begin{verbatim}
\begin{thm}
  \label{homotopy}
  If the maps $f\colon X \to Y$ and $g\colon X \to Y$ are homotopic,
  then the induced homomorphisms $f_{*} \colon \mathrm{H}_{*}X \to
  \mathrm{H}_{*}Y$ and $g_{*} \colon \mathrm{H}_{*}X \to
  \mathrm{H}_{*}Y$ are equal.
\end{thm}
\end{verbatim}
If we now type ``\verb"see Theorem~\ref{homotopy}",'' then it will be
printed as ``see Theorem~\ref{homotopy}.''

%--------------------------------------------------------------------
\subsubsection*{So, what exactly is the label labeling?}

The command \verb"\label{key}" assigns to \verb"key" the value of the
\emph{smallest enclosing structure}.  For example, we began this
section by typing
\begin{center}
  \begin{tabular}{l}
    \verb"\section{Cross-References}"\\
    \verb"\label{sec:xreferences}"
  \end{tabular}
\end{center}
and we began this subsection by typing
\begin{center}
  \begin{tabular}{l}
    \verb"\subsection{References to sections, theorems and equations}"\\
    \verb"\label{sec:thmrefs}"
  \end{tabular}
\end{center}
The phrase ``\verb"See section~\ref{sec:xreferences}"'' is printed as
``See section~\ref{sec:xreferences}'' while the phrase
%
``\verb"See section~\ref{sec:thmrefs}"'' is printed as ``See
section~\ref{sec:thmrefs}'' because the key \verb"sec:xreferences" was
defined inside of section~\ref{sec:xreferences} but outside of
section~\ref{sec:thmrefs}, while the key \verb"sec:thmrefs" was
defined inside of section~\ref{sec:thmrefs}.

%--------------------------------------------------------------------
\subsubsection{References to equations}
\label{sec:EqRef}

To make reference to a numbered equation, you assign the \emph{key} as
before, but you replace \verb"\ref" with \verb"\eqref", so that
parentheses will be printed around the equation number.  For example,
if you type
\begin{verbatim}
\begin{equation}
  \label{additivity}
  \mathrm{H}_{*} \bigvee_{\alpha\in A} X_{\alpha}   \approx
    \bigoplus_{\alpha\in A}\mathrm{H}_{*} X_{\alpha}
\end{equation}
\end{verbatim}
then you'll get
\begin{equation}
  \label{additivity}
  \mathrm{H}_{*} \bigvee_{\alpha\in A} X_{\alpha}   \approx
    \bigoplus_{\alpha\in A}\mathrm{H}_{*} X_{\alpha}
\end{equation}
If we now type
\begin{verbatim}
\begin{thm}
  Equation~\eqref{additivity} is true for all sorts of functors
  $\mathrm{H}$.
\end{thm}
\end{verbatim}
then we'll get
\begin{thm}
  Equation~\eqref{additivity} is true for all sorts of functors
  $\mathrm{H}$.
\end{thm}
Notice the parentheses around the equation number, and the fact that
even though the theorem is set in slanted type, the equation number is
set in an upright font?  This is the difference between \verb"\eqref"
and \verb"\ref"; the command \verb"\eqref" provides parentheses,
arranges it so that the number and surrounding parentheses are in an
upright font no matter what the surrounding font, and supplies an
italic correction if it's needed.

%---------------------------------------------------------------------
\subsection{References to page numbers}

If you want to make reference to the \emph{page} that contains a
label, rather than to the structure that is labeled, use the command
\verb"\pageref{key}".  For example, if you type
\begin{verbatim}
See page~\pageref{homotopy} to find Theorem~\ref{homotopy}.
\end{verbatim}
you'll get ``See page~\pageref{homotopy} to find
Theorem~\ref{homotopy}.''

%---------------------------------------------------------------------
\subsection{Bibliographic references}
\label{sec:bibreferences}

Each bibliography item receives a \emph{key} as part of its basic
structure, and you refer to that item using the command
\verb"\cite{key}".

When using the \texttt{amsrefs} package, each item in the bibliography
is begun with
\begin{verbatim}
\bib{key}{TypeOfItem}{
\end{verbatim}
For example, the bibliography of these instructions contains the entry
\begin{verbatim}
\bib{HA}{book}{
  author={Quillen, Daniel G.},
  title={Homotopical Algebra},
  series={Lecture Notes in Mathematics},
  volume={43},
  publisher={Springer-Verlag},
  address={Berlin-New York},
  date={1967}
}
\end{verbatim}
If we type ``\verb"This is the work of Quillen~\cite{HA}",'' then it
will be printed as ``This is the work of Quillen~\cite{HA}.''  Notice
that square brackets have been inserted around the bibliography item
number.

The \verb"\cite" command takes an optional argument, which allows you
to annotate the reference.  If we type
``\verb"see~\cite[Chapter I]{HA}"'', then it will be printed as
``see~\cite[Chapter I]{HA}''.  If you're using \texttt{amsrefs} (which
we strongly recommend), then there's an alternate form available: If
we type ``\verb"\cite{HA}*{Chapter I}"'', then we also get
``\cite{HA}*{Chapter I}'', and this second form is less likely to
cause errors when used, e.g., in the optional argument to a
\verb"\begin{theorem}" command (see section~\ref{sec:theorems}).

%--------------------------------------------------------------------
\subsubsection{Multiple references}
\label{sec:mulref}

If you're using \texttt{amsrefs} and when you refer to multiple
bibliography items you want to have the item numbers automatically
sorted and compressed (e.g., replacing ``7, 6, 5, 8'' with ``5--8''),
you can use the \verb"\cites" command.  For example, if we type
\verb"\cites{HA,yellowmonster}" then we get \cites{HA,yellowmonster}.

If you want some of the references in the list to have annotations,
you put your \verb"\cite" commands into the argument of a
\verb"\citelist" command.  For example, if we type
\verb"\citelist{\cite{HA}*{Chapter I} \cite{yellowmonster}}" then we
get \citelist{\cite{HA}*{Chapter I} \cite{yellowmonster}}.

%--------------------------------------------------------------------
\subsubsection{Author-year citations}
\label{sec:authyrcit}

If you're using \texttt{amsrefs} and you use the \texttt{author-year}
option to convert your references to the author-year format (see
section~\ref{sec:refoptions}), then there are two variations on the
\verb"\cite" command (\verb"\ycite" and \verb"\ocite") that are
useful.  If we were using the \texttt{author-year} option in this
document, then
\begin{center}
  \begin{tabular}{l@{ would be typeset as }l}
    ``\verb"\cite{yellowmonster}"''& ``(Bousfield and Kan, 1972)''\\
    ``\verb"\ycite{yellowmonster}"''& ``(1972)''\\
    ``\verb"\ocite{yellowmonster}"''& ``Bousfield and Kan (1972)''
  \end{tabular}
\end{center}
Thus, you could type ``For further details, see
\verb"\cite{yellowmonster}"'', or ``Bousfield and Kan
\verb"\ycite{yellowmonster}" showed\ldots'', or ``This can be found in
\verb"\ocite{yellowmonster}"''.


%---------------------------------------------------------------------
%---------------------------------------------------------------------
\section{Mathematics in running text}

This is pretty much exactly as it is in plain \TeX, except that you
have an extra option (which you can ignore).  The simplest thing is to
just enclose between dollar signs any material that should be in math
mode.  Thus, if you type
\begin{center}
  \verb"Let $f\colon X \to Y$ be a continuous function."
\end{center}
you'll get
\begin{center}
  Let $f\colon X \to Y$ be a continuous function.
\end{center}
(Note that if we had typed that as ``\verb"f: X \to Y"'' then the
spacing around the colon would be wrong, since the colon symbol is
typeset as a binary relation, with spaces on the two sides equal.)
The only novelty that \LaTeX{} introduces is that, instead of using a
dollar sign to toggle math mode on and off, you can use `\verb"\("' to
\emph{begin} math mode, and `\verb"\)"' to \emph{end} math mode.
Thus, the example above could also be typed as
\begin{center}
\verb"Let \(f\colon X \to Y\) be a continuous function."
\end{center}
This provides a tiny bit more error checking, but can
otherwise be safely ignored.

The reference \cite{mathguide} is an excellent concise summary of the
features of \AmS-\LaTeX{} for typesetting mathematics, both displayed
and in running text.  It also includes lists of symbols available for
mathematics.

%--------------------------------------------------------------------
%---------------------------------------------------------------------
\section{Displayed mathematics}

For a complete discussion of the environments for displayed
mathematics, including options for customizing equation numbers,
see~\cite[section~3]{amslatexusersguide}.

%--------------------------------------------------------------------
\subsection{Single line displays}
\label{sec:SngLne}

To display mathematics and number the display (so that you can refer
to it from elsewhere in the paper) you use the \verb"equation"
environment.  (\LaTeX{} calls all such numbers \emph{equation
  numbers}, whether or not the display has anything to do with
equations.)  If you type
\begin{verbatim}
\begin{equation}
  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
\end{equation}
\end{verbatim}
you'll get
\begin{equation}
  \label{pi1eqn}
  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
\end{equation}

If you'd like to be able to make reference to the equation number,
you need to \emph{label} the equation, using a \emph{key} that you
can use for referencing it:
\begin{verbatim}
\begin{equation}
  \label{pi1eqn}
  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
\end{equation}
\end{verbatim}
If you later type ``\verb"see formula~\eqref{pi1eqn}"'' you'll get 
``see formula~\eqref{pi1eqn}.''  (For more on cross-references to
formulas, see section~\ref{sec:thmrefs}.)

To display a single line of mathematics without an equation number,
you use the \verb"equation*" environment.  (This is a common \LaTeX
ism: Adding an asterisk to the name of a numbered \LaTeX{} environment
often gives the unnumbered equivalent.)  If you type
\begin{verbatim}
\begin{equation*}
  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
\end{equation*}
\end{verbatim}
then you'll get
\begin{equation*}
  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
\end{equation*}
The \texttt{displaymath} environment is equivalent to this; you can
produce the same unnumbered display by typing
\begin{verbatim}
\begin{displaymath}
  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
\end{displaymath}
\end{verbatim}
You can also produce that same unnumbered display by typing
\begin{verbatim}
\[
  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
\]
\end{verbatim}
but some people think that that makes your \LaTeX{} source file less
readable.  Although you can also produce the same thing by typing
\begin{verbatim}
$$
  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
$$
\end{verbatim}
using double dollar signs is definitely deprecated.

%--------------------------------------------------------------------
\subsection{Text in displayed mathematics}
\label{sec:text}

To include text in a displayed mathematics environment you use the
\verb"\text" command, as in \verb"\text{here's some text}".  The
\verb"\text" command is preferable to the standard \LaTeX{}
\verb"\mbox" command because \verb"\text" correctly adjusts its size
for use in subscripts and superscripts.  For example, if you type
\begin{verbatim}
\begin{equation*}
  \pi_{1}(X) \approx G * H
  \quad\text{where $G$ is torsion and $H$ is torsion free}
\end{equation*}
\end{verbatim}
then you'll get
\begin{equation*}
  \pi_{1}(X) \approx G * H
  \quad\text{where $G$ is torsion and $H$ is torsion free}
\end{equation*}
and if you type
\begin{verbatim}
\begin{equation*}
  X_{n} = \coprod_{\text{monomorphisms $[k] \to [n]$}} S_{k}
\end{equation*}
\end{verbatim}
then you'll get
\begin{equation*}
  X_{n} = \coprod_{\text{monomorphisms $[k] \to [n]$}} S_{k}
\end{equation*}


%--------------------------------------------------------------------
\subsection{Displaying multiple lines without alignment}
\label{sec:gather}

You can put several displayed lines together, each one centered, with
no alignment between the different lines, using the \verb"gather"
environment.  When typing this, the lines are separated by a double
backslash \verb"\\".  For example, if you type
\begin{verbatim}
\begin{gather}
  (X\otimes L) \amalg_{(X\otimes K)} (Y\otimes K)
    \longrightarrow Y\otimes L\\
  X^{L} \longrightarrow X^{K} \times_{Y^{K}} Y^{L}
\end{gather}
\end{verbatim}
then you'll get
\begin{gather}
  \label{eq:push}
  (X\otimes L) \amalg_{(X\otimes K)} (Y\otimes K)
    \longrightarrow Y\otimes L\\
  \label{eq:pull}
  X^{L} \longrightarrow X^{K} \times_{Y^{K}} Y^{L}
\end{gather}
The \verb"gather*" environment would produce the same thing without
the equation numbers.  You can also label each line so that you can
refer to them: If you had typed that as
\begin{verbatim}
\begin{gather}
  \label{eq:push}
  (X\otimes L) \amalg_{(X\otimes K)} (Y\otimes K)
    \longrightarrow Y\otimes L\\
  \label{eq:pull}
  X^{L} \longrightarrow X^{K} \times_{Y^{K}} Y^{L}
\end{gather}
\end{verbatim}
and then typed ``\verb"see \eqref{eq:push} or \eqref{eq:pull}"'',
you'd get ``see \eqref{eq:push} or \eqref{eq:pull}''.

There is also a \texttt{gathered} environment, which does not produce
a display of its own but rather produces a block of centered lines of
mathematics that can be used inside of another displayed mathematics
environment (see section~\ref{sec:gathered}).
%--------------------------------------------------------------------
\subsection{Displays with linebreaks}
\label{sec:multline}

For a long display that must be broken across several lines, you can
use the \texttt{multline} environment.  (There is also a
\texttt{multline*} environment, which is similar except that it omits
the equation number.)  When typing this, the lines are separated by a
double backslash \verb"\\".  The first line will be shifted left of
center, the last will be shifted right of center, and the lines in
between those will be centered.  For example, if you type
\begin{verbatim}
\begin{multline}
  \label{eq:BigComp}
  \mathrm{F} X\otimes\Delta[n]   \xrightarrow{1 \otimes D}
    \mathrm{F} X\otimes(\Delta[n]\times\Delta[n])\\
  \xrightarrow{\sigma} 
    \mathrm{F}\bigl(X\otimes(\Delta[n]\times\Delta[n])\bigr)
    \approx
    \mathrm{F}\bigl((X\otimes\Delta[n])\otimes\Delta[n]\bigr)\\
  \xrightarrow{\mathrm{F}(\alpha\otimes 1)}
    \mathrm{F}(Y\otimes\Delta[n]) \xrightarrow{\mathrm{F}(\beta)}
    \mathrm{F}(Z)
\end{multline}
\end{verbatim}
then you'll get
\begin{multline}
  \label{eq:BigComp}
  \mathrm{F} X\otimes\Delta[n]   \xrightarrow{1 \otimes D}
    \mathrm{F} X\otimes(\Delta[n]\times\Delta[n])\\
  \xrightarrow{\sigma} 
    \mathrm{F}\bigl(X\otimes(\Delta[n]\times\Delta[n])\bigr)
    \approx
    \mathrm{F}\bigl((X\otimes\Delta[n])\otimes\Delta[n]\bigr)\\
  \xrightarrow{\mathrm{F}(\alpha\otimes 1)}
    \mathrm{F}(Y\otimes\Delta[n]) \xrightarrow{\mathrm{F}(\beta)}
    \mathrm{F}(Z)
\end{multline}
You can then type ``\verb"the composition \eqref{eq:BigComp}"" and it
will appear as ``the composition \eqref{eq:BigComp}''.

%--------------------------------------------------------------------
\subsection{Displays with alignment}
\label{sec:align}

There are a number of ways to produce displays with multiple lines and
horizontal alignment between the lines.
\begin{itemize}
\item The \texttt{align} and \texttt{align*} environments allow
  horizontal alignment of characters chosen from each line (see
  section~\ref{sec:SingAlign}).
\item The \texttt{aligned} and \texttt{split} environments produce
  alignments that can be be part of a larger display (see
  section~\ref{sec:aligned}).
\item The \texttt{gathered} environment produces a block of several
  lines, each centered as in the \texttt{gather} environment (see
  section~\ref{sec:gather}), that can be part of a larger display (see
  section~\ref{sec:gathered}).  (Of course, this doesn't involve
  horizontal alignment, but the \texttt{gathered} environment is often
  used in conjunction with other display environments.)
\item The \texttt{align} and \texttt{align*} environments also allow
  for multiple columns with alignment in each column (see
  section~\ref{sec:MulAlign}).
\item The \texttt{flalign} and \texttt{flalign*} environments allow
  multiple alignment points and place the outermost aligned blocks
  flush against the margins (see section~\ref{sec:flalign}).
\item The \texttt{alignat} and \texttt{alignat*} environments allow
  multiple alignment points and allow you to choose the spacing
  between the columns (see section~\ref{sec:alignat}).
\item The \texttt{alignedat} environment is similar to the
  \texttt{alignat*} environment, except that it produces an alignment
  intended to be part of a larger display (see
  section~\ref{sec:alignedat}).
\end{itemize}
%--------------------------------------------------------------------
\subsubsection{Displays with a single alignment point}
\label{sec:SingAlign}

To display several lines of mathematics with horizontal alignment, you
use the \texttt{align} environment.  (There is also an \texttt{align*}
environment, which is similar except that it omits the equation
numbers.)  When typing this, the lines are separated by a double
backslash \verb"\\" and each line has an ampersand \verb"&"
immediately preceding the symbol to be aligned with the corresponding
symbols on the other lines.  For example, if you type
\begin{verbatim}
\begin{align}
  \label{eq:pi1}
  \pi_{1}(X\vee Y) &\approx \pi_{1}X * \pi_{1}Y\\
  \label{eq:additivity}
  \widetilde{\mathrm{H}}_{*}(X\vee Y) &\approx
    \widetilde{\mathrm{H}}_{*}X \oplus \widetilde{\mathrm{H}}_{*}Y
\end{align}
\end{verbatim}
then you'll get
\begin{align}
  \label{eq:pi1}
  \pi_{1}(X\vee Y) &\approx \pi_{1}X * \pi_{1}Y\\
  \label{eq:additivity}
  \widetilde{\mathrm{H}}_{*}(X\vee Y) &\approx
    \widetilde{\mathrm{H}}_{*}X \oplus \widetilde{\mathrm{H}}_{*}Y
\end{align}
and if you type ``\verb"see \eqref{eq:pi1} or \eqref{eq:additivity}"''
then you'll get ``see \eqref{eq:pi1} or \eqref{eq:additivity}''.  For
another example, if you type
\begin{verbatim}
\begin{align*}
  \mathcal{M}\bigl((\operatorname{colim} \boldsymbol{X})
    \otimes K,Y\bigr) &\approx
    \mathcal{M}(\operatorname{colim} \boldsymbol{X},Y^K)\\
  &\approx \lim \mathcal{M}(\boldsymbol{X},Y^K)\\
  &\approx \lim \mathcal{M}(\boldsymbol{X} \otimes K,Y)\\
  &\approx \mathcal{M}\bigl(\operatorname{colim}
    (\boldsymbol{X} \otimes K),Y\bigr)
\end{align*}
\end{verbatim}
then you'll get
\begin{align*}
  \mathcal{M}\bigl((\operatorname{colim} \boldsymbol{X})
    \otimes K,Y\bigr) &\approx
    \mathcal{M}(\operatorname{colim} \boldsymbol{X},Y^K)\\
  &\approx \lim \mathcal{M}(\boldsymbol{X},Y^K)\\
  &\approx \lim \mathcal{M}(\boldsymbol{X} \otimes K,Y)\\
  &\approx \mathcal{M}\bigl(\operatorname{colim}
    (\boldsymbol{X} \otimes K),Y\bigr)
\end{align*}
Both the \texttt{align} and \texttt{align*} environments allow you to
have multiple columns in the display; for this, see
section~\ref{sec:MulAlign}.

%--------------------------------------------------------------------
\subsubsection{Alignments that are part of a larger display}
\label{sec:aligned}

The \texttt{aligned} environment does not create a display of its own.
Instead, it allows you to have several lines of mathematics, with
alignment as in the \texttt{align*} environment (see
section~\ref{sec:SingAlign}), which can then be used inside of a
displayed mathematics environment.

The \textbf{aligned} environment doesn't produce any equation numbers
of its own.  If the enclosing display environment produces a number,
and if the \texttt{aligned} environment doesn't use the optional
argument of \texttt{[t]} or \texttt{[b]} (see \eqref{eq:dbltop} and
\eqref{eq:dblbot}), then that number is centered vertically in the
contents of the \texttt{aligned} environment.  For an example of this,
see \eqref{eq:longeq}.

There is also a \texttt{split} environment, which is similar to the
\texttt{aligned} environment except that the \texttt{aligned}
environment has options (changing the vertical alignment (see the
examples in this section) or having multiple alignment points (see
section~\ref{sec:MulAlign})) that are not possible with the
\texttt{split} environment.  Thus, a \texttt{split} environment can
always be replaced by an \texttt{aligned} environment.

For example, if you type
\begin{verbatim}
\begin{equation}
  \label{eq:dblaligned}
  \begin{aligned}
    a &= b + c\\
    d &= e + f
  \end{aligned}
  \qquad\text{which, of course, leads to}\qquad
  \begin{aligned}
    g &= h + i\\
    j &= k + l
  \end{aligned}
\end{equation}
\end{verbatim}
then you'll get
\begin{equation}
  \label{eq:dblaligned}
  \begin{aligned}
    a &= b + c\\
    d &= e + f
  \end{aligned}
  \qquad\text{which, of course, leads to}\qquad
  \begin{aligned}
    g &= h + i\\
    j &= k + l
  \end{aligned}
\end{equation}

The above example illustrates that, by default, when an
\texttt{aligned} (or \texttt{aligned*}) environment is used as part of
a display involving other elements, it is vertically aligned at its
center.  The \texttt{aligned} environment can take an optional
argument of \texttt{[t]} or \texttt{[b]} to cause the alignment to be
vertically aligned at either the top or the bottom row.  For example,
if you type
\begin{verbatim}
\begin{equation}
  \label{eq:dbltop}
  \begin{aligned}[t]
    a &= b + c\\
    d &= e + f
  \end{aligned}
  \qquad\text{which, of course, leads to}\qquad
  \begin{aligned}[t]
    g &= h + i\\
    j &= k + l
  \end{aligned}
\end{equation}
\end{verbatim}
then you'll get
\begin{equation}
  \label{eq:dbltop}
  \begin{aligned}[t]
    a &= b + c\\
    d &= e + f
  \end{aligned}
  \qquad\text{which, of course, leads to}\qquad
  \begin{aligned}[t]
    g &= h + i\\
    j &= k + l
  \end{aligned}
\end{equation}
and if you type
\begin{verbatim}
\begin{equation}
  \label{eq:dblbot}
  \begin{aligned}[b]
    a &= b + c\\
    d &= e + f
  \end{aligned}
  \qquad\text{which, of course, leads to}\qquad
  \begin{aligned}[b]
    g &= h + i\\
    j &= k + l
  \end{aligned}
\end{equation}
\end{verbatim}
then you'll get
\begin{equation}
  \label{eq:dblbot}
  \begin{aligned}[b]
    a &= b + c\\
    d &= e + f
  \end{aligned}
  \qquad\text{which, of course, leads to}\qquad
  \begin{aligned}[b]
    g &= h + i\\
    j &= k + l
  \end{aligned}
\end{equation}

The display created by an \texttt{aligned} environment can be used as
an element of a display.  For example, if you type
\begin{verbatim}
\begin{equation*}
  \left .
    \begin{aligned}
      f(x) &= \sin^{2} x\\
      g(x) &= \cos^{2} x\\
      h(x) &= \sin x + \cos x
    \end{aligned}
  \right \}
  \qquad \text{A nice collection of functions}
\end{equation*}
\end{verbatim}
then you'll get
\begin{equation*}
  \left .
    \begin{aligned}
      f(x) &= \sin^{2} x\\
      g(x) &= \cos^{2} x\\
      h(x) &= \sin x + \cos x
    \end{aligned}
  \right \}
  \qquad \text{A nice collection of functions}
\end{equation*}


%--------------------------------------------------------------------
\subsubsection{Several centered lines as part of a larger display}
\label{sec:gathered}

The \texttt{gathered} environment does not create a display of its
own.  Instead, it allows you to have several lines of mathematics,
each one centered as in the \texttt{gather} environment (see
section~\ref{sec:gather}), which can then be used inside of a
displayed mathematics environment.  For example, if you type
\begin{verbatim}
\begin{equation}
  \begin{gathered}
    x^{2} + y^{2} = z^{2}\\
    a = b + c
  \end{gathered}
  \qquad \text{and, in addition,}\qquad
  \begin{aligned}
    A &= B+C\\
    &= D+E
  \end{aligned}
\end{equation}
\end{verbatim}
then you'll get
\begin{equation}
  \begin{gathered}
    x^{2} + y^{2} = z^{2}\\
    a = b + c
  \end{gathered}
  \qquad \text{and, in addition,}\qquad
  \begin{aligned}
    A &= B+C\\
    &= D+E
  \end{aligned}
\end{equation}

The \texttt{gathered} environment can take the same optional argument
of \texttt{[t]} or \texttt{[b]} that the \texttt{aligned} argument can
take (see section~\ref{sec:aligned}) to cause the vertical alignment
to be on either the top or the bottom line.  For example, if you type
\begin{verbatim}
\begin{equation}
  \begin{gathered}[t]
    x^{2} + y^{2} = z^{2}\\
    a = b + c
  \end{gathered}
  \qquad \text{and, in addition,}\qquad
  \begin{aligned}[t]
    A &= B+C\\
    &= D+E
  \end{aligned}
\end{equation}
\end{verbatim}
then you'll get
\begin{equation}
  \begin{gathered}[t]
    x^{2} + y^{2} = z^{2}\\
    a = b + c
  \end{gathered}
  \qquad \text{and, in addition,}\qquad
  \begin{aligned}[t]
    A &= B+C\\
    &= D+E
  \end{aligned}
\end{equation}

%--------------------------------------------------------------------
\subsubsection{Multiple alignment points}
\label{sec:MulAlign}

The \texttt{align} and \texttt{align*} environments (see
section~\ref{sec:SingAlign}) as well as the \texttt{aligned}
environment (see section~\ref{sec:aligned}) can be used to create
displays in which each line is broken into several pieces, with each
piece horizontally aligned at a chosen character.  That is: they can
be used to create a display with several columns, with alignment
within each column.

When typing this, the lines are separated by a double backslash
\verb"\\", the different displays on each line are separated by an
ampersand \verb"&", and the symbols to be aligned are preceded by an
ampersand \verb"&".  For example, if you type
\begin{verbatim}
\begin{align*}
  K &\approx G * H&   i&= j+k&  B &\subset C\\
  H &\approx A_{0}*B_{0}&  i'&= j'+k'&  C &= D\cap E\\
  G &\approx \coprod_{\alpha\in A} L_{\alpha}&  i''&=j''+k''&
                                        A &= D \cup E
\end{align*}
\end{verbatim}
then you'll get
\begin{align*}
  K &\approx G * H&   i&= j+k&  B &\subset C\\
  H &\approx A_{0}*B_{0}&  i'&= j'+k'&  C &= D\cap E\\
  G &\approx \coprod_{\alpha\in A} L_{\alpha}&  i''&=j''+k''&
                                        A &= D \cup E
\end{align*}
When typing such an alignment with $n$ columns, each line will have at
most $2n-1$ ampersands.

For another example, if you type
\begin{verbatim}
\begin{align*}
  \pi_{1}(X\vee Y) &\approx \pi_{1}(X) * \pi_{1}(Y)
    &&\text{(by the van Kampen theorem)}\\
  &\approx G*H &&\text{(by the computation in the previous section)}
\end{align*}
\end{verbatim}
then you'll get
\begin{align*}
  \pi_{1}(X\vee Y) &\approx \pi_{1}(X) * \pi_{1}(Y)
    &&\text{(by the van Kampen theorem)}\\
  &\approx G*H &&\text{(by the computation in the previous section)}
\end{align*}
If you want to specify the separation between the columns in the
alignment, you should use the \texttt{alignat} environment (see
section~\ref{sec:alignat}).


%--------------------------------------------------------------------
\subsubsection{Alignments flush left and flush right}
\label{sec:flalign}

To produce alignments similar to those in section~\ref{sec:MulAlign}
except with the leftmost column flush left and the rightmost column
flush right, you use the \texttt{flalign} environment (or, to omit the
equation numbers, the \texttt{flalign*} environment).  For example, if
you type
\begin{verbatim}
\begin{flalign*}
  K &\approx G * H&   i&= j+k&  B &\subset C\\
  H &\approx A_{0}*B_{0}&  i'&= j'+k'&  C &= D\cap E\\
  G &\approx \coprod_{\alpha\in A} L_{\alpha}&  i''&=j''+k''&
                                        A &= D \cup E
\end{flalign*}
\end{verbatim}
then you'll get
\begin{flalign*}
  K &\approx G * H&   i&= j+k&  B &\subset C\\
  H &\approx A_{0}*B_{0}&  i'&= j'+k'&  C &= D\cap E\\
  G &\approx \coprod_{\alpha\in A} L_{\alpha}&  i''&=j''+k''&
                                        A &= D \cup E
\end{flalign*}

%--------------------------------------------------------------------
\subsubsection{Multiple alignment points with chosen spacing}
\label{sec:alignat}

To produce alignments as in section~\ref{sec:MulAlign} except with the
ability to choose the amount of horizontal space between the columns,
you use the \texttt{alignat} environment (or, to omit the equation
numbers, the \texttt{alignat*} environment).  These environments don't
insert any horizontal space between the columns, and so you can insert
the exact amount of space you want by including it at the beginning of
one of the columns.

The format of \texttt{alignat} is slightly different from that of the
\texttt{align} environment in that you must include an argument
specifying the number of columns.  For example, if you type
\begin{verbatim}
\begin{alignat}{2}
  K &\approx G*H&  \qquad&\text{(by an earlier theorem)}\\
  A &\approx \lim_{i\in I} A_{i}&  &\text{(by the definition of $A$)}
\end{alignat}
\end{verbatim}
then you'll get
\begin{alignat}{2}
  K &\approx G*H&  \qquad&\text{(by an earlier theorem)}\\
  A &\approx \lim_{i\in I} A_{i}&  &\text{(by the definition of $A$)}
\end{alignat}
Note that it's only necessary to insert the \verb"\qquad" space in one
row; the alignment forces the space to appear in all rows.

%--------------------------------------------------------------------
\subsubsection{Multiple alignment points with chosen spacing as part of a larger display}
\label{sec:alignedat}

The \texttt{alignedat} environment is similar to the \texttt{aligned}
environment (see section~\ref{sec:aligned}), in that neither of them
create a display of their own but are instead used as a part of a
larger display, but the \texttt{alignedat} environment allows you to
choose the spacing between the columns.  That is, the
\texttt{alignedat} environment doesn't insert any space between the
columns, so you can insert the exact amount of space you want by
including it at the beginning of one of the columns.  The
\texttt{alignedat} environment also has a required argument stating
the number of columns.  For example, if you type
\begin{verbatim}
\begin{equation*}
  \left \{
    \begin{alignedat}{2}
      a &= b + c& d &= e + f\\
      A &= B + C \qquad&  D &= E + F
    \end{alignedat}
  \right \}
  \qquad \text{Twin pairs of equations}
\end{equation*}
\end{verbatim}
then you'll get
\begin{equation*}
  \left \{
    \begin{alignedat}{2}
      a &= b + c& d &= e + f\\
      A &= B + C \qquad&  D &= E + F
    \end{alignedat}
  \right \}
  \qquad \text{Twin pairs of equations}
\end{equation*}
Note that it's only necessary to insert the \verb"\qquad" space in one
row; the alignment forces the space to appear in all rows.


%--------------------------------------------------------------------
\subsection{The \texttt{cases} environment}
\label{sec:cases}

There is a \texttt{cases} environment, which constructs the usual
display of several cases, and which is used as a part of one of the
displayed mathematics environments.  For example, if you type
\begin{verbatim}
\begin{equation}
  \label{eq:abs}
  |x| =
  \begin{cases}
    x& \text{if $x \ge 0$}\\
    -x& \text{if $x < 0$}
  \end{cases}
\end{equation}
\end{verbatim}
then you'll get
\begin{equation}
  \label{eq:abs}
  |x| =
  \begin{cases}
    x& \text{if $x \ge 0$}\\
    -x& \text{if $x < 0$}
  \end{cases}
\end{equation}

For another example, if you type
\begin{verbatim}
\begin{align*}
  d_{i} \sigma &=
  \begin{cases}
    \alpha_{1} \xrightarrow{\sigma_{1}} \alpha_{2}
          \xrightarrow{\sigma_{2}} \cdots
          \xrightarrow{\sigma_{n-1}} \alpha_{n} 
      &\text{if $i=0$}\\
    \alpha_0 \xrightarrow{\sigma_{0}} \cdots
          \xrightarrow{\sigma_{i-2}} \alpha_{i-1}
          \xrightarrow{\sigma_{i}\sigma_{i-1}} \alpha_{i+1} 
          \xrightarrow{\sigma_{i+1}} \cdots 
          \xrightarrow{\sigma_{n-1}} \alpha_{n}
      &\text{if $0<i<n$}\\
    \alpha_{0} \xrightarrow{\sigma_{0}} \alpha_{1}
          \xrightarrow{\sigma_1} \cdots
          \xrightarrow{\sigma_{n-2}} \alpha_{n-1} 
      &\text{if $i=n$}
  \end{cases}\\
  s_{i} \sigma &= \alpha_{0} \xrightarrow{\sigma_{0}} \cdots 
    \xrightarrow{\sigma_{i-1}} \alpha_{i}
    \xrightarrow{1_{\alpha_{i}}} \alpha_{i}
    \xrightarrow{\sigma_{i}} \alpha_{i+1}
    \xrightarrow{\sigma_{i+1}} \cdots 
    \xrightarrow{\sigma_{n-1}} \alpha_{n}
\end{align*}
\end{verbatim}
then you'll get
\begin{align*}
  d_{i} \sigma &=
  \begin{cases}
    \alpha_{1} \xrightarrow{\sigma_{1}} \alpha_{2}
          \xrightarrow{\sigma_{2}} \cdots
          \xrightarrow{\sigma_{n-1}} \alpha_{n} 
      &\text{if $i=0$}\\
    \alpha_0 \xrightarrow{\sigma_{0}} \cdots
          \xrightarrow{\sigma_{i-2}} \alpha_{i-1}
          \xrightarrow{\sigma_{i}\sigma_{i-1}} \alpha_{i+1} 
          \xrightarrow{\sigma_{i+1}} \cdots 
          \xrightarrow{\sigma_{n-1}} \alpha_{n}
      &\text{if $0<i<n$}\\
    \alpha_{0} \xrightarrow{\sigma_{0}} \alpha_{1}
          \xrightarrow{\sigma_1} \cdots
          \xrightarrow{\sigma_{n-2}} \alpha_{n-1} 
      &\text{if $i=n$}
  \end{cases}\\
  s_{i} \sigma &= \alpha_{0} \xrightarrow{\sigma_{0}} \cdots 
    \xrightarrow{\sigma_{i-1}} \alpha_{i}
    \xrightarrow{1_{\alpha_{i}}} \alpha_{i}
    \xrightarrow{\sigma_{i}} \alpha_{i+1}
    \xrightarrow{\sigma_{i+1}} \cdots 
    \xrightarrow{\sigma_{n-1}} \alpha_{n}
\end{align*}

%--------------------------------------------------------------------
\subsection{Multiple lines with equation numbers on some of them}
\label{sec:MuEqOneNo}

By default, most of the environments for displayed mathematics either
have an equation number on every line or no equation numbers at all.
In this section we discuss how to have multiple lines with either one
equation number for the entire display (see
sections~\ref{sec:MulLinOneNo} and \ref{sec:GatherCenter}) or equation
numbers on some of the lines but not all of them (see
section~\ref{sec:notag}).

%--------------------------------------------------------------------
\subsubsection{Multiple lines with horizontal alignment and one equation number}
\label{sec:MulLinOneNo}

To display multiple lines with horizontal alignment but with only one
equation number for the entire display, you use the \texttt{equation}
environment (see section~\ref{sec:SngLne}) to produce the equation
number and you put an \texttt{aligned} environment (see
section~\ref{sec:aligned}) inside of that to produce the alignment.
For example, if you type
\begin{verbatim}
\begin{equation}
  \label{eq:BothCoprod}
  \begin{aligned}
    \pi_{1}(X\vee Y) &\approx \pi_{1}X * \pi_{1}Y\\
    \widetilde{\mathrm{H}}_{*}(X\vee Y) &\approx
    \widetilde{\mathrm{H}}_{*}X \oplus \widetilde{\mathrm{H}}_{*}Y
  \end{aligned}
\end{equation}
\end{verbatim}
then you'll get
\begin{equation}
  \label{eq:BothCoprod}
  \begin{aligned}
    \pi_{1}(X\vee Y) &\approx \pi_{1}X * \pi_{1}Y\\
    \widetilde{\mathrm{H}}_{*}(X\vee Y) &\approx
    \widetilde{\mathrm{H}}_{*}X \oplus \widetilde{\mathrm{H}}_{*}Y
  \end{aligned}
\end{equation}
and if you type ``\verb"the isomorphisms~\eqref{eq:BothCoprod}"'' then
you'll get ``the isomorphisms~\eqref{eq:BothCoprod}''.

For another example, if you type
\begin{verbatim}
\begin{equation}
  \label{eq:longeq}
  \begin{aligned}
    F(a,b,c,d) &= \int_{a}^{b} \sin^{3}x \cos^{2}x \, dx
      + \int_{c}^{d} \sin^{3}x \cos x \, dx\\
    &\quad + \int_{a}^{b} \sin^{2} x \cos^{2} x \, dx + \int_{c}^{d}
      \sin x \cos x \, dx
  \end{aligned}
\end{equation}
\end{verbatim}
then you'll get
\begin{equation}
  \label{eq:longeq}
  \begin{aligned}
    F(a,b,c,d) &= \int_{a}^{b} \sin^{3}x \cos^{2}x \, dx
      + \int_{c}^{d} \sin^{3}x \cos x \, dx\\
    &\quad + \int_{a}^{b} \sin^{2} x \cos^{2} x \, dx + \int_{c}^{d}
      \sin x \cos x \, dx
  \end{aligned}
\end{equation}

If you'd like to specify the exact amount of horizontal space between
the columns, you'd use the \texttt{alignedat} environment (see
section~\ref{sec:alignedat}) in place of the \texttt{aligned}
environment.

%--------------------------------------------------------------------
\subsubsection{Multiple lines without horizontal alignment and with one equation number}
\label{sec:GatherCenter}

To create a display of multiple lines without horizontal alignment but
with each line centered, and to have a single equation number for the
entire display, you use the \texttt{equation} environment (see
section~\ref{sec:SngLne}) to produce the equation number and you put a
\texttt{gathered} environment (see section~\ref{sec:gathered}) inside
of that.  For example, if you type
\begin{verbatim}
\begin{equation}
  \label{eq:Corner}
  \begin{gathered}
    (X\otimes L) \amalg_{(X\otimes K)} (Y\otimes K)
    \longrightarrow Y\otimes L\\
    X^{L} \longrightarrow X^{K} \times_{Y^{K}} Y^{L}
  \end{gathered}
\end{equation}
\end{verbatim}
then you'll get
\begin{equation}
  \label{eq:Corner}
  \begin{gathered}
    (X\otimes L) \amalg_{(X\otimes K)} (Y\otimes K)
    \longrightarrow Y\otimes L\\
    X^{L} \longrightarrow X^{K} \times_{Y^{K}} Y^{L}
  \end{gathered}
\end{equation}
and if you type ``\verb"see~\eqref{eq:Corner}"'' then you'll get
``see~\eqref{eq:Corner}''.

%--------------------------------------------------------------------
\subsubsection{Suppressing some equation numbers: {\normalfont \texttt{\bs notag}}}
\label{sec:notag}

If you want to have equation numbers on some but not all of the lines
of a display, you can use an environment that numbers every line and
then suppress the equation number on particular lines by typing
\begin{center}
  \verb"\notag"
\end{center}
at the end of each of those lines, either just before the double
backslash that ends the line or at the end of the last line.  For
example, if you type
\begin{verbatim}
\begin{align}
  a^{2} + b^{2} &= c^{2} \notag\\
  \label{eq:ABC}
  A^{2} + B^{2} &= C^{2}\\
  p^{2} + q^{2} &- r^{2} \notag\\
  \label{eq:PQR}
  P^{2} + Q^{2} &= R^{2}
\end{align}
\end{verbatim}
then you'll get
\begin{align}
  a^{2} + b^{2} &= c^{2} \notag\\
  \label{eq:ABC}
  A^{2} + B^{2} &= C^{2}\\
  p^{2} + q^{2} &- r^{2} \notag\\
  \label{eq:PQR}
  P^{2} + Q^{2} &= R^{2}
\end{align}
and if you type ``\verb"equations~\eqref{eq:ABD} and \eqref{eq:PQR}"''
then you'll get``equations~\eqref{eq:ABC} and \eqref{eq:PQR}''.

%--------------------------------------------------------------------
\subsection{Overriding equation numbers}
\label{sec:EqNoOver}

There are commands \verb"\tag" and \verb"\tag*" that allow you to
override the normal equation numbering for any line of any of the
displayed mathematics environments.  You can also suppress the equation
number entirely using the \verb"\notag" command (see
section~\ref{sec:notag}).

If you put the command ``\verb"\tag{stuff}"'' at the end of a line of
a display (either just before the double backslash that ends the line
or at the end of the last line), where ``\texttt{stuff}'' is arbitrary
text, then that arbitrary text will be placed on the page, enclosed in
parentheses, in place of an equation number.  The command
``\verb"\tag*{stuff}"'' is similar, but it does not add the
parentheses.  If these commands are used in a display that normally
has unnumbered lines, then ``\texttt{stuff}'' will be inserted anyway.

For example, if you type
\begin{verbatim}
\begin{equation}
  \label{eq:add}
  \widetilde{\mathrm{H}}_{*}(X\vee Y) \approx
  \widetilde{\mathrm{H}}_{*}X \oplus \widetilde{\mathrm{H}}_{*}Y
  \tag*{$(*)$}
\end{equation}
\end{verbatim}
then you'll get
\begin{equation}
  \label{eq:add}
  \widetilde{\mathrm{H}}_{*}(X\vee Y) \approx
  \widetilde{\mathrm{H}}_{*}X \oplus \widetilde{\mathrm{H}}_{*}Y
  \tag*{$(*)$}
\end{equation}
and if you type
%
``\verb"equation~\ref{eq:add} is additivity"'' then you'll get
``equation~\ref{eq:add} is additivity''.  Note that we used the
command \verb"\tag*" and inserted the parentheses as part of the tag,
rather than using the command \verb"\tag", because we wanted cross
references to show the parentheses surrounding the asterisk.



%--------------------------------------------------------------------
%--------------------------------------------------------------------
\section{Commutative diagrams using \texorpdfstring{\Xy-pic}{Xy-pic}}
\label{sec:xypic}

\Xy-pic is a powerful package that enables you to draw very complex
commutative diagrams within your \LaTeX{} file, avoiding the need to
create the graphics separately and then import them.  \Xy-pic isn't a
part of \AmS-LaTeX, but \Xy-pic and \AmS-LaTeX{} work quite well
together.  We'll describe the most important features of \Xy-pic's
commutative diagram macros here.  The full documentation for
commutative diagrams can be found in the \emph{\Xy-pic User's Guide}
\cite{xyguide}, and the full documentation for all of \Xy-pic, which
can do quite a bit more than draw commutative diagrams, can be found
in the \emph{\Xy-pic Reference Manual} \cite{xyrefer}.

%--------------------------------------------------------------------
\subsection{Basic diagrams}
\label{sec:basicdiag}

To use \Xy-pic to draw commutative diagrams, you include the lines
\begin{center}
  \begin{tabular}{l}
    \verb"\usepackage[all,cmtip]{xy}"\\
    \verb"\let\objectstyle=\displaystyle"
  \end{tabular}
\end{center}
in the preamble of your document, i.e., after the
\verb"\documentclass" command and before the \verb"\begin{document}"
command.  You can omit the second of those two lines if you want the
nodes of your diagram to be in \verb"\textstyle" by default.

You create a diagram using the \verb"\xymatrix" command, immediately
followed by a pair of braces that enclose the diagram specification.
The diagram is described in a manner similar to the way matrices are
described:
\begin{enumerate}
\item There is a rectangular array of nodes of the diagram.  A node
  should not begin with a macro or with an asterisk, and so it's
  safest (although not always required) to enclose each node in braces
  \verb"{}".
\item The nodes in a row are separated by an ampersand \verb"&",
  and the rows are separated by a double backslash \verb"\\".
\item An arrow is specified immediately following the node that is its
  source (although this is not actually required; see
  section~\ref{sec:SrcTrgt}).  A straight line arrow is specified by
  \verb"\ar[target]", where \verb"target" is
  \begin{itemize}
  \item \verb"r" for the node one place to the right,
  \item \verb"l" for the node one place to the left,
  \item \verb"d" for the node one place down,
  \item \verb"u" for the node one place up,
  \item \verb"rr" for the node two places to the right,
  \item \verb"dr" for the node one place down and one place to the
    right,
  \item etc.
  \end{itemize}
\end{enumerate}
It's also possible to have labels on the arrows (see
section~\ref{sec:labelarrow}), arrows that are built of things other
than a single solid line with a single arrowhead at the end (see
section~\ref{sec:arstyle}), curved arrows (see
section~\ref{sec:arcurve}), and arrows that pass over or under other
arrows (see sections~\ref{sec:ArUnder} and~\ref{sec:CrossArrow}).

For example,
if you type
\begin{verbatim}
\begin{equation}
  \xymatrix{
    {A}\ar[r] \ar[d]
    &{B} \ar[d]\\
    {C} \ar[r]
    &{D}
  }
\end{equation}
\end{verbatim}
then you'll get
\begin{equation}
  \xymatrix{
    {A}\ar[r] \ar[d]
    &{B} \ar[d]\\
    {C} \ar[r]
    &{D}
  }
\end{equation}
If you'd like the equation number to be centered vertically with
respect to the diagram, you should enclose the \verb"\xymatrix"
command in a \verb"\vcenter", as in
\begin{verbatim}
\begin{equation}
  \vcenter{
    \xymatrix{
      {A}\ar[r] \ar[d]
      &{B} \ar[d]\\
      {C} \ar[r]
      &{D}
    }
  }
\end{equation}
\end{verbatim}
in which case you'll get
\begin{equation}
  \vcenter{
    \xymatrix{
      {A}\ar[r] \ar[d]
      &{B} \ar[d]\\
      {C} \ar[r]
      &{D}
    }
  }
\end{equation}

Not all nodes need appear.  To omit a node in the middle of a line,
just type the \verb"&" that moves you on to the next node.  To omit a
node at the end of the line, just type the line-ending \verb"\\" and
go on to the next line.  For example, if you type
\begin{verbatim}
\begin{equation*}
  \xymatrix{
    {A} \ar[rr] \ar[dr] \ar[d]
    && {B} \ar[dd]\\
    {C} \ar[d] \ar[drr]
    & {D} \ar[l] \ar[ur]\\
    {E} \ar[rr]
    &&{F}
  }
\end{equation*}
\end{verbatim}
you'll get
\begin{equation*}
  \xymatrix{
    {A} \ar[rr] \ar[dr] \ar[d]
    && {B} \ar[dd]\\
    {C} \ar[d] \ar[drr]
    & {D} \ar[l] \ar[ur]\\
    {E} \ar[rr]
    &&{F}
  }
\end{equation*}
Note that that diagram has three rows and three columns, but two of
the nodes in the second column and one of the nodes in the third
column are missing.

%--------------------------------------------------------------------
\subsection{Changing the spacing}
\label{sec:ChgSpc}

You can change the space between the rows of a diagram and the space
between the columns of a diagram.  The following commands can be
inserted following the command \verb"\xymatrix" and before the left
brace that begins the diagram specification:
\begin{center}
  \begin{tabular}{ll}
    \texttt{@=dimen}& set row and column spacing to \texttt{dimen}\\
    \texttt{@R=dimen}& set row spacing to \texttt{dimen}\\
    \texttt{@C=dimen}& set column spacing to \texttt{dimen}
  \end{tabular}
\end{center}
The \texttt{dimen} in all of these commands can be positive, zero, or
negative.  (Negative \texttt{dimen}'s can be useful in diagrams with
very wide entries; see \eqref{diag:wide}.)  Although you can use
whatever units you like for \texttt{dimen} (e.g., inches
(\texttt{in}), centimeters (\texttt{cm}), points (\texttt{pt}), etc.),
it's a good idea to measure vertical distances in \texttt{ex} (roughly
the height of an upper case ``X'') and horizontal distances in
\texttt{em} (roughly the width of an upper case ``M''), so that the
distance will be roughly correct even if you change font sizes.

For example, if you type
\begin{verbatim}
\begin{displaymath}
  \vcenter{
    \xymatrix@=2ex{
      {A} \ar[r] \ar[d]
      & {B} \ar[d]\\
      {C} \ar[r]
      & {D}
    }% \xymatrix
  }% \vcenter
  \qquad\text{or}\qquad
  \vcenter{
    \xymatrix{
      {A} \ar[r] \ar[d]
      & {B} \ar[d]\\
      {C} \ar[r]
      & {D}
    }% \xymatrix
  }% \vcenter
  \qquad\text{or}\qquad
  \vcenter{
    \xymatrix@C=10ex{
      {A} \ar[r] \ar[d]
      & {B} \ar[d]\\
      {C} \ar[r]
      & {D}
    }% \xymatrix
  }% \vcenter
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \vcenter{
    \xymatrix@=2ex{
      {A} \ar[r] \ar[d]
      & {B} \ar[d]\\
      {C} \ar[r]
      & {D}
    }% \xymatrix
  }% \vcenter
  \qquad\text{or}\qquad
  \vcenter{
    \xymatrix{
      {A} \ar[r] \ar[d]
      & {B} \ar[d]\\
      {C} \ar[r]
      & {D}
    }% \xymatrix
  }% \vcenter
  \qquad\text{or}\qquad
  \vcenter{
    \xymatrix@C=10ex{
      {A} \ar[r] \ar[d]
      & {B} \ar[d]\\
      {C} \ar[r]
      & {D}
    }% \xymatrix
  }% \vcenter
\end{displaymath}
For an example of the usefulness of negative column spacing: If you
type
\begin{verbatim}
\newcommand{\pushout}[3]{#1\mathbin{\mathord{\amalg}_{#2}}#3}
\begin{equation}
  \label{diag:wide}
  \vcenter{
    \xymatrix@C=-12em{
      {\pushout{(A\otimes L)}
               {A\otimes K}
               {(B\otimes K)}
             } \ar[r] \ar[dr]
      & {B\otimes L}
      & {\pushout{(P\otimes L)}
                 {P\otimes M}
                 {(Q\otimes M)}
             } \ar[l] \ar[dl]\\
      & {\pushout{\bigl(\pushout{(A\otimes L)}
                                {A\otimes K}
                                {(B\otimes K)}\bigr)}
                 {\pushout{(X\otimes L)}
                          {X\otimes K}
                          {(Y\otimes K)}}
                 {\bigl(\pushout{(P\otimes L)}
                                {P\otimes M}
                                {(Q\otimes M)}\bigr)}
             } \ar[u]
    }
  }       
\end{equation}
\end{verbatim}
then you'll get
\newcommand{\pushout}[3]{#1\mathbin{\mathord{\amalg}_{#2}}#3}
\begin{equation}
  \label{diag:wide}
  \vcenter{
    \xymatrix@C=-12em{
      {\pushout{(A\otimes L)}
               {A\otimes K}
               {(B\otimes K)}
             } \ar[r] \ar[dr]
      & {B\otimes L}
      & {\pushout{(P\otimes L)}
                 {P\otimes M}
                 {(Q\otimes M)}
             } \ar[l] \ar[dl]\\
      & {\pushout{\bigl(\pushout{(A\otimes L)}
                                {A\otimes K}
                                {(B\otimes K)}\bigr)}
                 {\pushout{(X\otimes L)}
                          {X\otimes K}
                          {(Y\otimes K)}}
                 {\bigl(\pushout{(P\otimes L)}
                                {P\otimes M}
                                {(Q\otimes M)}\bigr)}
             } \ar[u]
    }
  }       
\end{equation}
which would never fit on the page without negative column spacing.

%--------------------------------------------------------------------
\subsection{Shifting nodes sideways}
\label{sec:ShiftSide}

You can shift a node to the right by enclosing it in braces and
preceding the opening brace with \verb"*+[r]" and shift it to the left
by enclosing it in braces and preceding the opening brace with
\verb"*+[l]".  This can be useful, for example, if you want a node to
consist of both the name of something and its definition, but you want
the arrows to the node to point to the name, rather than to the center
of the node.  For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar[r] \ar[d]
    & *+[r]{B = \{x \in W \mid f(x) > 0\}} \ar[d]\\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar[r] \ar[d]
    & *+[r]{B = \{x \in W \mid f(x) > 0\}} \ar[d]\\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}
Note that the large node at the upper right caused the columns to be
far apart, even though that node was shifted to the right before being
placed onto the page.  To counteract that, you can decrease the column
spacing: If you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix@C=-2em{
    {A} \ar[r] \ar[d]
    & *+[r]{B = \{x \in W \mid f(x) > 0\}} \ar[d]\\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix@C=-2em{
    {A} \ar[r] \ar[d]
    & *+[r]{B = \{x \in W \mid f(x) > 0\}} \ar[d]\\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}



%--------------------------------------------------------------------
\subsection{Arrows passing under nodes; crossing arrows}
\label{sec:ArUnder}

Arrows can be created that go to a sequence of nodes, passing under
(i.e., leaving a small gap at) all the intermediate nodes.  If an
arrow is created that passes under an empty node while a second arrow
crosses that node in the normal way, the effect is that the first
arrow passes under the second arrow.  (For another way to have one
arrow pass under another, see section~\ref{sec:CrossArrow}.)

To draw an arrow that passes under a sequence of nodes and then goes
on to a final node, the \verb"\ar" is followed by \verb"'[node]" for
each node that you pass under, followed by \verb"[finalnode]".  For
example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar '[r] '[rr] [rrr]
    & {B}
    & {}
    & {D}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar '[r] '[rr] [rrr]
    & {B}
    & {}
    & {D}
  }
\end{displaymath}
For an example of crossing arrows, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix@=3ex{
    {A} \ar[rr] \ar[dd] \ar'[dr][ddrr]
    && {B} \ar[dd] \ar[ddll]\\
    & {}\\
    {C} \ar[rr]
    && {D}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix@=3ex{
    {A} \ar[rr] \ar[dd] \ar'[dr][ddrr]
    && {B} \ar[dd] \ar[ddll]\\
    & {}\\
    {C} \ar[rr]
    && {D}
  }
\end{displaymath}
Note that that diagram has three rows and three columns, but nothing
appears in either the second row or the second column.  (For an
explanation of the command \verb"@=3ex", which changes the size of the
diagram, see section~\ref{sec:ChgSpc}.)  For a more elaborate example,
if you type
\begin{verbatim}
\begin{displaymath} 
  \xymatrix@=2ex{ 
    {A} \ar[rr] \ar[dr] \ar[dd]
    && {B} \ar[dr] \ar'[d][dd]\\ 
    & {A'} \ar[rr] \ar[dd]
    && {B'} \ar[dd]\\ 
    {C} \ar'[r][rr] \ar[dr] 
    && {D} \ar[dr]\\ 
    & {C'} \ar[rr]
    && {D'} 
    } 
\end{displaymath} 
\end{verbatim}
then you'll get
\begin{displaymath} 
  \xymatrix@=2ex{ 
    {A} \ar[rr] \ar[dr] \ar[dd]
    && {B} \ar[dr] \ar'[d][dd]\\ 
    & {A'} \ar[rr] \ar[dd]
    && {B'} \ar[dd]\\ 
    {C} \ar'[r][rr] \ar[dr] 
    && {D} \ar[dr]\\ 
    & {C'} \ar[rr]
    && {D'} 
    } 
\end{displaymath} 



%--------------------------------------------------------------------
\subsection{Labeling the arrows}
\label{sec:labelarrow}

It's possible to label an arrow, on one or both sides of the arrow.
(It's also possible to have the label ``break'' the arrow; for this,
see section~\ref{sec:arbreak}.)  By default, the label is located
halfway from the center of the source to the center of the target.
This will often be halfway along the arrow, but not if the source and
the target are of different sizes.  There are also options to place
the label along the midpoint of the arrow (see
section~\ref{sec:arcent}) or at an arbitrary point between the center
of the source and the center of the target (see
section~\ref{sec:LblArb}).

To put a label above an arrow (where ``above'' means when the paper is
oriented so that the arrow goes left to right), you type
\verb"^{thelabel}" either before or after the target.  To put a label
below an arrow (where ``below'' means when the paper is oriented so
that the arrow goes left to right), you type \verb"_{thelabel}" either
before or after the target.  Thus, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A}\ar[r]^{f} \ar[d]_{g}
    &{B} \ar[d]^{h}\\
    {C} \ar[r]_{k}^{\text{above}}
    &{D}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A}\ar[r]^{f} \ar[d]_{g}
    &{B} \ar[d]^{h}\\
    {C} \ar[r]_{k}^{\text{above}}
    &{D}
  }
\end{displaymath}

%--------------------------------------------------------------------
\subsubsection{Centering the labels on the arrows}
\label{sec:arcent}

If you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {X\cup_{Y}Z} \ar[r]^{f} \ar[d]_{g}
    &{B} \ar[d]^{h}\\
    {C} \ar[r]_{k}
    &{D}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {X\cup_{Y}Z} \ar[r]^{f} \ar[d]_{g}
    &{B} \ar[d]^{h}\\
    {C} \ar[r]_{k}
    &{D}
  }
\end{displaymath}
Note that the label $f$ is halfway from the center of the arrow's
source to the center of the arrow's target, but it is not centered
along the arrow.  To have that label centered along the arrow, we
insert a \verb"-" immediately following the \verb"^", so that we type
it as
\begin{center}
  \verb"\ar[r]^-{f}"
\end{center}
and we then get
\begin{displaymath}
  \xymatrix{
    {X\cup_{Y}Z} \ar[r]^-{f} \ar_{g}[d]
    &{B} \ar^{h}[d]\\
    {C} \ar_{k}[r]
    &{D}
  }
\end{displaymath}
For another example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A \amalg_{B} C} \ar[r]^{\text{Add}}_{\sigma} \ar[d]
    & {X} \ar[d]^{\psi}\\
    {Y} \ar[r]^{\alpha}_{\text{Lifted}} \ar[r]
    & {P\times_{Q}R}
  }
\end{displaymath}
\end{verbatim}
you'll get
\begin{displaymath}
  \xymatrix{
    {A \amalg_{B} C} \ar[r]^{\text{Add}}_{\sigma} \ar[d]
    & {X} \ar[d]^{\psi}\\
    {Y} \ar[r]^{\alpha}_{\text{Lifted}} \ar[r]
    & {P\times_{Q}R}
  }
\end{displaymath}
but if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A \amalg_{B} C} \ar[r]^-{\text{Add}}_-{\sigma} \ar[d]
    & {X} \ar[d]\\
    {Y} \ar[r]^-{\alpha}_-{\text{Lifted}}
    & {P\times_{Q}R}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A \amalg_{B} C} \ar[r]^-{\text{Add}}_-{\sigma} \ar[d]
    & {X} \ar[d]\\
    {Y} \ar[r]^-{\alpha}_-{\text{Lifted}}
    & {P\times_{Q}R}
  }
\end{displaymath}

%--------------------------------------------------------------------
\subsubsection{Arbitrary placement of labels}
\label{sec:LblArb}

In addition to the possibility of centering a label along an arrow
(see section~\ref{sec:arcent}), it is possible to place a label at an
arbitrary point between the center of the source and the center of the
target.  If \texttt{a} is a number between $0$ and $1$, then you can
place a label \texttt{a} of the way from the center of the source to
the center of the target by typing \verb"(a)" immediately following
the \verb"^" or \verb"_".  For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix@C=8em{
    {A} \ar[r]^(.3){f} \ar[d]
    & {B} \ar[d]\\
    {C} \ar[r]^{g} \ar[d]
    & {D} \ar[d]\\
    {E} \ar[r]_(.7){h}
    & {F}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix@C=8em{
    {A} \ar[r]^(.3){f} \ar[d]
    & {B} \ar[d]\\
    {C} \ar[r]^{g} \ar[d]
    & {D} \ar[d]\\
    {E} \ar[r]_(.7){h}
    & {F}
  }
\end{displaymath}
For a more elaborate example, with labels along segments of segmented
arrows, see diagram~\ref{diag:LblCube}.


%--------------------------------------------------------------------
\subsubsection{Labeling each segment of a segmented arrow}
\label{sec:LblSeg}

An arrow that uses the ``arrows passing under'' feature (see
section~\ref{sec:ArUnder}) will be in several segments, and a label
can be attached to each of the segments.  For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar '[r]^{f} '[rr]^{g} [rrr]^{h}
    & {B}
    & {}
    & {D}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar '[r]^{f} '[rr]^{g} [rrr]^{h}
    & {B}
    & {}
    & {D}
  }
\end{displaymath}
For a more elaborate example, if you type
\begin{verbatim}
\begin{equation} 
  \label{diag:LblCube}
  \vcenter{
    \xymatrix@=2ex{ 
      {A} \ar[rr]^{f} \ar[dr] \ar[dd]_{i}
      && {B} \ar[dr] \ar'[d][dd]^(.3){j}\\ 
      & {A'} \ar[rr]^(.3){f'} \ar[dd]^(.25){i'}
      && {B'} \ar[dd]^{j'}\\ 
      {C} \ar'[r]^{g}[rr] \ar[dr] 
      && {D} \ar[dr]\\ 
      & {C'} \ar[rr]_{g'}
      && {D'} 
    }
  }
\end{equation} 
\end{verbatim}
then you'll get
\begin{equation} 
  \label{diag:LblCube}
  \vcenter{
    \xymatrix@=2ex{ 
      {A} \ar[rr]^{f} \ar[dr] \ar[dd]_{i}
      && {B} \ar[dr] \ar'[d][dd]^(.3){j}\\ 
      & {A'} \ar[rr]^(.3){f'} \ar[dd]^(.25){i'}
      && {B'} \ar[dd]^{j'}\\ 
      {C} \ar'[r]^{g}[rr] \ar[dr] 
      && {D} \ar[dr]\\ 
      & {C'} \ar[rr]_{g'}
      && {D'} 
    }
  }
\end{equation} 


%--------------------------------------------------------------------
\subsubsection{Breaking an arrow with a label}
\label{sec:arbreak}

Instead of placing a label to the side of an arrow, you can have the
label ``break'' the arrow.  For this, you use the vertical bar
character \verb"|" in place of either \verb"^" or \verb"_".

For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar[r]|{f}
    & {B} \ar[r]|{g}
    & {C}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar[r]|{f}
    & {B} \ar[r]|{g}
    & {C}
  }
\end{displaymath}

A label that breaks an arrow can be positioned anywhere along the
arrow, just as for labels placed alongside the arrow (see
sections~\ref{sec:arcent} and \ref{sec:LblArb}).  For example, if you
type
\begin{verbatim}
\begin{equation*}
  \xymatrix{
    {A \amalg_{B} C} \ar[r]|-{f} \ar[d]_{i}
    & {X} \ar[r]|(.7){m} \ar[d]^{p}
    & {Y} \ar[dl]|(.3)n\\
    {D} \ar[r]|-{g}
    & {W \times_{Z} Y}
  }
\end{equation*}
\end{verbatim}
then you'll get
\begin{equation*}
  \xymatrix{
    {A \amalg_{B} C} \ar[r]|-{f} \ar[d]_{i}
    & {X} \ar[r]|(.7){m} \ar[d]^{p}
    & {Y} \ar[dl]|(.3)n\\
    {D} \ar[r]|-{g}
    & {W \times_{Z} Y}
  }
\end{equation*}
%--------------------------------------------------------------------
\subsection{More crossing arrows}
\label{sec:CrossArrow}

In section~\ref{sec:ArUnder} we presented a method of having an arrow
pass over another arrow by having them cross at an empty node.  We
present here another method of having one arrow pass over another.

To have one arrow appear to cross over another, we create a small gap
in the second arrow at the spot at which the first arrow will cross.
We do this by breaking that second arrow with a label, as in
section~\ref{sec:arbreak}, using the special label \verb"\hole", which
just leaves an empty hole in the arrow.  For example, if you type
\begin{verbatim}
\begin{equation*}
  \xymatrix{
    {A} \ar[dr]|{\hole}
    & {B} \ar[dl]\\
    {C}
    & {D}
  }
\end{equation*}
\end{verbatim}
then you'll get
\begin{equation*}
  \xymatrix{
    {A} \ar[dr]|{\hole}
    & {B} \ar[dl]\\
    {C}
    & {D}
  }
\end{equation*}

For another example, if you type
\begin{verbatim}
\begin{equation*}
  \xymatrix{
    {A} \ar[dr] \ar[r] \ar[d]
    & {B} \ar[r]
    & {C}\\
    {D} \ar[urr]|(.33){\hole} \ar[r]
    & {E}
  }
\end{equation*}
\end{verbatim}
then you'll get
\begin{equation*}
  \xymatrix{
    {A} \ar[dr] \ar[r] \ar[d]
    & {B} \ar[r]
    & {C}\\
    {D} \ar[urr]|(.33){\hole} \ar[r]
    & {E}
  }
\end{equation*}
(see section~\ref{sec:LblArb}).  It took some trial and error to
figure out that \texttt{.33} was the number to use in the arrow with
the hole, but there's a much better way specify the location of the
hole: Instead of inserting that number between 0 and 1 that specifies
how far from the center of the source to the center of the target the
break should appear, you can specify the place to put the break using
the notation \verb"!{[node1];[node2]}" to denote the point on the
arrow at which a straight line from \verb"[node1]" to \verb"[node2]"
would cross the arrow.  (\emph{Note:} This method only works for
straight line arrows.)  That is, if you type
\begin{verbatim}
\begin{equation*}
  \xymatrix{
    {A} \ar[dr] \ar[r] \ar[d]
    & {B} \ar[r]
    & {C}\\
    {D} \ar[urr]|!{[u];[r]}{\hole} \ar[r]
    & {E}
  }
\end{equation*}
\end{verbatim}
then you'll get
\begin{equation*}
  \xymatrix{
    {A} \ar[dr] \ar[r] \ar[d]
    & {B} \ar[r]
    & {C}\\
    {D} \ar[urr]|!{[u];[r]}{\hole} \ar[r]
    & {E}
  }
\end{equation*}
The position of the hole is specified as \verb"!{[u];[r]}" because the
arrow with the hole begins at $D$ and, relative to that node, the
arrow that crosses over goes from the node \verb"[u]" to the node
\verb"[r]".

For another example, if you type
\begin{verbatim}
\begin{equation*}
  \xymatrix{
    {A} \ar[d] \ar[dr]
    & {B} \ar[d] \ar[dr]
    & {C} \ar[d]\\
    {D}
    \ar[urr]|!{[u];[r]}{\hole}
            |!{[ur];[r]}{\hole}
            |!{[ur];[rr]}{\hole}
    & {E}
    & {F}
  }
\end{equation*}
\end{verbatim}
then you'll get
\begin{equation*}
  \xymatrix{
    {A} \ar[d] \ar[dr]
    & {B} \ar[d] \ar[dr]
    & {C} \ar[d]\\
    {D}
    \ar[urr]|!{[u];[r]}{\hole}
            |!{[ur];[r]}{\hole}
            |!{[ur];[rr]}{\hole}
    & {E}
    & {F}
  }
\end{equation*}


%--------------------------------------------------------------------
\subsection{Making room for large labels}
\label{sec:labelroom}

Although \Xy-pic automatically leaves enough room for large nodes, it
ignores the size of the labels on the arrows when constructing the
diagram.  Thus, if you have a particularly large label, it may
overwrite the nodes if you don't do something to make the diagram
larger.  You can do that either by enlarging the entire diagram (see
section~\ref{sec:LblEnlrgeDiag}) or by creating a row (or column) all
of whose nodes are empty and have the arrows span the empty row (or
column) (see section~\ref{sec:emptynodes}).

%--------------------------------------------------------------------
\subsubsection{Enlarging the entire diagram}
\label{sec:LblEnlrgeDiag}

To enlarge the entire diagram, we use the spacing commands described
in section~\ref{sec:ChgSpc}.  For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A\amalg B} \ar[d]
    \ar[r]^-{\operatorname{amalgamate}}
    & {X} \ar[r] \ar[d]
    & {P} \ar[d]\\
    {C \amalg D}
    \ar[r]_-{\operatorname{amalgamate}}
    & {Y} \ar[r]
    & {Q}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A\amalg B} \ar[d]
    \ar[r]^-{\operatorname{amalgamate}}
    & {X} \ar[r] \ar[d]
    & {P} \ar[d]\\
    {C \amalg D}
    \ar[r]_-{\operatorname{amalgamate}}
    & {Y} \ar[r]
    & {Q}
  }
\end{displaymath}
but if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix@C=6em{
    {A\amalg B} \ar[d] \ar[r]^-{\operatorname{amalgamate}}
    & {X} \ar[r] \ar[d]
    & {P} \ar[d]\\
    {C \amalg D} \ar[r]_-{\operatorname{amalgamate}}
    & {Y} \ar[r]
    & {Q}
  }
\end{displaymath}
\end{verbatim}
(see section~\ref{sec:ChgSpc}) then you'll get
\begin{displaymath}
  \xymatrix@C=6em{
    {A\amalg B} \ar[d] \ar[r]^-{\operatorname{amalgamate}}
    & {X} \ar[r] \ar[d]
    & {P} \ar[d]\\
    {C \amalg D} \ar[r]_-{\operatorname{amalgamate}}
    & {Y} \ar[r]
    & {Q}
  }
\end{displaymath}
If you don't want to have to make the entire diagram larger, you can
just create a column of empty nodes (see
section~\ref{sec:emptynodes}).
%--------------------------------------------------------------------
\subsubsection{Creating a column of empty nodes}
\label{sec:emptynodes}

We continue the discussion of section~\ref{sec:LblEnlrgeDiag}: Instead
of enlarging the entire diagram, you can create an additional column
in between the first two columns and have all the nodes in the new
column be empty.  For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A\amalg B} \ar[d] \ar[rr]^-{\operatorname{amalgamate}}
    && {X} \ar[r] \ar[d]
    & {P} \ar[d]\\
    {C \amalg D} \ar[rr]_-{\operatorname{amalgamate}}
    && {Y} \ar[r]
    & {Q}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A\amalg B} \ar[d] \ar[rr]^-{\operatorname{amalgamate}}
    && {X} \ar[r] \ar[d]
    & {P} \ar[d]\\
    {C \amalg D} \ar[rr]_-{\operatorname{amalgamate}}
    && {Y} \ar[r]
    & {Q}
  }
\end{displaymath}
Note that that diagram has four columns, but both of the nodes in the
second column are empty, and the labelled arrows are created as
\verb"\ar[rr]".  If you'd like to create even more space for that
large label, you can put an \verb"\hspace" command into one of the
nodes in the second column.  For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A\amalg B} \ar[d] \ar[rr]^-{\operatorname{amalgamate}}
    & {\hspace{3em}}
    & {X} \ar[r] \ar[d]
    & {P} \ar[d]\\
    {C \amalg D} \ar[rr]_-{\operatorname{amalgamate}}
    && {Y} \ar[r]
    & {Q}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A\amalg B} \ar[d] \ar[rr]^-{\operatorname{amalgamate}}
    & {\hspace{3em}}
    & {X} \ar[r] \ar[d]
    & {P} \ar[d]\\
    {C \amalg D} \ar[rr]_-{\operatorname{amalgamate}}
    && {Y} \ar[r]
    & {Q}
  }
\end{displaymath}
Note that we only put the \verb"\hspace" command into the first row;
there's no need to put it into every row.

%--------------------------------------------------------------------
\subsection{Different arrow styles}
\label{sec:arstyle}

It's possible to have arrows with tails, multiple heads, dotted,
dashed, or multiple shafts, and any combination of these (see
Table~\ref{tab:ArSty}).  You can even omit both the head and the tail,
or omit the arrow entirely, which is useful for placing things into
the diagram in places outside of the grid of nodes (see
Diagram~\ref{diag:CentEq}).
\begin{table}
  \centering
  \begin{tabular}{c@{\qquad\qquad}l}
    To produce:& \multicolumn{1}{c}{Type:}\\[1ex]
    $\xymatrix{\ar@{.>}[r]&}$& \verb"\ar@{.>}[r]"\\
    $\xymatrix{\ar@{-->}[r]&}$& \verb"\ar@{-->}[r]"\\
    $\xymatrix{\ar@{=>}[r]&}$& \verb"\ar@{=>}[r]"\\
    $\xymatrix{\ar@{:>}[r]&}$& \verb"\ar@{:>}[r]"\\
    $\xymatrix{\ar@{->>}[r]&}$& \verb"\ar@{->>}[r]"\\
    $\xymatrix{\ar@{>->}[r]&}$& \verb"\ar@{>->}[r]"\\
    $\xymatrix{\ar@{<->}[r]&}$& \verb"\ar@{<->}[r]"\\
    $\xymatrix{\ar@{-->>}[r]&}$& \verb"\ar@{-->>}[r]"\\
    $\xymatrix{\ar@{>-->}[r]&}$& \verb"\ar@{>-->}[r]"\\
    $\xymatrix{\ar@{<-->}[r]&}$& \verb"\ar@{<-->}[r]"\\
    $\xymatrix{\ar@{|->}[r]&}$& \verb"\ar@{|->}[r]"\\
    $\xymatrix{\ar@{^{(}->}[r]&}$& \verb"\ar@{^{(}->}[r]"\\
    $\xymatrix{\ar@{_{(}->}[r]&}$& \verb"\ar@{_{(}->}[r]"\\
    $\xymatrix{\ar@{=>>}[r]&}$& \verb"\ar@{=>>}[r]"\\
    $\xymatrix{\ar@{<=>}[r]&}$& \verb"\ar@{<=>}[r]"\\
    $\xymatrix{\ar@{:>>}[r]&}$& \verb"\ar@{:>>}[r]"\\
    $\xymatrix{\ar@{<:>}[r]&}$& \verb"\ar@{<:>}[r]"\\
    $\xymatrix{\ar@{-}[r]&}$& \verb"\ar@{-}[r]"\\
    $\xymatrix{\ar@{.}[r]&}$& \verb"\ar@{.}[r]"\\
    $\xymatrix{\ar@{=}[r]&}$& \verb"\ar@{=}[r]"\\
    $\xymatrix{\ar@{}[r]&}$& \verb"\ar@{}[r]"\\[1ex]
  \end{tabular}
  \caption{Arrow Styles}
  \label{tab:ArSty}
\end{table}

All of these arrows can point in whatever direction you choose; we
used \verb"[r]" in the table just for readability.  For an example of
the use of invisible arrows: If you type
\begin{verbatim}
\begin{equation}
  \label{diag:CentEq}
  \vcenter{
    \xymatrix{
      {A} \ar@{.>}[r] \ar@{.>}[d] \ar@{}[dr]|{=}
      & {B} \ar[d]\\
      {C} \ar[r]
      & {D}
    }
  }
\end{equation}
\end{verbatim}
then you'll get
\begin{equation}
  \label{diag:CentEq}
  \vcenter{
    \xymatrix{
      {A} \ar@{.>}[r] \ar@{.>}[d] \ar@{}[dr]|{=}
      & {B} \ar[d]\\
      {C} \ar[r]
      & {D}
    }
  }
\end{equation}
Note that the equals sign is inserted by having it break the middle of
the invisible arrow from the upper left node to the lower right node.

%--------------------------------------------------------------------
\subsection{Curved arrows}
\label{sec:arcurve}

It's possible to have arrows curve, either by specifying the amount
that they curve or by specifying the direction in which they leave
their source and the direction from which they arrive at their
target.

%--------------------------------------------------------------------
\subsubsection{Specifying the amount of the curve}

To have an arrow curve in the up direction (where ``up'' means when
the paper is oriented so that the arrow goes left to right) by a
default amount you follow the \verb"\ar" with \verb"@/^/"; that is,
you type \verb"\ar@/^/[target]".  To have it curve in the down
direction by a default amount you follow the \verb"\ar" with
\verb"@/_/"; that is, you type \verb"\ar@/_/[target]" (or, if you want
to label the arrow, \verb"\ar@/_/[target]^{label}").  Thus, if you
type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar@/^/[drr]^{p} \ar@{.>}[dr]|{\exists!} \ar@/_/[ddr]_{q}\\
    & {B} \ar[r] \ar[d]
    & {C} \ar[d]\\
    & {D} \ar[r]
    & {E}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar@/^/[drr]^{p} \ar@{.>}[dr]|{\exists!} \ar@/_/[ddr]_{q}\\
    & {B} \ar[r] \ar[d]
    & {C} \ar[d]\\
    & {D} \ar[r]
    & {E}
  }
\end{displaymath}
If you'd like to specify the amount of curve, you can specify a
dimension following the \verb"^" or the \verb"_", as in
\verb"\ar@/^1ex/[target]".  (Although you can use whatever units you
like for \texttt{dimen} (e.g., inches (\texttt{in}), centimeters
(\texttt{cm}), points (\texttt{pt}), etc.), it's a good idea to
measure vertical distances in \texttt{ex} (roughly the height of an
upper case ``X'') and horizontal distances in \texttt{em} (roughly the
width of an upper case ``M''), so that the distance will be roughly
correct even if you change font sizes.)  For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar@/^3ex/[drr]^{p} \ar@{.>}[dr]|{\exists!}
        \ar@/_3ex/[ddr]_{q}\\
    & {B} \ar[r] \ar[d]
    & {C} \ar[d]\\
    & {D} \ar[r]
    & {E}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar@/^3ex/[drr]^{p} \ar@{.>}[dr]|{\exists!}
        \ar@/_3ex/[ddr]_{q}\\
    & {B} \ar[r] \ar[d]
    & {C} \ar[d]\\
    & {D} \ar[r]
    & {E}
  }
\end{displaymath}
and if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar@{>->}[r]^{i}
    & {B} \ar@{->>}[r]^{p}
    & {C} \ar@{.>}@/_4ex/[l]_{s}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar@{>->}[r]^{i}
    & {B} \ar@{->>}[r]^{p}
    & {C} \ar@{.>}@/_4ex/[l]_{s}
  }
\end{displaymath}


%--------------------------------------------------------------------
\subsubsection{Specifying the start and end directions}
\label{sec:CrvStEd}

To specify the start and end directions of a curved arrow you type
\verb"\ar@(start,end)[target]" where ``\verb"start"'' is the direction
towards which the arrow begins and ``\verb"end"'' is the direction
from which the arrow ends.  For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar[dr]|{p} \ar@(r,u)[dr]^{f} \ar@(d,l)[dr]_{g}\\
    & {B}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar[dr]|{p} \ar@(r,u)[dr]^{f} \ar@(d,l)[dr]_{g}\\
    & {B}
  }
\end{displaymath}
and if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar@(ur,ul)[rr]^{1_{A}} \ar[r]_{f}
    & {B} \ar[r]_{g} \ar@(dr,dl)[rr]_{1_{B}}
    & {A} \ar[r]_{f}
    & {B}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar@(ur,ul)[rr]^{1_{A}} \ar[r]_{f}
    & {B} \ar[r]_{g} \ar@(dr,dl)[rr]_{1_{B}}
    & {A} \ar[r]_{f}
    & {B}
  }
\end{displaymath}
Since the current node is denoted \verb"[]" (i.e., a pair of square
brackets with nothing inside), if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar@(ur,dr)[]^{f}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar@(ur,dr)[]^{f}
  }
\end{displaymath}

%--------------------------------------------------------------------
\subsection{Sliding arrows sideways; multiple arrows}
\label{sec:arslide}

It's possible to slide arrows sideways, so that you can have more than
one straight line arrow between a single pair of nodes.  To do this,
you type \verb"\ar@<dimen>" to move the arrow ``upwards'' by
\verb"dimen" (where ``upwards'' means when the paper is held so that
the arrow goes from left to right).  (A negative \verb"dimen" will
move the arrow ``downwards''.)  Although you can use whatever units
you like for \texttt{dimen} (e.g., inches (\texttt{in}), centimeters
(\texttt{cm}), points (\texttt{pt}), etc.), it's a good idea to
measure vertical distances in \texttt{ex} (roughly the height of an
upper case ``X'') and horizontal distances in \texttt{em} (roughly the
width of an upper case ``M''), so that the distance will be roughly
correct even if you change font sizes.  For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar@<1ex>[r]
    & {B} \ar[r]
    & {C} \ar@<-1ex>[r]
    & {D}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar@<1ex>[r]
    & {B} \ar[r]
    & {C} \ar@<-1ex>[r]
    & {D}
  }
\end{displaymath}
For another example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{ 
    {\coprod_{i \in I} A_{i}\quad}
      \ar@<+.7ex>[r]^-{\phi} \ar@<-.7ex>[r]_-{\psi} 
    & {\quad\coprod_{i \in I} B_{i}}
  } 
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{ 
    {\coprod_{i \in I} A_{i}\quad}
      \ar@<+.7ex>[r]^-{\phi} \ar@<-.7ex>[r]_-{\psi} 
    & {\quad\coprod_{i \in I} B_{i}}
  } 
\end{displaymath}
It's also possible to slide curved arrows sideways.  For example, if
you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar@/^ 1ex/[r] \ar@/^ 1ex/@<1ex>[r]
    & {B}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar@/^ 1ex/[r] \ar@/^ 1ex/@<1ex>[r]
    & {B}
  }
\end{displaymath}


%--------------------------------------------------------------------
\subsection{Specifying both the source and the target of an arrow}
\label{sec:SrcTrgt}

Up until now, all of our examples of arrows have gone from the current
node to a node specified in the code for the arrow.  It's possible,
however, to specify both the source and the target of an arrow, rather
than letting the source be the current node.  This allows you to save
the description of all of the arrows of a diagram until the end.

To draw an arrow who's source may not be at the current node, you type
\begin{center}
  \verb"\ar[source];[target]"
\end{center}
instead of just \verb"\ar[target]".  Since the current node is denoted
\verb"[]", typing \verb"\ar[target]" is the same as typing
\verb"\ar[];[target]".

For example, if you
type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A}
    & {B}\\
    {C}
    & {D}
    \ar^{f}[ul];[u]
    \ar_{g}[l];[]
    \ar_{h}[ul];[l]
    \ar^{k}[u];[]
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A}
    & {B}\\
    {C}
    & {D}
    \ar^{f}[ul];[u]
    \ar_{g}[l];[]
    \ar_{h}[ul];[l]
    \ar^{k}[u];[]
  }
\end{displaymath}
Note that when we drew the arrows the current node was $D$, and so,
e.g., the target of the arrow labelled $g$ was specified as \verb"[]"
(which denotes the current node).

%--------------------------------------------------------------------
\subsection{Absolute specification of the source and target of an arrow}
\label{sec:AbsDesc}

Up until now, the target and source (see section~\ref{sec:SrcTrgt}) of
an arrow were always specified relative to the current node (see
section~\ref{sec:basicdiag}).  It is possible to give instead an
absolute specification of the target and source of an arrow.

Each node of a diagram can be specified as
\begin{center}
  \verb!"p,q"!
\end{center}
(including the double quote symbols) where \verb"p" is the row number
(counting from the top, where the first row is row number $1$) and
\verb"q" is the column number (counting from the left, where the first
column is column number $1$).  For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar^{f}"1,2" \ar_{h}"2,1"
    & {B} \ar^{k}"2,2"\\
    {C} \ar_{g}"2,2"
    & {D}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar^{f}"1,2" \ar_{h}"2,1"
    & {B} \ar^{k}"2,2"\\
    {C} \ar_{g}"2,2"
    & {D}
  }
\end{displaymath}
In that example, each arrow began at the current node.  For example,
the arrow labelled $f$ began at the current node $A$ (and so it's
source was not specified) and its target $B$ was specified as
\verb!"1,2"! because it is in the first row, second column.

As described in section~\ref{sec:SrcTrgt}, you can also specify both
the source and target of an arrow, and this allows you to save the
specification of all the arrows until the end of the diagram.  For
example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A}
    & {B}
    & {C}\\
    {D}
    & {E}
    & {F}
    \ar_{f} "1,1";"1,2"
    \ar_{g} "1,2";"1,3"
    \ar@(ur,ul)^{h} "1,1";"1,3"
    \ar"1,1";"2,1"
    \ar"1,2";"2,2"
    \ar"1,3";"2,3"
    \ar^{p} "2,1";"2,2"
    \ar^{q} "2,2";"2,3"
    \ar@(dr,dl)_{r} "2,1";"2,3"
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A}
    & {B}
    & {C}\\
    {D}
    & {E}
    & {F}
    \ar_{f} "1,1";"1,2"
    \ar_{g} "1,2";"1,3"
    \ar@(ur,ul)^{h} "1,1";"1,3"
    \ar"1,1";"2,1"
    \ar"1,2";"2,2"
    \ar"1,3";"2,3"
    \ar^{p} "2,1";"2,2"
    \ar^{q} "2,2";"2,3"
    \ar@(dr,dl)_{r} "2,1";"2,3"
  }
\end{displaymath}
(For a description of the curved arrows, see
section~\ref{sec:CrvStEd}.)  Note that, e.g., the arrow labelled $f$
is specified as
\begin{center}
  \verb!\ar_{f} "1,1";"1,2"!
\end{center}
because it starts at $A$, which is in row $1$ column $1$, and goes to
$B$, which is in row $1$ column $2$.

%--------------------------------------------------------------------
\subsection{Nodes off of the grid}
\label{sec:NdOffGrd}

It's possible to place a node at an arbitrary point in the diagram
without affecting the positioning of the other nodes.  You can also
assign a \emph{name} to the node that you can use later to draw arrows
to or from that node.

You place a node off the grid using the \emph{excursion} commands
\begin{center}
  \verb"\save"\qquad and\qquad\verb"\restore"
\end{center}
In between those commands you can put commands to
\begin{itemize}
\item move from the current node using a displacement vector
  ``\verb"[]+<Dx,Dy>"'' (where \verb"Dx" is the horizontal
  displacement and \verb"Dy" is the vertical displacement),
\item typeset a node there using ``\verb"*+{stuff}"'',
\item optionally name that node using ``\verb!*+{stuff}="name"!'', and
\item optionally draw arrows from whatever you've typeset there to
  other nodes in the diagram.  If you've named the new node, you can
  specify the arrows later on, and the arrows can either start from
  this node or go to this node (or both, as in the second example in
  this section).
\end{itemize}
Although you can use whatever units you like for the displacements
(e.g., inches (\texttt{in}), centimeters (\texttt{cm}), points
(\texttt{pt}), etc.), it's a good idea to measure vertical distances
in \texttt{ex} (roughly the height of an upper case ``X'') and
horizontal distances in \texttt{em} (roughly the width of an upper
case ``M''), so that the displacement will be roughly correct even if
you change font sizes.

For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar[r] \ar[d]
    & {B} \ar[d]
    \save []+<3em, 3em>*+{H}
      \ar@(l,ur)_{p}[l] \ar@(d,ur)^{r}[d] \ar_{q}[]
    \restore
    \\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar[r] \ar[d]
    & {B} \ar[d]
    \save []+<3em, 3em>*+{H}
      \ar@(l,ur)_{p}[l] \ar@(d,ur)^{r}[d] \ar_{q}[]
    \restore
    \\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}
We created the node $H$ off of the grid by starting at the node $B$
(in the first row, second column) and moving \verb"3em" to the right
and \verb"3em" up.  We then drew three arrows starting at that node:
\begin{itemize}
\item The arrow $p$ goes to the node $A$, and so its target is
  specified as \verb"[l]" since that is one node to the left of the
  node $B$ (which is the current node from which we were on an
  excursion),
\item the arrow $q$ goes to the node $B$, and so its target is
  specified as \verb"[]" (a pair of square brackets with nothing
  inside) since that is the current node, and
\item the arrow $r$ goes to the node $D$, and so its target is
  specified as \verb"[d]" since it is one node below the current
  node.
\end{itemize}

For another example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar[r] \ar[d]
    & {B} \ar[d]\\
    {C} \ar[r]
    & {D}
    \save []+<4em,1em>*+{H}="firstthing" \restore
    \save []+<-1em,-3em>*+{K}="secondthing" \restore
    \ar"firstthing";"1,2"_{p}
    \ar"firstthing";"2,2"_{q}
    \ar@(d,r)"firstthing";"secondthing"^{r}
    \ar"secondthing";"2,2"_{s}
    \ar"secondthing";"2,1"^{t}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar[r] \ar[d]
    & {B} \ar[d]\\
    {C} \ar[r]
    & {D}
    \save []+<4em,1em>*+{H}="firstthing" \restore
    \save []+<-1em,-3em>*+{K}="secondthing" \restore
    \ar"firstthing";"1,2"_{p}
    \ar"firstthing";"2,2"_{q}
    \ar@(d,r)"firstthing";"secondthing"^{r}
    \ar"secondthing";"2,2"_{s}
    \ar"secondthing";"2,1"^{t}
  }
\end{displaymath}
In this diagram, the only nodes on the grid are $A$, $B$, $C$, and
$D$.  After creating the nodes and arrows on the grid, we created the
nodes $H$ and $K$ and the arrows that go to and from these nodes.
\begin{itemize}
\item We created each of the nodes $H$ and $K$ with an excursion from
  the node $D$, giving the name \texttt{firstthing} to the node $H$
  and the name \texttt{secondthing} to the node \verb"K".
\item Each of the arrows to or from the nodes $H$ and $K$ specify both
  their source and their target (a semicolon separates the source from
  the target; see section~\ref{sec:SrcTrgt}).
\item Each of the arrows from either $H$ or $K$ to one of the nodes on
  the grid describes that node on the grid absolutely (see
  section~\ref{sec:AbsDesc}).
\end{itemize}
The arrow from $H$ to $K$ begins \verb"\ar@(d,r)" because it is a
curved arrow (see section~\ref{sec:CrvStEd}).  Since that arrow goes
from the node $H$ (named \texttt{firsthing}) to the node $K$ (named
\texttt{secondthing}), the arrow is specified as
\begin{center}
  \verb!\ar@(d,r)"firstthing";"secondthing"!
\end{center}
Finally, that arrow is labelled $r$, and so the complete specification
is
\begin{center}
  \verb!\ar@(d,r)"firstthing";"secondthing"^{r}!
\end{center}

For another example, we show how to annotate ``diagram chase'' proofs
in homological algebra.  If you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar[r]^{\alpha} \ar[d]_{f}
    & {B} \ar[r]^{\beta} \ar[d]_{g}
    % Add the element b of B:
    \save []+<2ex,3ex>*+{b}="b"\restore
    & {C} \ar[r]^{\gamma} \ar[d]_{h}
    % Add the element c of C:
    \save []+<0ex,3ex>*+{c}="c"\restore
    & {D} \ar[r]^{\delta} \ar[d]_{j}
    % Add the element d of D:
    \save []+<2ex,3ex>*+{d}="d"\restore
    & {E} \ar[d]_{k}\\
    {A'} \ar[r]^{\alpha'}
    & {B'} \ar[r]^{\beta'}
    % Add the element b' of B':
    \save []+<2ex,-5ex>*+{b'}="bprime"\restore
    & {C'} \ar[r]^{\gamma'}
    % Add the element c' of C':
    \save []+<0ex,-3ex>*+{c'}="cprime"\restore
    % Add the element c'-gamma(c) of C':
    \save []+<1.5ex,-5ex>*+{(c'-\gamma(c))}="cprimeminus"\restore
    & {D'} \ar[r]^{\delta'}
    % Add the element \gamma'(c') of D':
    \save []+<2ex,-3ex>*+{\gamma'(c')}="gammacprime"\restore
    & {E'}
    % Add the arrows connecting the elements:
    \ar@{|->}"b";"bprime"
    \ar@{|->}"c";"d"
    \ar@{|->}"d";"gammacprime"
    \ar@{|->}"cprime";"gammacprime"
    \ar@{|->}"bprime";"cprimeminus"
  }
\end{displaymath}
\end{verbatim}
then you'll get the following diagram, which that might be used in
part of the proof of the ``five lemma''.
\begin{displaymath}
  \xymatrix{
    {A} \ar[r]^{\alpha} \ar[d]_{f}
    & {B} \ar[r]^{\beta} \ar[d]_{g}
    % Add the element b of B:
    \save []+<2ex,3ex>*+{b}="b"\restore
    & {C} \ar[r]^{\gamma} \ar[d]_{h}
    % Add the element c of C:
    \save []+<0ex,3ex>*+{c}="c"\restore
    & {D} \ar[r]^{\delta} \ar[d]_{j}
    % Add the element d of D:
    \save []+<2ex,3ex>*+{d}="d"\restore
    & {E} \ar[d]_{k}\\
    {A'} \ar[r]^{\alpha'}
    & {B'} \ar[r]^{\beta'}
    % Add the element b' of B':
    \save []+<2ex,-5ex>*+{b'}="bprime"\restore
    & {C'} \ar[r]^{\gamma'}
    % Add the element c' of C':
    \save []+<0ex,-3ex>*+{c'}="cprime"\restore
    % Add the element c'-gamma(c) of C':
    \save []+<1.5ex,-5ex>*+{(c'-\gamma(c))}="cprimeminus"\restore
    & {D'} \ar[r]^{\delta'}
    % Add the element \gamma'(c') of D':
    \save []+<2ex,-3ex>*+{\gamma'(c')}="gammacprime"\restore
    & {E'}
    % Add the arrows connecting the elements:
    \ar@{|->}"b";"bprime"
    \ar@{|->}"c";"d"
    \ar@{|->}"d";"gammacprime"
    \ar@{|->}"cprime";"gammacprime"
    \ar@{|->}"bprime";"cprimeminus"
  }
\end{displaymath}

%--------------------------------------------------------------------
\subsection{Text objects}
\label{sec:TextObj}

You can insert a text object into a diagram by typing
\begin{center}
  \verb"*\txt{Some text}"\quad or\quad \verb"*\txt<width>{Some text}"
\end{center}
In the first form you create linebreaks with a double backslash
``\verb"\\"'' and in the second form the lines are broken
automatically.  Text objects by default don't have any blank space
around them, but you can add a bit of space by typing them as
\verb"*+\txt{Some text}", or even more space by typing them as
\verb"*++\txt{Some text}".

For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    *+\txt{Roses are red,\\ Violets are blue.\\
      This is a contrived example,\\ And the rhyme is bad too.}
    \ar@{=>}[r]
    & *\txt<2in>{Four score and seven years ago our fathers brought
      forth on this continent a new nation, conceived in liberty,
      and dedicated to the proposition that all men are created
      equal.}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    *+\txt{Roses are red,\\ Violets are blue.\\
      This is a contrived example,\\ And the rhyme is bad too.}
    \ar@{=>}[r]
    & *\txt<2in>{Four score and seven years ago our fathers brought
      forth on this continent a new nation, conceived in liberty,
      and dedicated to the proposition that all men are created
      equal.}
  }
\end{displaymath}
We typed that first one as \verb"*+\txt" so that the arrow wouldn't
start too close to the text.

You can also shift a text object to the left (so that arrows to or
from the object will go to its right end instead of its center) by
typing it as
%
\verb"*+[l]\txt{Some text}" or to the right (so that arrows to or from
the object will go to its left end instead of its center) by typing it
as
%
\verb"*+[r]\txt{Some text}" (see section~\ref{sec:ShiftSide}).  For
example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    *+[l]\txt{Interior} \ar[r] \ar[d]
    & *+[r]\txt{Exterior} \ar[d]\\
    {A} \ar[r]
    & {B}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    *+[l]\txt{Interior} \ar[r] \ar[d]
    & *+[r]\txt{Exterior} \ar[d]\\
    {A} \ar[r]
    & {B}
  }
\end{displaymath}

Text objects are mostly used for adding annotations to a diagram (see
section~\ref{sec:Drop}).

%--------------------------------------------------------------------
\subsection{Arrows off of the grid}
\label{sec:OffGrid}

It's possible to draw arrows that begin or end at points other than
nodes.  To do this, you specify the point by choosing a node and then
adding a displacement vector ``\verb"+<Dx,Dy>"'' from that node (where
\verb"Dx" is the horizontal displacement and \verb"Dy" is the vertical
displacement).  (Although you can use whatever units you like for the
displacements (e.g., inches (\texttt{in}), centimeters (\texttt{cm}),
points (\texttt{pt}), etc.), it's a good idea to measure vertical
distances in \texttt{ex} (roughly the height of an upper case ``X'')
and horizontal distances in \texttt{em} (roughly the width of an upper
case ``M''), so that the displacement will be roughly correct even if
you change font sizes.)  For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar[d]+<4em,0ex>\\
    {B}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar[d]+<4em,0ex>\\
    {B}
  }
\end{displaymath}
since the arrow is pointing to the spot 4em to the right of the $B$.

For another example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A}
    & {B}\\
    {C}
    & {D}
    \ar[ul]+<-2em,0ex>;[]+<2em,0ex>
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A}
    & {B}\\
    {C}
    & {D}
    \ar[ul]+<-2em,0ex>;[]+<2em,0ex>
  }
\end{displaymath}
(see section~\ref{sec:SrcTrgt}) because we specified the source as
being \verb"[ul]+<-2em,0ex>", i.e., 2em to the left of the node that's
one up and one to the left of the current node ($D$) and the target as
being \verb"[]+<2em,0ex>", i.e., 2em to the right of the current node
($D$).

For an example that combines displacements of source and target along
with curving arrows and labelled arrows: if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar[r]+<0em,-3ex>^{f}
    & {B} \ar@(ur,ur)[d]+<2em,0ex>^{g}\\
    {C}
    & {D} \ar@/^ 2ex/[l];[]^{h}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar[r]+<0em,-3ex>^{f}
    & {B} \ar@(ur,ur)[d]+<2em,0ex>^{g}\\
    {C}
    & {D} \ar@/^ 2ex/[l];[]^{h}
  }
\end{displaymath}

Realistic examples of arrows off of the grid can be found in
section~\ref{sec:Drop}, where we show how to drop objects into a
diagram at either the source or the target of an arrow.

%--------------------------------------------------------------------
\subsection{Dropping objects into a diagram; annotations}
\label{sec:Drop}

You can drop an object at an arbitrary point of a diagram (i.e., not
necessarily at a node) by drawing an arrow to or from that point (see
section~\ref{sec:OffGrid}) and dropping the object at one of the ends
of the arrow.  The object can be either mathematics or a text object
(see section~\ref{sec:TextObj}).

For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar[r] \ar[d]
    & {B} \ar[d]
    \ar[]+<5em,-5ex>*+\txt{This space\\is compact.};[]\\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar[r] \ar[d]
    & {B} \ar[d]
    \ar[]+<5em,-5ex>*+\txt{This space\\is compact.};[]\\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}
Note that the text and the arrow from it was typed by first creating
the arrow
\begin{center}
  \verb"\ar[]+<5em,-5ex>;[]"
\end{center}
whose source is displaced from the current node (see
sections~\ref{sec:SrcTrgt} and \ref{sec:OffGrid}) and whose target is
the current node, and then dropping the text object (see
section~\ref{sec:TextObj}) at the source by inserting
\begin{center}
  \verb"*+\txt{This space\\is compact.}"
\end{center}
immediately after the specification of the source.

Note that the arrows to or from a text object by default go to its
center, and if a text object is dropped at a specified point by
default it's the center of the text object that lands at that point.
Thus, if you're typing a wide text object, you should be sure either
to leave enough room so that it doesn't overlap other parts of the
diagram or to shift the text object sideways (as in
section~\ref{sec:ShiftSide}).  For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar[r] \ar[d]
    & {B} \ar[d]
    \ar[]+<5em,-5ex>*+\txt{This topological space is compact.};[]\\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar[r] \ar[d]
    & {B} \ar[d]
    \ar[]+<5em,-5ex>*+\txt{This topological space is compact.};[]\\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}
but if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar[r] \ar[d]
    & {B} \ar[d]
    \ar[]+<10em,-5ex>*+\txt{This topological space is compact.};[]\\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}
\end{verbatim}
(in which we increased the horizontal displacement to 10em) then
you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar[r] \ar[d]
    & {B} \ar[d]
    \ar[]+<10em,-5ex>*+\txt{This topological space is compact.};[]\\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}
and if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar[r] \ar[d]
    & {B} \ar[d]
    \ar[]+<5em,-5ex>*+[r]\txt{This topological space is compact.};[]\\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}
\end{verbatim}
(in which we typed
\begin{center}
  \verb"*+[r]\txt{This topological space is compact.}"
\end{center}
to create the text object, i.e., we added ``\verb"[r]"'' to shift the
text object to the right; see section~\ref{sec:ShiftSide}) then you'll
get
\begin{displaymath}
  \xymatrix{
    {A} \ar[r] \ar[d]
    & {B} \ar[d]
    \ar[]+<5em,-5ex>*+[r]\txt{This topological space is compact.};[]\\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}

If you want to drop an object onto a spot other than at a node without
printing an arrow to or from that spot, you can draw an invisible
arrow going to that spot and drop the object at the end of the
invisible arrow.  For example, if you type
\begin{verbatim}
\begin{displaymath}
  \xymatrix{
    {A} \ar[r] \ar[d]_{f}
    & {B} \ar[d]^{g}
    \ar@{}[]+<8em,-4ex>*+\txt{Both $f$ and $g$\\are fibrations.}\\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar[r] \ar[d]_{f}
    & {B} \ar[d]^{g}
    \ar@{}[]+<8em,-4ex>*+\txt{Both $f$ and $g$\\are fibrations.}\\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}
The arrow to the text object is invisible because we typed it starting
with \verb"\ar@{}" (see section~\ref{sec:arstyle} and
Diagram~\ref{diag:CentEq}).  After that, we typed
\begin{center}
  \verb"[]+<8em,-4ex>*+\txt{Both $f$ and $g$\\are fibrations.}"
\end{center}
so that the arrow would go to the point 8em to the right and 4ex below
the current node, and we dropped the text object at that point.


%--------------------------------------------------------------------
%--------------------------------------------------------------------
\section{Including graphics}
\label{sec:graphics}

If you have graphics files that you want to include in your document,
then you should load the \verb"graphicx" package by putting the
command
\begin{center}
\verb"\usepackage{graphicx}"
\end{center}
after your \verb"\documentclass" command.  (Note: The spelling
\verb"graphicx" is correct; the package graphicx.sty is the updated
version of the older package graphics.sty.)  You can then include
graphics using the \verb"\includegraphics" command.

The standard practice is to include graphics inside of a \verb"figure"
environment, so that it can float to the next page if there isn't
enough room for it on the current page.  This also allows you to give
a caption (printed below the graphics) and a label (for use in a
\verb"\ref" command).  Thus, if you want to include the contents of
\verb"mygraphic.eps", you might type
\begin{verbatim}
\begin{figure}[h]
  \centering
  \includegraphics{mygraphics.eps}
  \caption{A pretty picture}
  \label{fig:pretty}
\end{figure}
\end{verbatim}
and you can then refer to it as \verb"figure~\ref{fig:pretty}".  (The
\verb"\centering" command centers the graphics in the figure; this is
preferable to using a \verb"center" environment, which would add
additional vertical space.)  It's important to note that the
\verb"\label" command must come \emph{after} the \verb"\caption"
command, since it's the \verb"\caption" command that creates the
number that the \verb"\label" command labels.  You can also put the
\verb"\label" command \emph{inside} the caption, as in
\begin{center}
\verb"\caption{A pretty picture\label{fig:pretty}}"
\end{center}
and some people find that this helps prevent errors.


%--------------------------------------------------------------------
\subsection{Graphic formats: Postscript or pdf}

One problem to watch out for is that different versions of \LaTeX{}
require different types of graphics files:
\begin{itemize}
\item Standard \verb"latex" (which produces dvi files) requires
  encapsulationed postscript graphics files, while
\item \verb"pdflatex" (which directly produces pdf files) requires pdf
  graphics files.
\end{itemize}
This can be confusing, because some installations of \LaTeX{} give you
a \verb"latex" command that is actually running \verb"pdflatex".  In
particular, the standard installation of \TeX Shop for Mac OS X gives
you a \verb"latex" command that actually runs \verb"pdflatex" and
produces a pdf file.  \TeX Shop can be configured to run standard
\verb"latex" to produce dvi files and then convert them to pdf; for
this, see the list of frequently asked questions at
\begin{center}
  \url{http://www.ams.org/authors/author-faq.html}
\end{center}
and search that page for ``texshop''.

On most systems, if you need to include encapsulated postscript
graphics and produce a pdf file, you can use plain old \verb"latex" to
produce a dvi file and then use \verb"dvipdf" to create the pdf file.
Another option is to use the \verb"epstopdf" program to create a
postscript file from your eps file.


%---------------------------------------------------------------------
%---------------------------------------------------------------------
\section{Macro definitions, a.k.a.\ \texttt{\bs newcommand}}
\label{sec:definitions}

\LaTeX{} allows you to use the same \verb"\def" command that you use
in plain \TeX, but it's considered bad style.  Instead, \LaTeX{} has
the \verb"\newcommand" and \verb"\renewcommand" commands, which do a
little error checking for you.  In plain \TeX, you might use the
command
\begin{center}
  \verb"\def\tensor{\otimes}"
\end{center}
but in \LaTeX{} the preferred form is
\begin{center}
  \verb"\newcommand{\tensor}{\otimes}"
\end{center}
The advantage of this is that \LaTeX{} will check to see if there
already is a command with the name \verb"\tensor" and give you an
error message if there is.  If you know that there is a previous
definition of \verb"\tensor" but you \emph{want} to override it,
then you use the command
\begin{center}
  \verb"\renewcommand{\tensor}{\otimes}"
\end{center}

If you want to use macros with replaceable parameters, both
\verb"\newcommand" and \verb"\renewcommand" allow this.  For the
equivalent of the plain \TeX{} command
\begin{center}
  \verb"\def\pushout#1#2#3{#1\cup_{#2}#3}"
\end{center}
you use the \LaTeX{} command
\begin{center}
  \verb"\newcommand{\pushout}[3]{#1\cup_{#2}#3}"
\end{center}
i.e., the command name is enclosed in braces, and the number of
parameters is enclosed in square brackets.


%---------------------------------------------------------------------
%---------------------------------------------------------------------
\section{Lists: \texttt{itemize, enumerate, and description}}

There are three list making environments:
\begin{itemize}
\item \texttt{itemize},
\item \texttt{enumerate}, and
\item \texttt{description}.
\end{itemize}
The \texttt{itemize} environment just lists the items with a marker in
front of each one.  If you type
\begin{verbatim}
\begin{itemize}
\item This is the first item in the list, which runs on long enough
  to spill over onto a second line.
\item This is the second item in the list, which is a bit shorter.
\item This is the last item.
\end{itemize}
\end{verbatim}
then you'll get
\begin{itemize}
\item This is the first item in the list, which runs on long enough
  to spill over onto a second line.
\item This is the second item in the list, which is a bit shorter.
\item This is the last item.
\end{itemize}

The \texttt{enumerate} environment looks the same, except that the
items in the list are numbered.  If you type
\begin{verbatim}
\begin{enumerate}
\item This is the first item in the list, which runs on long enough
  to spill over onto a second line.
\item This is the second item in the list, which is a bit shorter.
\item This is the last item.
\end{enumerate}
\end{verbatim}
then you'll get
\begin{enumerate}
\item This is the first item in the list, which runs on long enough
  to spill over onto a second line.
\item This is the second item in the list, which is a bit shorter.
\item This is the last item.
\end{enumerate}

The \texttt{description} environment requires an argument for each
\verb"\item" command, which will be printed at the beginning of the
item.  If you type
\begin{verbatim}
\begin{description}
\item[sedge] A green plant, found in both wetlands and uplands.
  Sedges are often confused with grasses and rushes.
\item[grass] A green plant, found in both wetlands and uplands.
  Grasses are often confused with sedges and rushes.
\item[rush] A green plant, found in both wetlands and uplands.
  Rushes are often confused with sedges and grasses
\end{description}
\end{verbatim}
you'll get
\begin{description}
\item[sedge] A green plant, found in both wetlands and uplands.
  Sedges are often confused with grasses and rushes.
\item[grass] A green plant, found in both wetlands and uplands.
  Grasses are often confused with sedges and rushes.
\item[rush] A green plant, found in both wetlands and uplands.
  Rushes are often confused with sedges and grasses
\end{description}

These environments can be inserted within each other, and the
\verb"enumerate" environment keeps track of what level it's at, and
numbers its items accordingly.  If you type
\begin{verbatim}
\begin{enumerate}
\item I went to the dry cleaners.
\item I went to the supermarket.  I bought
  \begin{enumerate}
  \item bread,
  \item cheese, and
  \item Tabasco sauce.
  \end{enumerate}
\item I went to the bank.
\end{enumerate}
\end{verbatim}
you'll get
\begin{enumerate}
\item I went to the dry cleaners.
\item I went to the supermarket.  I bought
  \begin{enumerate}
  \item bread,
  \item cheese, and
  \item Tabasco sauce.
  \end{enumerate}
\item I went to the bank.
\end{enumerate}

%--------------------------------------------------------------------

\subsection{Referring to enumerated items by number}
\label{sec:enumref}

If you want to refer to an item in an enumerated list by item number,
then you need to label the item using the \verb"\label" command and
then refer to it using the \verb"\ref" command, just as you do for
sections, subsections, theorems, etc.\ (see
section~\ref{sec:xreferences}).  For example, if you type
\begin{verbatim}
\begin{enumerate}
\item \label{step:trans} Put the transmission into drive.
\item \label{step:brake} Release the parking brake.
\item \label{step:move} Start moving.
\end{enumerate}
You must complete steps \ref{step:trans} and \ref{step:brake}
before beginning step~\ref{step:move}.
\end{verbatim}
then you'll get
\begin{enumerate}
\item \label{step:trans} Put the transmission into drive.
\item \label{step:brake} Release the parking brake.
\item \label{step:move} Start moving.
\end{enumerate}
You must complete steps \ref{step:trans} and \ref{step:brake} before
beginning step~\ref{step:move}.

%---------------------------------------------------------------------
%---------------------------------------------------------------------
\section{The bibliography}
\label{sec:bibliography}

The recommended way to create a bibliography is to use the
\texttt{amsrefs} package (see section~\ref{sec:amsrefs}).  This makes
it simple to type the bibliography items and have them correctly
formatted and automatically numbered, and makes it possible to change
the style of the bibliography without retyping the items.  At the same
time, it allows you to preserve the structure of the information
contained in the items, and (optionally) to create a database of
possible references that can be reused in other documents.

The \texttt{amsrefs} package is fairly new.  The classical method of
creating a bibliography from a database of possible references uses
Bib\TeX{}, which is a separate program that must be run in between the
runs of \LaTeX.  If you already have one or more BiB\TeX{} databases,
\texttt{amsrefs} can make use of them through a special Bib\TeX{}
style file (see section~\ref{sec:bibtex}).

If for some reason you want to avoid the use of \texttt{amsrefs} and
of Bib\TeX, you can type and format the bibliography items any way you
choose by creating a \texttt{thebibliography} environment (see
section~\ref{sec:thebibliography}).



%--------------------------------------------------------------------
\subsection{Using \texttt{amsrefs}}
\label{sec:amsrefs}

To use \texttt{amsrefs}, you load it by putting the command
\begin{center}
  \verb"\usepackage[lite]{amsrefs}"
\end{center}
into the preamble of your document, i.e., after the
\verb"\documentclass" command and before the \verb"\begin{document}"
command.  You can then
\begin{enumerate}
\item begin the bibliography with the commands
\begin{verbatim}
\begin{bibdiv}
  \begin{biblist}
\end{verbatim}
\item define the bibliography items using a \verb"\bib" command for
  each item (see section~\ref{sec:bib}), and then
\item end the bibliography with the commands
\begin{verbatim}
  \end{biblist}
\end{bibdiv}
\end{verbatim}
\end{enumerate}
The reason there are two different environments (the outer environment
\texttt{bibdiv} containing the inner environment \texttt{biblist}) is
that the \texttt{bibdiv} environment produces the chapter or section
heading for the bibliography (depending on what's appropriate for the
type of your document) and the \texttt{biblist} environment produces
the actual list of references.  You can also type material in between
the \verb"\begin{bibdiv}" and \verb"begin{biblist}" commands and it
will be printed in between the heading for the list and the actual
list.

%--------------------------------------------------------------------
\subsection{The \texttt{\bs bib} command}
\label{sec:bib}

The information for each bibliography item inside the \texttt{bibdiv}
environment (see section~\ref{sec:amsrefs}) is entered using a
\verb"\bib" command.  The format of a \verb"\bib" command is
\begin{verbatim}
\bib{ReferenceKey}{ReferenceType}{
  comma separated list of keyword={value} statements
}
\end{verbatim}
where \texttt{ReferenceKey} is the key that will be used to refer to
the item with a \verb"\cite" command (see
section~\ref{sec:bibreferences}) and \texttt{ReferenceType} is either
\begin{quote}
  \texttt{article}, \texttt{book}, \texttt{misc}, \texttt{report}, or
  \texttt{thesis}.
\end{quote}
For example, the bibliography of these notes contains the following
\verb"\bib" commands:
\begin{verbatim}
\bib{yellowmonster}{book}{
  author={Bousfield, A.K.},
  author={Kan, D.M.},
  title={Homotopy Limits, Completions and Localizations},
  date={1972},
  series={Lecture Notes in Mathematics},
  volume={304},
  publisher={Springer-Verlag},
  address={Berlin-New York}
}

\bib{quil:rht}{article}{
  author={Quillen, Daniel G.},
  title={Rational Homotopy Theory},
  journal={Ann. of Math. (2)},
  volume={90},
  date={1969},
  pages={205--295}
}
\end{verbatim}
If those \verb"\bib" commands are in your \texttt{biblist}
environment, you can type
\begin{center}
  \verb"see \cite{yellowmonster} and \cite{quil:rht}"
\end{center}
and it will be typeset as ``see \cite{yellowmonster} and
\cite{quil:rht}''.

Some of the rules governing the \verb"\bib" command are:
\begin{itemize}
\item The keywords must all be typed in lower case.  (This is
  different from in Bib\TeX{} databases, in which the case of field
  names is ignored.)
\item The value in each ``\verb"keyword={value}"'' statement must
  always be enclosed in braces.  (This is different from in Bib\TeX{}
  databases, in which the braces can sometimes be omitted.)
\item Author names must be typed in the form ``von Last, First, Jr.'',
  as in ``\verb"author={Jones, John Paul}"'', or
  ``\verb"author={van Beethoven, Ludwig}"'', or
  ``\verb"author={Ford, Henry, Jr.}"''.  This ensures there won't be
  errors in reversing the first and last names, or when
  \texttt{amsrefs} replaces the first and middle names with initials
  (when the \texttt{initials} option is used; see
  section~\ref{sec:refoptions}), or when \texttt{amsrefs} creates a
  label based on the last name (when the \texttt{alphabetic} option is
  used; see section~\ref{sec:refoptions}).
\item Multiple authors must each be listed in a separate \verb"author"
  field.
\item The capitalization in the title should be exactly as you want it
  to appear in the bibliography.  (This is different from in Bib\TeX{}
  databases, in which capitalization is often changed to fit a chosen
  style.)
\item The date must be written in the form \verb"date={1776}", or
  \verb"date={1776-07}", or \verb"date={1776-07-04}".
\end{itemize}
The full list of simple fields (i.e., fields that can appear at most
once) is
\begin{quote}
  \texttt{address}, \texttt{booktitle}, \texttt{date},
  \texttt{edition}, \texttt{eprint}, \texttt{hyphenation},
  \texttt{journal}, \texttt{label}, \texttt{language}, \texttt{note},
  \texttt{number}, \texttt{organization}, \texttt{pages},
  \texttt{part}, \texttt{publisher}, \texttt{series}, \texttt{status},
  \texttt{subtitle}, \texttt{title}, \texttt{type}, \texttt{volume},
  and \texttt{xref}
\end{quote}
The full list of repeatable fields is
\begin{quote}
  \texttt{author}, \texttt{editor}, \texttt{translator},
  \texttt{isbn}, \texttt{issn}, and \texttt{review}.
\end{quote}
The full list of compound fields (for which the value of each is a
comma separated list of \verb"keyword={value}" statements) is
\begin{quote}
  \texttt{book}, \texttt{conference}, \texttt{contribution},
  \texttt{partial}, \texttt{reprint}, and \texttt{translation}.
\end{quote}
A description of the less obvious ones can be found in
\cite[section~5.2]{amsrefsguide}.

%--------------------------------------------------------------------
\subsection{Obtaining bibliographic information from MathSciNet}
\label{sec:mathscinet}

If you have access to MathSciNet (at
\url{http://www.ams.org/mathscinet}), then you can obtain a completely
filled out \verb"\bib" command (see section~\ref{sec:bib}) for any of
the publications listed there.  To do this:
\begin{itemize}
\item Use MathSciNet to find the Mathematical Reviews review of the
  publication.
\item Look near the upper left corner of the page for a pulldown menu
  titled ``Select Alternative Format''.
\item Select ``AMSRefs'' from that menu; you'll get a new page showing
  the \verb"\bib" command for the publication.
\item Copy that \verb"\bib" command and paste it into your \LaTeX{}
  file.
\end{itemize}


%--------------------------------------------------------------------
\subsection{Using a database of possible references}
\label{sec:refbase}

If your list of \verb"\bib" commands (see section~\ref{sec:bib}) grows
so long that you don't want to include it in your \LaTeX{} file, or if
you want to reuse a collection of \verb"\bib" commands from other
papers, you can put all of your \verb"\bib" commands into a file and
have \texttt{amsrefs} extract only the ones used in your current
paper.  To do this, you create a file whose name ends in \verb".ltb",
for example: \verb"myrefs.ltb", and put all of your \verb"\bib"
commands into that file.  In such a file, it's important that each
\verb"\bib" command begins on a new line, and that that line contains
the first two arguments and the following open brace, as in
\begin{center}
  \verb"\bib{ReferenceKey}{ReferenceType}{"
\end{center}
You then delete all of the \verb"\bib" commands from your \LaTeX{}
file and replace them with the single line
\begin{center}
  \verb"\bibselect{myrefs}"
\end{center}
and \texttt{amsrefs} will automatically read \verb"myrefs.ltb" and
extract only those items that are cited in the current paper.  Thus,
if you have the \verb"\bib" commands for all of your possible
references in the file \verb"myrefs.ltb", and you've loaded
\texttt{amsrefs} with the command \verb"\usepackage[lite]{amsrefs}" in
your preamble (see section~\ref{sec:amsrefs}), then you would create
the bibliography with the lines
\begin{verbatim}
\begin{bibdiv}
  \begin{biblist}
    \bibselect{myrefs}
  \end{biblist}
\end{bibdiv}
\end{verbatim}
If your \LaTeX{} file contains the above lines and your file is named
\verb"mypaper.tex", then \LaTeX{} will create the file
\verb"mypaper.bbl" which will contain the \verb"\bib" commands for
only those references cited in your paper, and \verb"mypaper.bbl" will
automatically be read and used in your \LaTeX{} file.

If your \verb"\bib" commands are spread over several \verb".ltb"
files, you can either list them all, separated by commas, in the
argument of the \verb"\bibselect" command, as in
\begin{center}
  \verb"\bibselect{myfirstrefs,mysecondrefs}"
\end{center}
or just use multiple \verb"\bibselect" commands, as in
\begin{center}
  \begin{tabular}{l}
    \verb"\bibselect{myfirstrefs}"\\
    \verb"\bibselect{mysecondrefs}"
  \end{tabular}
\end{center}

As described above, a \verb"\bibselect" command uses only the
references actually cited in your document.  To list \emph{all} the
references in an \verb".ltb" file, whether or not they're cited in
your document, use the command
\begin{center}
  \verb"\bibselect*{myrefs}"
\end{center}

%--------------------------------------------------------------------
\subsection{Using \texttt{amsrefs} with Bib\TeX}
\label{sec:bibtex}

If you already have a Bib\TeX{} database of possible references, you
can use that by combining Bib\TeX{} with \texttt{amsrefs}.  If your
references are in the Bib\TeX{} file \verb"myrefs.bib", then you would
load the \texttt{amsrefs} package by putting the command
\verb"\usepackage[lite]{amsrefs}" in your preamble (in the same way
that you would load it if you were using \texttt{amsrefs} without
Bib\TeX; see section~\ref{sec:amsrefs}) and put the single line
\begin{center}
  \verb"\bibliography{myrefs}"
\end{center}
where you want the bibliography to appear.  (That is, you don't create
the \texttt{bibdiv} or \texttt{biblist} environments.)  You then run
Bib\TeX{} on your file in the normal way, and Bib\TeX{} will create
the \verb".bbl" file.  That \verb".bbl" file will contain the
\texttt{bibdiv} and \texttt{biblist} environments, and \LaTeX{} will
read it the next time that you run \LaTeX{} and create the
bibliography.

If you use this method, you don't use the \verb"\bibliographystyle"
command (and any such command in your \LaTeX{} file will be ignored).

%--------------------------------------------------------------------
\subsection{Options for the \texttt{amsrefs} package}
\label{sec:refoptions}

There are a number of optional arguments that you can use when loading
the \texttt{amsrefs} package (see section~\ref{sec:amsrefs}), most of
which affect the formatting of the bibliography.  You use them by
listing them as optional arguments to the \verb"\usepackage" command
(see section~\ref{sec:amsrefs}).  For example, we suggested that you
always use the optional argument \texttt{lite}, as in
\begin{center}
  \verb"\usepackage[lite]{amsrefs}"
\end{center}
We'll describe the most important options here; the full list can be
found in \cite[section~6]{amsrefsguide}.
\begin{description}
\item[lite] We suggest that you always use this option in order to
  avoid a conflict between \texttt{amsrefs} and the \Xy-pic package
  (see section~\ref{sec:xypic}).  If you don't use the \texttt{lite}
  option, then \texttt{amsrefs} automatically loads the packages
  \texttt{mathscinet} (which defines a number of special characters
  and accents that are sometimes encountered when downloading data
  from MathSciNet) and \texttt{txtcmds} (which provides shorthand
  commands for a number of characters that are usually specified via
  ligatures).  The problem is that both \Xy-pic and the
  \texttt{txtcmds} package define the command \verb"\cir" (they define
  it to be entirely different things).  If you load \texttt{txtcmds}
  \emph{before} loading \Xy-pic then \Xy-pic will redefine \verb"\cir"
  as it pleases while putting a warning into your \verb".log" file,
  but if you load \texttt{txtcmds} \emph{after} loading \Xy-pic then
  \LaTeX{} will stop with an error, since the \texttt{txtcmds} package
  uses \verb"\newcommand" to define \verb"\cir".  Unfortunately, if
  you need to use the \texttt{mathscinet} package and you try to load
  it by putting the command \verb"\usepackage{mathscinet}" after your
  \verb"\documentclass" command, it will also load the
  \texttt{txtcmds} package, so this doesn't avoid the conflict.

  If you need to use both \Xy-pic and the definitions in the
  \texttt{mathscinet} and \texttt{txtcmds} packages, you should load
  \texttt{amsrefs} \emph{before} \Xy-pic, as in
  \begin{center}
    \begin{tabular}{l}
      \verb"\usepackage{amsrefs}"\\
      \verb"\usepackage[all,cmtip]{xy}"
    \end{tabular}
  \end{center}
  to keep \LaTeX{} from generating an error when loading
  \texttt{txtcmds}.  (Of course, the command \verb"\cir" as defined in
  \texttt{textcmds} will not be available.)
\item[initials] If you use this option, then all authors, editors, and
  translators will have their first and middle names replaced by their
  initials.
\item[alphabetic] If you use this option, as in
  \begin{center}
    \verb"\usepackage[alphabetic,lite]{amsrefs}"
  \end{center}
  then instead of numbers being used to label the bibliography
  entries, you will get alphabetic labels similar to the
  \texttt{alpha} style used by Bib\TeX, consisting of the first
  letter(s) of each author name plus the year of publication.
\item[shortalphabetic] This is similar to the \texttt{alphabetic}
  option, except that you'll get a shorter alphabetic label using only
  the first letter of each author name.
\item[y2k] If you use the \texttt{alphabetic} option, only the last
  two digits of the year are normally used in the label.  If you also
  use the \texttt{y2k} option, as in
  \begin{center}
    \verb"\usepackage[y2k,alphabetic,lite]{amsrefs}"
  \end{center}
  then the full year will be used.
\item[author-year] If you use this option, then bibliography items
  will not be labelled at all, and references to them will be in the
  author-year format similar to that described in
  \verb"The Chicago Manual of Style".  If you do use this option, you
  may sometimes want to use the \verb"\ycite" and \verb"\ocite"
  commands when referring to bibliography items (see
  section~\ref{sec:authyrcit}).
\end{description}



%--------------------------------------------------------------------
\subsection{Avoiding both \texttt{amsrefs} and Bib\TeX}
\label{sec:thebibliography}

If for some reason you don't want to use \texttt{amsrefs} or Bib\TeX,
you can type each bibliography item exactly as you want it to appear
and have the items numbered automatically.  To do this, you
\begin{enumerate}
\item  begin the bibliography with the command
  \begin{center}
    \verb"\begin{thebibliography}{number}"
  \end{center}
  where \emph{number} is any number that, when printed, is as wide as
  the widest number of any item in the bibliography,
\item define the bibliography items using a \verb"\bibitem" command
  for each item (see section~\ref{sec:bibitem}), and then
\item end the bibliography with the command
    \begin{center}
      \verb"\end{thebibliography}"
  \end{center}
\end{enumerate}
The only use made of the \verb"number" in
\verb"\begin{thebibliography}{number}" is that \LaTeX{} assumes that
its width when printed is at least as large as the width of any number
of an item in the bibliography.  For example, if the bibliography will
contain between 10 and~99 items, you can use
\verb"\begin{thebibliography}{99}".


%--------------------------------------------------------------------
\subsubsection{Bibliography items}
\label{sec:bibitem}

When using the commands \verb"\begin{thebibliography}" and
\verb"\end{thebibliography}" to create the bibliography (see
section~\ref{sec:thebibliography}), each item is begun with a
\verb"\bibitem" command.  The format is
\begin{center}
  \verb"\bibitem{ReferenceKey}Item entry"
\end{center}
For example, had we not been using \texttt{amsrefs}, the bibliography
in these instructions might have contained the entry
\begin{verbatim}
\bibitem{yellowmonster}
A. K. Bousfield and D. M. Kan, \emph{Homotopy Limits, Completions
and Localizations,} Lecture Notes in Mathematics number 304,
Springer-Verlag, New York, 1972.
\end{verbatim}

The above entry would allow us to type
\begin{verbatim}
Homotopy inverse limits are discussed
in~\cite[Chapter 11]{yellowmonster}.
\end{verbatim}
and have it print as ``Homotopy inverse limits are discussed
in~\cite[Chapter 11]{yellowmonster}.''  For more on the \verb"\cite"
command, see section~\ref{sec:bibreferences}.











%---------------------------------------------------------------------
%---------------------------------------------------------------------
\section{The template file}
\label{sec:template}

The following is the text of the file \verb"template.tex".

\begin{verbatim}
%%% template.tex
%%% This is a template for making up an AMS-LaTeX file
%%% Version of February 12, 2011
%%%---------------------------------------------------------
%%% The following command chooses the default 10 point type.
%%% To choose 12 point, change it to
%%% \documentclass[12pt]{amsart}
\documentclass{amsart}

%%% The following command loads the amsrefs package, which will be
%%% used to create the bibliography:
\usepackage[lite]{amsrefs}

%%% The following command defines the standard names for all of the
%%% special symbols in the AMSfonts package, listed in
%%% http://www.ctan.org/tex-archive/info/symbols/math/symbols.pdf
\usepackage{amssymb}

%%% The following commands allow you to use \Xy-pic to draw
%%% commutative diagrams.  (You can omit the second line if you want
%%% the default style of the nodes to be \textstyle.)
\usepackage[all,cmtip]{xy}
\let\objectstyle=\displaystyle

%%% If you'll be importing any graphics, uncomment the following
%%% line.  (Note: The spelling is correct; the package graphicx.sty is
%%% the updated version of the older graphics.sty.)
% \usepackage{graphicx}



%%% This part of the file (after the \documentclass command,
%%% but before the \begin{document}) is called the ``preamble''.
%%% This is where we put our macro definitions.

%%% Comment out (or delete) any of these that you don't want to use.
\newcommand{\tensor}{\otimes}
\newcommand{\homotopic}{\simeq}
\newcommand{\homeq}{\cong}
\newcommand{\iso}{\approx}

\DeclareMathOperator{\ho}{Ho}
\DeclareMathOperator*{\colim}{colim}

\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}

\newcommand{\M}{\mathcal{M}}
\newcommand{\W}{\mathcal{W}}

\newcommand{\itilde}{\tilde{\imath}}
\newcommand{\jtilde}{\tilde{\jmath}}
\newcommand{\ihat}{\hat{\imath}}
\newcommand{\jhat}{\hat{\jmath}}



%%%-------------------------------------------------------------------
%%%-------------------------------------------------------------------
%%% The Theorem environments:
%%%
%%%
%%% The following commands set it up so that:
%%% 
%%% All Theorems, Corollaries, Lemmas, Propositions, Definitions,
%%% Remarks, Examples, Notations, and Terminologies  will be numbered
%%% in a single sequence, and the numbering will be within each
%%% section.  Displayed equations will be numbered in the same
%%% sequence. 
%%% 
%%% 
%%% Theorems, Propositions, Lemmas, and Corollaries will have the most
%%% formal typesetting.
%%% 
%%% Definitions will have the next level of formality.
%%% 
%%% Remarks, Examples, Notations, and Terminologies will be the least
%%% formal.
%%% 
%%% Theorem:
%%% \begin{thm}
%%% 
%%% \end{thm}
%%% 
%%% Corollary:
%%% \begin{cor}
%%% 
%%% \end{cor}
%%% 
%%% Lemma:
%%% \begin{lem}
%%% 
%%% \end{lem}
%%% 
%%% Proposition:
%%% \begin{prop}
%%% 
%%% \end{prop}
%%% 
%%% Definition:
%%% \begin{defn}
%%% 
%%% \end{defn}
%%% 
%%% Remark:
%%% \begin{rem}
%%% 
%%% \end{rem}
%%% 
%%% Example:
%%% \begin{ex}
%%% 
%%% \end{ex}
%%% 
%%% Notation:
%%% \begin{notation}
%%% 
%%% \end{notation}
%%% 
%%% Terminology:
%%% \begin{terminology}
%%% 
%%% \end{terminology}
%%% 
%%%       Theorem environments

% The following causes equations to be numbered within sections
\numberwithin{equation}{section}

% We'll use the equation counter for all our theorem environments, so
% that everything will be numbered in the same sequence.

%       Theorem environments

\theoremstyle{plain} %% This is the default, anyway
\newtheorem{thm}[equation]{Theorem}
\newtheorem{cor}[equation]{Corollary}
\newtheorem{lem}[equation]{Lemma}
\newtheorem{prop}[equation]{Proposition}

\theoremstyle{definition}
\newtheorem{defn}[equation]{Definition}

\theoremstyle{remark}
\newtheorem{rem}[equation]{Remark}
\newtheorem{ex}[equation]{Example}
\newtheorem{notation}[equation]{Notation}
\newtheorem{terminology}[equation]{Terminology}

%%%-------------------------------------------------------------------
%%%-------------------------------------------------------------------
%%%-------------------------------------------------------------------
%%%-------------------------------------------------------------------
%%%-------------------------------------------------------------------
%%%-------------------------------------------------------------------
%%%-------------------------------------------------------------------
\begin{document}

%%% In the title, use a double backslash "\\" to show a linebreak:
%%% Use one of the following two forms:
%%% \title{Text of the title}
%%% or
%%% \title[Short form for the running head]{Text of the title}
\title{}


%%% If there are multiple authors, they're described one at a time:
%%% First author: \author{} \address{} \curraddr{} \email{} \thanks{}
%%% Second author: \author{} \address{} \curraddr{} \email{} \thanks{}
%%% Third author: \author{} \address{} \curraddr{} \email{} \thanks{}
\author{}

%%% In the address, show linebreaks with double backslashes:
\address{}

%%% Current address is optional.
% \curraddr{}

%%% Email address is optional.
% \email{}


%%% If there's a second author:
% \author{}
% \address{}
% \curraddr{}
% \email{}


%%% To have the current date inserted, use \date{\today}:
\date{}

%%% To include an abstract, uncomment the following two lines and type
%%% the abstract in between them:
% \begin{abstract}
% \end{abstract}


\maketitle

%%% To include a table of contents, uncomment the following line:
% \tableofcontents


%%%-------------------------------------------------------------------
%%%-------------------------------------------------------------------
%%% Start the body of the paper here!  E.G., maybe use:
%%% \section{Introduction}
%%% \label{sec:intro}

%%% For a numbered display, use
%%% \begin{equation}
%%%   \label{something}
%%%   The display goes here
%%% \end{equation}
%%% and you can refer to it as \eqref{something}.

%%% For an unnumbered display, use
%%% \begin{equation*}
%%%   The display goes here
%%% \end{equation*}

%%% To import a graphics file, you must have said
%%% \usepackage{graphicx}
%%% in the preamble (i.e., before the \begin{document}).
%%% Putting it into a figure environment enables it to float to the
%%% next page if there isn't enough room for it on the current page.
%%% The \label command must come after the \caption command.
% \begin{figure}[h]
%   \includegraphics{filename}
%   \caption{Some caption}
%   \label{somelabel}
% \end{figure}















%%% -------------------------------------------------------------------
%%% -------------------------------------------------------------------
%%% This is where we create the bibliography.

\begin{bibdiv}
  \begin{biblist}

%%% The format of bibliography items is as in the following examples:
%%% 
%%% \bib{yellowmonster}{book}{
%%%   author={Bousfield, A.K.},
%%%   author={Kan, D.M.},
%%%   title={Homotopy Limits, Completions and Localizations},
%%%   date={1972},
%%%   series={Lecture Notes in Mathematics},
%%%   volume={304},
%%%   publisher={Springer-Verlag},
%%%   address={Berlin-New York}
%%% }

%%% \bib{HA}{book}{
%%%   author={Quillen, Daniel G.},
%%%   title={Homotopical Algebra},
%%%   series={Lecture Notes in Mathematics},
%%%   volume={43},
%%%   publisher={Springer-Verlag},
%%%   address={Berlin-New York},
%%%   date={1967}
%%% }

%%% \bib{serre:shfs}{article}{
%%%   author={Serre, Jean-Pierre},
%%%   title={Homologie Singuli\`ere des Espaces Fibr\'es.  Applications},
%%%   journal={Ann. of Math. (2)},
%%%   date={1951},
%%%   volume={54},
%%%   pages={425--505}
%%% }





  \end{biblist}
\end{bibdiv}

\end{document}
\end{verbatim}

%---------------------------------------------------------------------
%---------------------------------------------------------------------





\begin{bibdiv}
  \begin{biblist}

\bib{amslatexusersguide}{report}{
  author={American Mathematical Society},
  title={User's Guide for the \texttt{amsmath} Package},
  edition={version 2.0},
  date={2002-02-25},
  eprint={ftp://ftp.ams.org/pub/tex/doc/amsmath/amsldoc.pdf},
  note={also available from CTAN
  (the Comprehensive \TeX{} Archive Network) at
  \url{http://www.ctan.org/tex-archive/macros/latex/required/amslatex/math/amsldoc.pdf}}
}

\bib{instr-l}{report}{
  author={American Mathematical Society},
  title={Instructions for Preparation of Papers and Monographs:
    \AmS-\LaTeX},
  date={2004-08},
  eprint={ftp://ftp.ams.org/pub/tex/doc/amscls/instr-l.pdf}
}

\bib{testmath}{misc}{
  author={American Mathematical Society},
  title={Sample Paper for the \texttt{amsmath} Package},
  date={1996-11},
  note={available at \url{ftp://ftp.ams.org/pub/tex/amslatex/math/testmath.tex}}
}

\bib{yellowmonster}{book}{
  author={Bousfield, A.K.},
  author={Kan, D.M.},
  title={Homotopy Limits, Completions and Localizations},
  date={1972},
  series={Lecture Notes in Mathematics},
  volume={304},
  publisher={Springer-Verlag},
  address={Berlin-New York}
}

\bib{mathguide}{report}{
  author={Downes, Michael},
  title={Short Math Guide for \LaTeX},
  edition={version 1.09},
  publisher={American Mathematical Society},
  eprint={ftp://ftp.ams.org/pub/tex/doc/amsmath/short-math-guide.pdf}
}

\bib{amsrefsguide}{report}{
  author={Jones, David M.},
  title={User's Guide to the \texttt{amsrefs} Package},
  publisher={American Mathematical Society},
  date={2007-10-16},
  eprint={ftp://ftp.ams.org/pub/tex/amsrefs/amsrdoc.pdf}
}

\bib{latex}{book}{
  author={Lamport, Leslie},
  title={\LaTeX: A Document Preparation System},
  edition={2},
  publisher={Addison-Wesley},
  date={1994}
}

\bib{NotShort}{report}{
  author={Oetiker, Tobias},
  author={Partl, Hubert},
  author={Hyna, Irene},
  author={Schlegl, Elisabeth},
  title={The Not So Short Introduction to \LaTeXe},
  edition={version 4.24},
  eprint={http://www.ctan.org/tex-archive/info/lshort/english/lshort.pdf}
}

\bib{HA}{book}{
  author={Quillen, Daniel G.},
  title={Homotopical Algebra},
  series={Lecture Notes in Mathematics},
  volume={43},
  publisher={Springer-Verlag},
  address={Berlin-New York},
  date={1967}
}

\bib{quil:rht}{article}{
  author={Quillen, Daniel G.},
  title={Rational Homotopy Theory},
  journal={Ann. of Math. (2)},
  volume={90},
  date={1969},
  pages={205--295}
}

\bib{xyguide}{report}{
  author={Rose, Kristoffer H.},
  title={\Xy-pic User's Guide},
  edition={version 3.85},
  eprint={http://www.ctan.org/tex-archive/macros/generic/diagrams/xypic/xy-3.8.5/doc/xyguide.pdf}
}

\bib{xyrefer}{report}{
  author={Rose, Kristoffer H.},
  author={Moore, Ross},
  title={\Xy-pic Reference Manual},
  edition={version 3.85},
  eprint={http://www.ctan.org/tex-archive/macros/generic/diagrams/xypic/xy-3.8.5/doc/xyrefer.pdf}
}

  \end{biblist}
\end{bibdiv}



\end{document}