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

%%% Copyright (c) 2000, 2008, 2009 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@poincare.wellesley.edu
%%% psh@math.mit.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}{June 19, 2009}
%\newcommand{\filedate}{\today}
\newcommand{\fileversion}{Version 2.1}

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

\usepackage{url}

% 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 suppress reading textcmds.sty
\usepackage[lite]{amsrefs}

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




%\usepackage{hyperref}




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

\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 with it 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 may get you started quickly enough so that
you'll only need to refer to the main documentation occasionally,
which should eliminate most of the pain.

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
\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{\char`\\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 actually the choice of document class that determines whether
you're using \AmS-\LaTeX{} or just plain old \LaTeX.  There are three
document classes that are a part of \AmS-\LaTeX: \texttt{amsart},
\texttt{amsproc}, and \texttt{amsbook}.  There is also the
\texttt{amsmath} package that can be used with the standard \LaTeX{}
document classes.  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{\char`\\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.

Another important package is for when you want to use some of the
special symbols contained in the \AmS-Fonts package.  These are listed
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 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{\char`\\begin\{document\}} and
  \texttt{\char`\\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}
I refuse to even experiment to see if there are subsubsubsections.

%--------------------------------------------------------------------
\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 switch explicitly 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, 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{\char`\\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 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{\char`\\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".

%--------------------------------------------------------------------
\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 want to refer to multiple bibliography items, and 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 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}

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 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*}
There are several essentially equivalent ways of producing this
unnumbered display: The \verb"displaymath" environment, used as in
\begin{verbatim}
\begin{displaymath}
  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
\end{displaymath}
\end{verbatim}
accomplishes the same thing, as will either
\begin{verbatim}
$$
  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
$$
\end{verbatim}
or
\begin{verbatim}
\[
  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
\]
\end{verbatim}
It's generally thought to be a better idea to use either the
\verb"equation*" environment or the \verb"displaymath" environment for
displays, and to avoid the double dollar signs and the
\verb"\[",~\verb"\]" pair.

%--------------------------------------------------------------------
\subsection{Displaying several equations 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}''.

%--------------------------------------------------------------------
\subsection{Displays over several lines without alignment}
\label{sec:multline}

For a long display that must be broken across several lines, you can
use the \verb"multline" environment.  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}''.

The \verb"multline*" environment is similar, except that it omits the
equation number.


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

To display several lines of mathematics with alignment between the
lines, you use the \verb"align" environment.  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}''.

The \verb"align*" environment is similar, except that it omits the
equation numbers.  For 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*}

Another way to have multiple line displays with alignment is to use
the \verb"split" environment, which must go inside of some other
displayed mathematics environment.  For example, if you type
\begin{verbatim}
\begin{equation}
  \label{eq:split}
  \begin{split}
    a &= b + c\\
    d &= e + f
  \end{split}
\end{equation}
\end{verbatim}
then you'll get
\begin{equation}
  \label{eq:split}
  \begin{split}
    a &= b + c\\
    d &= e + f
  \end{split}
\end{equation}
The \verb"split" environment doesn't produce any equation numbers of
its own, and if the enclosing display environment produces a number
then that number is centered vertically in the contents of the
\verb"split".  This can also be useful if you need to display one very
long equation that you need to split over multiple lines and you want
the equation number to be centered vertically.  For example, if you
type
\begin{verbatim}
\begin{equation}
  \label{eq:longeq}
  \begin{split}
    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{split}
\end{equation}
\end{verbatim}
then you'll get
\begin{equation}
  \label{eq:longeq}
  \begin{split}
    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{split}
\end{equation}


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

The \verb"align" and \verb"align*" environments can be used to put
multiple displays on each of multiple lines, with alignment between
the lines.  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*}
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 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 space in one row; the
alignment forces the space to appear in all rows.


%--------------------------------------------------------------------
\subsection{Displays that are part of a larger display}
\label{sec:LgDsp}

In addition to the environments that create an entire display
(\texttt{equation}, \texttt{multline}, \texttt{align},
\texttt{flalign}, and \texttt{alignat}), there are environments that
can be used as components of a larger display:
\begin{center}
  \begin{tabular}{l@{\quad is similar to\quad}l}
    \texttt{gathered}& \texttt{gather*}\\
    \texttt{aligned}& \texttt{align*}\\
    \texttt{alignedat}& \texttt{aligned*}
  \end{tabular}
\end{center}
(There is also a \texttt{cases} environment, described in
section~\ref{sec:cases}, and we've already discussed the \verb"split"
environment in section~\ref{sec:align}.)

For example, if you type
\begin{verbatim}
\begin{equation}
  \begin{aligned}
    A &= B+C\\
    &= D+E
  \end{aligned}
  \qquad \text{and, in addition,}\qquad
  \begin{gathered}
    x^{2} + y^{2} = z^{2}\\
    a=b+c
  \end{gathered}
\end{equation}
\end{verbatim}
then you'll get
\begin{equation}
  \begin{aligned}
    A &= B+C\\
    &= D+E
  \end{aligned}
  \qquad \text{and, in addition,}\qquad
  \begin{gathered}
    x^{2} + y^{2} = z^{2}\\
    a=b+c
  \end{gathered}
\end{equation}

%--------------------------------------------------------------------
\subsection{Cases}
\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*}

%--------------------------------------------------------------------
%--------------------------------------------------------------------
\section{Commutative diagrams using \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 can be
found in the \emph{\Xy-pic Reference Manual} \cite{xyrefer}.

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.
\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.  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 section~\ref{sec:ArUnder}).

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*}

%--------------------------------------------------------------------
\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}.)

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}
  \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}
  \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{Arrows passing under}
\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.

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 halfway along the arrow (see section~\ref{sec:arcent}) or at
an arbitrary point along the arrow (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 along an arrow.  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 \eqref{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}
  \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}
  \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}



%--------------------------------------------------------------------
\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.
\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]"\\[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{displaymath}
  \xymatrix{
    {A} \ar@{.>}[r] \ar@{.>}[d] \ar@{}[dr]|{=}
    & {B} \ar[d]\\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}
\end{verbatim}
then you'll get
\begin{displaymath}
  \xymatrix{
    {A} \ar@{.>}[r] \ar@{.>}[d] \ar@{}[dr]|{=}
    & {B} \ar[d]\\
    {C} \ar[r]
    & {D}
  }
\end{displaymath}



%--------------------------------------------------------------------
\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]".  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}

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}

%--------------------------------------------------------------------
\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@<distance>" to move the arrow ``upwards'' by
\verb"distance" (a negative distance will move the arrow
``downwards'').

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@<2ex>[r]^-{\phi} \ar@<+.6ex>[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@<2ex>[r]^-{\phi} \ar@<+.6ex>[r]_-{\psi} 
    & {\quad\coprod_{i \in I} B_{i}}
  } 
\end{displaymath}


%--------------------------------------------------------------------
%--------------------------------------------------------------------
\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 this less prone to error.


%--------------------------------------------------------------------
\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 produce dvi
files; for this, see the list of frequently asked questions at
\url{http://www.ams.org/authors/author-faq.html} and search that page
for ``texshop''.



%---------------------------------------------------------------------
%---------------------------------------------------------------------
\section{Macro definitions, a.k.a.\ \texttt{\char`\\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, the
\verb"newcommand" command allows 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: \texttt{itemize},
\texttt{enumerate}, and \texttt{description}.  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 section 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.

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{\char`\\bib} command}
\label{sec:bib}

The information for each bibliography item is entered into the
argument of 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{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 load it if you're 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 \texttt{amsrefs}}
\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 May 14, 2009
%%%---------------------------------------------------------
%%% 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},
  eprint={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 3.7},
  eprint={http://www.ctan.org/tex-archive/macros/generic/diagrams/xypic/xyguide.pdf}
}

\bib{xyrefer}{report}{
  author={Rose, Kristoffer H.},
  author={Moore, Ross},
  title={\Xy-pic Reference Manual},
  edition={version 3.7},
  eprint={http://www.ctan.org/tex-archive/macros/generic/diagrams/xypic/xyrefer.pdf}
}

  \end{biblist}
\end{bibdiv}



\end{document}