From f073e7f029adc3b13fa4c46602e641c64b600a9d Mon Sep 17 00:00:00 2001
From: Karl Berry <karl@freefriends.org>
Date: Mon, 22 Jun 2009 23:28:14 +0000
Subject: amslatex-primer 2.1 (22jun09)

git-svn-id: svn://tug.org/texlive/trunk@13894 c570f23f-e606-0410-a88d-b1316a301751
---
 .../doc/latex/amslatex-primer/amshelp.tex          | 3107 +++++++++++++++-----
 1 file changed, 2385 insertions(+), 722 deletions(-)

(limited to 'Master/texmf-dist/doc/latex/amslatex-primer/amshelp.tex')

diff --git a/Master/texmf-dist/doc/latex/amslatex-primer/amshelp.tex b/Master/texmf-dist/doc/latex/amslatex-primer/amshelp.tex
index 5235334a9c4..9a732626d6c 100644
--- a/Master/texmf-dist/doc/latex/amslatex-primer/amshelp.tex
+++ b/Master/texmf-dist/doc/latex/amslatex-primer/amshelp.tex
@@ -1,16 +1,15 @@
 %%% amshelp.tex
 %%% This is an AMS-LaTeX file
 
-%%% Copyright (c) 2000 Philip S. Hirschhorn
+%%% Copyright (c) 2000, 2008, 2009 Philip S. Hirschhorn
 %
-% This document may be distributed and/or modified under the
-% conditions of the LaTeX Project Public License, either version 1.2
+% 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 1.2 or later is part of all distributions of LaTeX 
-% version 1999/12/01 or later.
-%
+% and version 1.3 or later is part of all distributions of LaTeX
+% version 2003/12/01 or later.
 
 
 %%% Philip Hirschhorn
@@ -22,36 +21,36 @@
 
 
 %%% This is an attempt to explain how to get up and running with
-%%% AmS-LaTeX, if you've already got some familiarity with either 
-%%% TeX or AMS-TeX or LaTeX.
+%%% AmS-LaTeX for someone having some familiarity with TeX,
+%%% AMS-TeX, or LaTeX.
 
 
 
 
-\newcommand{\filedate}{August 10, 2000}
-\newcommand{\fileversion}{Version 2.0}
+\newcommand{\filedate}{June 19, 2009}
+%\newcommand{\filedate}{\today}
+\newcommand{\fileversion}{Version 2.1}
 
 %---------------------------------------------------------------------
-\documentclass[12pt]{amsart}
-%\documentclass{amsart}
-\usepackage{amscd}
+%\documentclass[12pt]{amsart}
+\documentclass{amsart}
 
+\usepackage{url}
 
-\newcommand{\tensor}{\otimes}
-\newcommand{\homotopic}{\simeq}
-\newcommand{\homeq}{\cong}
-\newcommand{\iso}{\approx}
-\newcommand{\ho}{\operatorname{Ho}}
+% 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
 
 
 
-\newcommand{\C}{{\cal C}}
-\newcommand{\M}{{\cal M}}
-\newcommand{\W}{{\cal W}}
 
+%\usepackage{hyperref}
 
-\newenvironment{technical}{\begingroup \small}{\endgroup}
 
 
 
@@ -90,9 +89,8 @@
 \begin{document}
 
 
-\title[Running \AmS-\LaTeX]
-{Getting up and running\\
-with \AmS-\LaTeX}
+\title[Running \AmS-\LaTeX]{Getting up and running\\
+                            with \AmS-\LaTeX}
 
 \author{Philip S. Hirschhorn}
 
@@ -102,67 +100,64 @@ with \AmS-\LaTeX}
 
 \email{psh@math.mit.edu}
 
-
 \date{\filedate, \fileversion}
 
-
 \begin{abstract}
-  This is an attempt to tell you enough about \LaTeX{} and
-  \AmS-\LaTeX{} so that you can get started with it \emph{without}
-  having to read the book.
+  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{} with
-the least possible pain.  These instructions won't be a substitute for
-the User's Guide, but they may get you started quickly enough so that
-you'll only need to refer to the guide occasionally, which should
-eliminate most of the pain.
-
-The current version of \AmS-\LaTeX{} (version 1.2) is really just an
-optional package for the new standard \LaTeX.  \AmS-\LaTeX{} provides
-the document classes \verb"amsart" and \verb"amsbook" (see
-Section~\ref{sec:DocClsCom}) to replace the standard document classes
-\verb"article" and \verb"book", and an optional package \verb"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.
-
+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}.
-
-
-If you've never used \emph{any} version of \TeX, 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.
-
-I've also given you a template file \verb"template.tex", which is an
-attempt to give you enough to mostly fake your way through an
-\AmS-\LaTeX{} file, \emph{almost} without even reading these
+\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 get more information on whatever in that file
+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
@@ -175,8 +170,8 @@ In case you haven't guessed, these instructions were printed using
 \section{Basic \LaTeX{} stuff}
 \label{sec:basicstuff}
 
-In this section, we'll describe the three commands that have to be
-part of any \LaTeX{} document: \verb"\documentclass",
+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
@@ -187,7 +182,8 @@ need to run a file through \LaTeX{} \emph{twice} to be sure that all
 the cross-references are correct).
 
 
-\subsection{The  \texttt{documentclass} command}
+%--------------------------------------------------------------------
+\subsection{The  \texttt{\char`\\documentclass} command}
 \label{sec:DocClsCom}
 
 Before you type anything that actually appears in the paper, you must
@@ -196,87 +192,95 @@ include a \verb"\documentclass" command.  It's easiest to just put the
 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 two
-document classes that are a part of \AmS-\LaTeX: \verb"amsart" and
-\verb"amsbook".  There is also the \verb"amsmath" package that can be
-used with the standard \LaTeX{} \verb"article" document class.  I'll
-only be discussing the \verb"amsart" document class here.  For the
-others, see the \AmS-\LaTeX{} User's Guide~\cite{amslatexusersguide}.
-
-
-The simplest version of the \verb"documentclass" command is
+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}"
+  \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}"
+  \verb"\documentclass[12pt]{amsart}"
 \end{center}
 
+%--------------------------------------------------------------------
+\subsection{Loading optional packages}
+\label{sec:optpack}
 
-There are at least two optional packages that might be of interest.
-The first is for when you want to include the macros that make it
-easier to draw commutative diagrams.  (These aren't included
-automatically, since they take up a lot of memory, and not everyone
-wants to use them.)  If you want 10~point type and you want to use the
-commutative diagram macros, then the commands to use are
+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}
-  \begin{tabular}{l}
-    \verb"\documentclass{amsart}"\\
-    \verb"\usepackage{amscd}"
-  \end{tabular}
+  \verb"\usepackage[lite]{amsrefs}"
 \end{center}
-If you want 12~point type and you want to use the commutative
-diagram macros, then the commands are
+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}
-  \begin{tabular}{l}
-    \verb"\documentclass[12pt]{amsart}"\\
-    \verb"\usepackage{amscd}"
-  \end{tabular}
+  \url{http://www.ctan.org/tex-archive/info/symbols/math/symbols.pdf}
 \end{center}
-
-
-The other optional package is for use when you want to use some of the
-special symbols contained in the \AmS-Fonts package.  (These are the
-fonts \verb"msam" and \verb"msbm".)  If you want the standard names
-for these symbols to be defined for your use, then you need to use the
-optional package \verb"amssymb".  Thus, to use the default 10~point
-type and have the special symbols defined, use the commands
+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}
-If you want to use 12~point type, the commutative diagram macros, and
-special symbols from the \AmS-Fonts collection, then use the commands
+
+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"\documentclass[12pt]{amsart}"\\
-    \verb"\usepackage{amssymb}"
+    \verb"\usepackage[all,cmtip]{xy}"\\
+    \verb"\let\objectstyle=\displaystyle"
   \end{tabular}
 \end{center}
-This document uses both of these packages, and so we used the commands
+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[12pt]{amsart}"\\
-    \verb"\usepackage{amscd}"\\
-    \verb"\usepackage{amssymb}"
+    \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{begin\{document\}} and \texttt{end\{document\}}}
+\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.  You \emph{are} allowed to have macro definitions (i.e.,
-newcommands; see Section~\ref{sec:definitions}) before the
+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.
 
@@ -286,11 +290,11 @@ newcommands; see Section~\ref{sec:definitions}) before the
 
 To begin a new section, you give the command
 \begin{center}
-\verb"\section{Section name}"
+  \verb"\section{Section name}"
 \end{center}
 To begin the present section, I gave the command
 \begin{center}
-\verb"\section{Basic \LaTeX{} stuff}"
+  \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.
@@ -302,16 +306,16 @@ I actually began this section with the lines
     \verb"\label{sec:basicstuff}"
   \end{tabular}
 \end{center}
-This allows me to say ``\verb"Section~\ref{sec:basicstuff}"'' and have
-it printed as Section~\ref{sec:basicstuff}.
+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}"
+  \verb"\subsection{Subsection name}"
 \end{center}
 To begin the present subsection, I gave the command
 \begin{center}
-\verb"\subsection{Sections and subsections}"
+  \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}
@@ -322,42 +326,39 @@ it.  This subsection was begun with the lines
     \verb"\label{sec:sections}"
   \end{tabular}
 \end{center}
-so if we say ``\verb"Section~\ref{sec:sections}",'' it is printed as
-Section~\ref{sec:sections}.
+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}.
+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}"
+  \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}"
+  \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}"
+  \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}}
 
@@ -365,16 +366,16 @@ 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{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."
+  \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.
+  The whole is \emph{more} than the sum of its parts.
 \end{center}
 but if you type
 \begin{verbatim}
@@ -387,28 +388,26 @@ you'll get
   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
+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).
-
-
+have to do that anymore.
 
 %---------------------------------------------------------------------
-\subsection{Once is never enough}
+\subsection{Cross references and the table of contents}
 
 This is an explanation of how \LaTeX{} manages to fill in
-cross-references to parts of the file it hasn't processed yet, and
-what those \verb".aux" and \verb".toc" files are.
-
+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}
@@ -464,60 +463,47 @@ 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
+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{maketitle} command}
+\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" documentclass.
+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 announce the title with the
-command
+You specify the title with the command
 \begin{center}
-\verb"\title[Optional running title]{Actual title}"
+  \verb"\title[Optional running title]{Actual title}"
 \end{center}
 These instructions used the title command
-\begin{center}
-  \begin{tabular}{l}
-    \verb"\title[Running \AmS-\LaTeX]"\\
-    \verb"{Getting up and running\\"\\
-    \verb"with \AmS-\LaTeX}"
-  \end{tabular}
-\end{center}
+\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 let the \emph{full} title be printed
-in the head of the odd numbered pages, I would have used the command
+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}
-  \begin{tabular}{l}
-    \verb"\title{Getting up and running\\"\\
-    \verb"with \AmS-\LaTeX}"
-  \end{tabular}
+  \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}"
@@ -536,7 +522,7 @@ command
 \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 say
+with the command \verb"\curraddr".  For example, you might type
 \begin{center}
   \begin{tabular}{l}
     \verb"\curraddr{Department of Mechanics\\"\\
@@ -556,28 +542,28 @@ To acknowledge support, use the command \verb"\thanks", e.g.,
 \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 (and current
-address, and email address) following that individual's \verb"\author"
-command, in its own \verb"\address" command (and \verb"\curraddr"
-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, 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.  (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.)
+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:
@@ -588,56 +574,51 @@ To have the date of processing used, use the command
 \verb"\date{\today}".
 
 %---------------------------------------------------------------------
-\subsection{\texttt{maketitle}}
-
+\subsection{\texttt{\char`\\maketitle}}
 
 After you've given all of the commands mentioned in this section, you
-can give the command \verb"\maketitle".  The exact arrangement 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.  If you \emph{don't} give the command \verb"\maketitle", a
-title won't be made.
-
-
+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"\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.
+  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.
+  \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.
+  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.
+  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
@@ -647,109 +628,97 @@ 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.
+  \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}.
+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}[Pythagorus]
-The square of the hypotenuse of a right triangle
-is equal to the sum of the squares of the two
-adjacent sides.
+\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}[Pythagorus]
-The square of the hypotenuse of a right triangle
-is equal to the sum of the squares of the two
-adjacent sides.
+\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.
-
-$$
-\begin{tabular}{c@{\hspace{4em}}l@{\hspace{4em}}c}
-
-\multicolumn{3}{c}{Theorem Environments}\\*[8pt]
-
-\hspace{1em}Name&  Printed Form& Body font\\*[6pt]
-
-\texttt{thm}&           \textbf{Theorem}&      Italic\\
-\texttt{cor}&           \textbf{Corollary}&    Italic\\
-\texttt{lem}&           \textbf{Lemma}&        Italic\\
-\texttt{prop}&          \textbf{Proposition}&  Italic\\
-\texttt{defn}&          \textbf{Definition}&   Normal\\
-\texttt{rem}&           \textit{Remark}&       Normal\\
-\texttt{ex}&            \textit{Example}&      Normal\\
-\texttt{notation}&      \textit{Notation}&     Normal\\
-\texttt{terminology}&   \textit{Terminology}&  Normal\\*[3pt]
-\end{tabular}
-$$
-
+\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
+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!
+  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!
+  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!
+  Well, it's \emph{obviously} sufficient!
 \end{proof}
 \end{verbatim}
 and you'll get
 \begin{proof}[Proof (sufficiency)]
-Well, it's \emph{obviously} sufficient!
+  Well, it's \emph{obviously} sufficient!
 \end{proof}
-that is, the \verb"proof" environment allows you to use an optional
+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!
+  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!
+  As any fool can plainly see, it's true!
 \end{proof}
 
-
-
 %---------------------------------------------------------------------
 %---------------------------------------------------------------------
 \section{Cross-References}
@@ -765,41 +734,50 @@ reference numbers filled in automatically by \LaTeX.
 
 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 and numbered equations, you
-assign the key using the \verb"\label" command.  This command takes
-one argument, which is the \emph{key} you're assigning to the object.
-The command \verb"\ref{key}" then produces the number that was
-assigned to that structure.  If the structure is an equation, then the
-command \verb"\eqref{key}" should be used instead of the command
-\verb"\ref{key}".
+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 $f_* = g_* \colon \mathrm{H}_*X \to \mathrm{H}_*Y$.
+  \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 this theorem as follows.
+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 
-$f_* = g_* \colon \mathrm{H}_*X \to \mathrm{H}_*Y$.
+  \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?}
 
-We began this section by typing
+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}"\\
@@ -813,61 +791,56 @@ and we began this subsection by typing
     \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}.''
-
-
-The command \verb"\label{key}" assigns to \verb"key" the value of the
-\emph{smallest enclosing structure}.  That's why the command
-\verb"\ref{sec:xreferences}" is printed as~\ref{sec:xreferences},
-while \verb"\ref{sec:thmrefs}" is printed as~\ref{sec:thmrefs}: the
-key \verb"sec:xreferences" was defined inside of
-Section~\ref{sec:xreferences} but \emph{outside} of
-Section~\ref{sec:thmrefs}, while the key \verb"sec:thmrefs" was
-defined \emph{inside} of Section~\ref{sec:thmrefs}.
-
+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}
+%--------------------------------------------------------------------
+\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".  For
-example, if you type
+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   \iso
-          \bigoplus_{\alpha\in A}\mathrm{H}_* X_\alpha
+  \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   \iso
-          \bigoplus_{\alpha\in A}\mathrm{H}_* X_\alpha
+  \label{additivity}
+  \mathrm{H}_{*} \bigvee_{\alpha\in A} X_{\alpha}   \approx
+    \bigoplus_{\alpha\in A}\mathrm{H}_{*} X_{\alpha}
 \end{equation}
-If we now say
+If we now type
 \begin{verbatim}
 \begin{thm}
-Equation~\eqref{additivity} is true for all sorts of
-functors $\mathrm{H}$.
+  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 $\mathrm{H}$.
+  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 advantage of using
-\verb"\eqref" instead of \verb"\ref"; the command \verb"\eqref"
+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}
 
@@ -875,414 +848,1623 @@ 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}
-Look at page~\pageref{homotopy} to find
-Theorem~\ref{homotopy}.
+See page~\pageref{homotopy} to find Theorem~\ref{homotopy}.
 \end{verbatim}
-you'll get ``Look at page~\pageref{homotopy} to find
+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}".
 
-Bibliography entries receive a \emph{key} as part of their basic
-structure.  Each item in the bibliography is entered as
+When using the \texttt{amsrefs} package, each item in the bibliography
+is begun with
 \begin{verbatim}
-\bibitem{key} The actual bibliography item goes here.
+\bib{key}{TypeOfItem}{
 \end{verbatim}
-(For more detail on this, see Section~\ref{sec:bibliography}.)
-
-You refer to bibliography items using the \verb"\cite" command.  For
-example, the bibliography of these instructions contains the entry
-%
+For example, the bibliography of these instructions contains the entry
 \begin{verbatim}
-\bibitem{HA}
-D. G. Quillen, \emph{Homotopical Algebra,} Lecture Notes in
-Mathematics number 43, Springer-Verlag, Berlin, 1967.
+\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 say ``\verb"This is the work of Quillen~\cite{HA}",'' then it
+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 say
+to annotate the reference.  If we type
 ``\verb"see~\cite[Chapter I]{HA}"'', then it will be printed as
-``see~\cite[Chapter I]{HA}''.
+``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}
-
-
-\subsection{Mathematics in running text}
+\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 (\verb"$") any material that should
-be in math mode.  Thus, if you type
+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."
+  \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.
+  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 is equivalent to typing
+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.
 
+%--------------------------------------------------------------------
 %---------------------------------------------------------------------
-\subsection{Displayed mathematics}
+\section{Displayed mathematics}
 
-For simple displayed mathematics without an equation number, this is
-very much like plain \TeX, again with extra choices that can be
-ignored.  If you enclose material between double dollar
-signs(\verb"$$"), it will be interpreted in math mode and displayed.
-Thus, if you've previously given the command
-\begin{center}
-  \verb"\newcommand{\iso}{\approx}"
-\end{center}
-(see Section~\ref{sec:definitions}), and you type
+For a complete discussion of the environments for displayed
+mathematics, including options for customizing equation numbers,
+see~\cite[section~3]{amslatexusersguide}.
+
+%--------------------------------------------------------------------
+\subsection{Single line displays}
+\label{sec:SngLne}
+
+To display mathematics and number the display (so that you can refer
+to it from elsewhere in the paper) you use the \verb"equation"
+environment.  (\LaTeX{} calls all such numbers \emph{equation
+  numbers}, whether or not the display has anything to do with
+equations.)  If you type
 \begin{verbatim}
-$$
-\pi_1(X \vee Y) \iso \pi_1X * \pi_1Y
-$$
+\begin{equation}
+  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
+\end{equation}
 \end{verbatim}
 you'll get
+\begin{equation}
+  \label{pi1eqn}
+  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
+\end{equation}
+
+If you'd like to be able to make reference to the equation number,
+you need to \emph{label} the equation, using a \emph{key} that you
+can use for referencing it:
+\begin{verbatim}
+\begin{equation}
+  \label{pi1eqn}
+  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
+\end{equation}
+\end{verbatim}
+If you later type ``\verb"see formula~\eqref{pi1eqn}"'' you'll get 
+``see formula~\eqref{pi1eqn}.''  (For more on cross-references to
+formulas, see section~\ref{sec:thmrefs}.)
+
+To display mathematics without an equation number, you use the
+\verb"equation*" environment.  (This is a common \LaTeX ism: Adding an
+asterisk to the name of a numbered \LaTeX{} environment often gives
+the unnumbered equivalent.)  If you type
+\begin{verbatim}
+\begin{equation*}
+  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
+\end{equation*}
+\end{verbatim}
+then you'll get
+\begin{equation*}
+  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
+\end{equation*}
+There are several essentially equivalent ways of producing this
+unnumbered display: The \verb"displaymath" environment, used as in
+\begin{verbatim}
+\begin{displaymath}
+  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
+\end{displaymath}
+\end{verbatim}
+accomplishes the same thing, as will either
+\begin{verbatim}
 $$
-\pi_1(X \vee Y) \iso \pi_1X * \pi_1Y
+  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
 $$
-The new choices are that exactly the same thing will be obtained by
-either
+\end{verbatim}
+or
 \begin{verbatim}
 \[
-\pi_1(X \vee Y) \iso \pi_1X * \pi_1Y
+  \pi_{1}(X \vee Y) \approx \pi_{1}X * \pi_{1}Y
 \]
 \end{verbatim}
-or by
+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{displaymath}
-\pi_1(X \vee Y) \iso \pi_1X * \pi_1Y
-\end{displaymath}
+\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}
-or by
+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{equation*}
-\pi_1(X \vee Y) \iso \pi_1X * \pi_1Y
-\end{equation*}
+\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.
 
 
-If you'd like the displayed formula to be \emph{numbered}, then you
-should use the \verb"equation" environment.  (\LaTeX{} calls all
-formula numbers \emph{equation numbers}, whether or not the
-mathematics has anything to do with equations.)  If you type
+%--------------------------------------------------------------------
+\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}
-\pi_1(X \vee Y) \iso \pi_1X * \pi_1Y
+  \label{eq:split}
+  \begin{split}
+    a &= b + c\\
+    d &= e + f
+  \end{split}
 \end{equation}
 \end{verbatim}
-you'll get
+then you'll get
 \begin{equation}
-\label{pi1eqn}
-\pi_1(X \vee Y) \iso \pi_1X * \pi_1Y
+  \label{eq:split}
+  \begin{split}
+    a &= b + c\\
+    d &= e + f
+  \end{split}
 \end{equation}
-(Notice that the \verb"equation" environment produces an equation
-number, while the \verb"equation*" environment doesn't.  This is a
-standard \LaTeX ism: Adding an asterisk to the name of a numbered 
-\LaTeX{} environment often gives the unnumbered equivalent.)
-
-
-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:
+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{pi1eqn}
-\pi_1(X \vee Y) \iso \pi_1X * \pi_1Y
+  \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}
-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}.)
-
-\AmS-\LaTeX{} has several environments that make it easier to typeset
-complicated multi-line displays.  These are explained in the
-\AmS-\LaTeX{} User's Guide~\cite{amslatexusersguide} and illustrated
-in \verb"testmath.tex"~\cite[pages 29--40]{testmath}.
+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}).
 
-%---------------------------------------------------------------------
-\subsection{Commutative diagrams}
-
-\AmS-\LaTeX{} provides the \verb"CD" environment for drawing
-commutative diagrams.  These only allow for \emph{rectangular}
-diagrams, but they're very convenient to use.  If you need diagonal
-arrows, or curving arrows, or arrows that make a number of right angle
-turns, I recommend the XY-Pic package, which can do all of these
-things (and more).
-
-\textbf{Important}: To use the \verb"CD" environment, you need to load
-it with the command \verb"\usepackage{amscd}".  For example, the
-template file gives the commands
-\begin{center}
-  \begin{tabular}{l}
-    \verb"\documentclass[12pt]{amsart}"\\
-    \verb"\usepackage{amscd}"
-  \end{tabular}
-\end{center}
-which select the \verb"amsart" document class with twelve point type,
-and loads the macros for the \verb"CD" environment.
 
-\subsubsection*{A simple example}
+%--------------------------------------------------------------------
+\subsubsection{Alignments flush left and flush right}
+\label{sec:flalign}
 
-To produce the diagram
-$$
-\begin{CD}
-A    @>>>  B   @=     B\\
-@AAA       @|         @VVV\\
-X    @<<<  B   @>>>   Y
-\end{CD}
-$$
+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}
-$$
-\begin{CD}
-A    @>>>  B   @=     B\\
-@AAA       @|         @VVV\\
-X    @<<<  B   @>>>   Y
-\end{CD}
-$$
+\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}
-This illustrates several things.  First of all, the \verb"CD"
-environment must be inside of a displayed mathematics environment.
-(Here we used the standard \verb"$$" toggle to get displayed
-mathematics.  If we had used, e.g., \verb"\begin{equation}" and
-  \verb"\end{equation}", we would have had an equation number assigned
-to the display.)  Right pointing arrows are obtained with \verb"@>>>",
-left pointing arrows with \verb"@<<<", up pointing arrows with
-\verb"@AAA", down pointing arrows with \verb"@VVV", horizontal equals
-signs with \verb"@=", and vertical equals signs with \verb"@|".  Every
-line except the last is ended with a double backslash (\verb"\\").
-
-\subsubsection*{Labeling the arrows}
-
-The arrows can also be labeled.  For horizontal arrows, anything
-between the first and second inequality sign goes above the arrow,
-and anything between the second and third inequality sign goes below
-it.  For downward arrows, anything between the first and second
-\verb"V" goes to the left, and anything between the second and third
-goes to the right (and similarly for upward arrows).  Thus, if you
-type
+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{displaymath}
-\begin{CD}
-\mathrm{H}_iX    @>f_*>>   \mathrm{H}_iY   @<g_*<\iso<    E_fA\\
-@V\phi VV                  @V\psi VV                 @AA\Omega A\\
-\pi_iQ  @>\alpha\beta\gamma>>  \pi_i(R,S)
-                   @<<\text{A long arrow}<   \prod_{k=1}^i H_kZ
-\end{CD}
-\end{displaymath}
+\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}
-you'll get
-\begin{displaymath}
-\begin{CD}
-\mathrm{H}_iX  @>f_*>>  \mathrm{H}_iY   @<g_*<\iso<    E_fA\\
-@V\phi VV               @V\psi VV                @AA\Omega A\\
-\pi_iQ   @>\alpha\beta\gamma>>    \pi_i(R,S)
-                   @<<\text{A long arrow}<   \prod_{k=1}^i H_kZ
-\end{CD}
-\end{displaymath}
+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.
 
 
-\subsubsection*{Leaving out part of the rectangle}
+%--------------------------------------------------------------------
+\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}.)
 
-If you want to end a line in the diagram early (omitting the right
-end of that line), just type the double backslash.  If you want to
-leave out the \emph{beginning} of a line, you type ``\verb"@."''
-(the ``at'' sign followed by a period) to denote an invisible arrow. 
-(It's the arrows that are the column markers in the \verb"CD"
-environment.)  Thus, if you type
+For example, if you type
 \begin{verbatim}
-\begin{displaymath}
-\begin{CD}
-X\\
-@VfVV\\
-Y  @=     Y\\
-@.       @VVgV\\
-  @.     Z
-\end{CD}
-\end{displaymath}
+\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}
-you'll get
-\begin{displaymath}
-\begin{CD}
-X\\
-@VfVV\\
-Y  @=     Y\\
-@.       @VVgV\\
-  @.     Z
-\end{CD}
-\end{displaymath}
-
-
-
+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}
 
-%---------------------------------------------------------------------
-%---------------------------------------------------------------------
-\section{Macro definitions, a.k.a.\ \texttt{newcommand}}
-\label{sec:definitions}
+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*}
 
-\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
+%--------------------------------------------------------------------
+%--------------------------------------------------------------------
+\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}
-\verb"\newcommand{\tensor}{\otimes}"
+  \begin{tabular}{l}
+    \verb"\usepackage[all,cmtip]{xy}"\\
+    \verb"\let\objectstyle=\displaystyle"
+  \end{tabular}
 \end{center}
-The advantage of this is that \LaTeX{} will check to see if there
+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}"
+  \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}"
+  \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}"
+  \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. 
+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.
+\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.
+\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
+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.
+\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.
+\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 extra argument for each
+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
+\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
+\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
@@ -1290,92 +2472,446 @@ numbers its items accordingly.  If you type
 \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}
+  \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}
+  \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}
 
-\subsection{\texttt{begin\{thebibliography\}} and 
-  \texttt{end\{thebibliography\}}}
+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.
 
-The bibliography is begun with the command
-\begin{center}
-\verb"\begin{thebibliography}{number}"
-\end{center}
-where \emph{number} is a random number that, when printed, is as
-wide as the widest number of any item in the bibliography.  (The only
-use made of \verb"number" is that \LaTeX{} assumes that the numbers
-that it will assign to the actual items in the bibliography will be
-no wider (when printed) than \verb"number".) For example, if the
-bibliography will contain between 10 and~19 items, you can use
-\verb"\begin{thebibliography}{10}".
+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}).
 
-After listing each item in the bibliography, you end the bibliography
-with the \verb"\end{thebibliography}" command.
 
 
-\subsection{Bibliography items}
+%--------------------------------------------------------------------
+\subsection{Using \texttt{amsrefs}}
+\label{sec:amsrefs}
 
-Each item is begun with a \verb"\bibitem" command.  The format is
+To use \texttt{amsrefs}, you load it by putting the command
 \begin{center}
-\verb"\bibitem{key for cross-references}Item entry"
+  \verb"\usepackage[lite]{amsrefs}"
 \end{center}
-For example, the bibliography in these instructions contains the
-entry
-%
+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}
-\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.
+\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 above entry allows you to say
+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}
-Homotopy inverse limits are discussed
-in~\cite[Chapter 11]{yellowmonster}.
+\bib{ReferenceKey}{ReferenceType}{
+  comma separated list of keyword={value} statements
+}
 \end{verbatim}
-and have it print as
-``Homotopy inverse limits are discussed
-in~\cite[Chapter 11]{yellowmonster}.''
-For more on this, see Section~\ref{sec:bibreferences}.
+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}.
 
 
 
@@ -1397,25 +2933,40 @@ 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 August 10, 2000
+%%% Version of May 14, 2009
 %%%---------------------------------------------------------
-%%% The following commands choose 12 point type (instead
-%%% of the default 10 point), allow us to use the
-%%% commutative diagram macros, and define the standard
-%%% names for all of the special symbols in the AMSfonts
-%%% package:
-\documentclass[12pt]{amsart}
-\usepackage{amscd}
+%%% 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 a good place to  put our macro definitions.
+%%% 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}
@@ -1424,8 +2975,6 @@ The following is the text of the file \verb"template.tex".
 \DeclareMathOperator{\ho}{Ho}
 \DeclareMathOperator*{\colim}{colim}
 
-
-
 \newcommand{\R}{\mathbb{R}}
 \newcommand{\C}{\mathbb{C}}
 \newcommand{\Z}{\mathbb{Z}}
@@ -1433,8 +2982,6 @@ The following is the text of the file \verb"template.tex".
 \newcommand{\M}{\mathcal{M}}
 \newcommand{\W}{\mathcal{W}}
 
-
-
 \newcommand{\itilde}{\tilde{\imath}}
 \newcommand{\jtilde}{\tilde{\jmath}}
 \newcommand{\ihat}{\hat{\imath}}
@@ -1525,7 +3072,6 @@ The following is the text of the file \verb"template.tex".
 \newtheorem{lem}[equation]{Lemma}
 \newtheorem{prop}[equation]{Proposition}
 
-
 \theoremstyle{definition}
 \newtheorem{defn}[equation]{Definition}
 
@@ -1535,7 +3081,6 @@ The following is the text of the file \verb"template.tex".
 \newtheorem{notation}[equation]{Notation}
 \newtheorem{terminology}[equation]{Terminology}
 
-
 %%%-------------------------------------------------------------------
 %%%-------------------------------------------------------------------
 %%%-------------------------------------------------------------------
@@ -1550,40 +3095,76 @@ The following is the text of the file \verb"template.tex".
 %%% \title{Text of the title}
 %%% or
 %%% \title[Short form for the running head]{Text of the title}
-\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{}
 
 
-%%% 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 next line:
+%%% 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}
 
 
 
@@ -1598,92 +3179,174 @@ The following is the text of the file \verb"template.tex".
 
 
 
-%%%-------------------------------------------------------------------
-%%%-------------------------------------------------------------------
-%%% The number "10" that appears in the next command is a TOTALLY
-%%% RANDOM NUMBER which is chosen so that if it was printed, it would
-%%% be at least as wide as any number of an item in the bibliography:
-
-\begin{thebibliography}{10}
-
 
+%%% -------------------------------------------------------------------
+%%% -------------------------------------------------------------------
+%%% This is where we create the bibliography.
 
+\begin{bibdiv}
+  \begin{biblist}
 
 %%% The format of bibliography items is as in the following examples:
 %%% 
-%%% \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.
-%%% 
-%%% \bibitem{HA}
-%%% D. G. Quillen, \emph{Homotopical Algebra,} Lecture Notes in
-%%% Mathematics number 43, Springer-Verlag, Berlin, 1967.
-
-
-
-
+%%% \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{thebibliography}
 \end{document}
 \end{verbatim}
 
-
-
-
-
-
-
-
-
-
-
 %---------------------------------------------------------------------
 %---------------------------------------------------------------------
 
-\begin{thebibliography}{10}
-
-
-\bibitem{amslatexusersguide} 
-  American Mathematical Society, \emph{\AmS-\LaTeX{} Version 1.2 User's
-    Guide,} November, 1996.  This is the file \texttt{amsldoc.dvi},
-  available from the AMS ftp site \texttt{e-math.ams.org}.
 
-\bibitem{instr-l} 
-  American Mathematical Society, \emph{Instructions for preparation of
-    papers and monographs: \AmS-\LaTeX,} November, 1966.  This is the
-  file \texttt{instr-l.dvi}, available from the AMS ftp site
-  \texttt{e-math.ams.org}.
 
-\bibitem{testmath}
-  American Mathematical Society, \emph{Sample paper for the amsmath
-    package,} November, 1996.  This is the file \texttt{testmath.tex},
-  available from the AMS ftp site \texttt{e-math.ams.org}.
 
-\bibitem{latex}
-  Leslie Lamport, \emph{\LaTeX{} User's Guide and Reference Manual,}
-  Addison-Wesley, 1986.
 
-\bibitem{NotShort} 
-  Tobias Oetiker, Hubert Partl, Irene Hyna, and Elisabeth Schlegl
-  \emph{The not so short introduction to \LaTeXe}, available by ftp
-  from CTAN (the Comprehensive \TeX{} archive network), at
-  \verb"ftp.tex.ac.uk", \verb"ftp.dante.de", and many mirrors, under
-  the name \verb"lshort2e.tex" or \verb"lshort2e.dvi".
+\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}
 
-\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.
-
-\bibitem{HA}
-  D. G. Quillen, \emph{Homotopical Algebra,} Lecture Notes in
-  Mathematics number 43, Springer-Verlag, Berlin, 1967.
 
 
-\end{thebibliography}
 \end{document}
-
-- 
cgit v1.2.3

