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-%%% amshelp.tex
-%%% This is an AMS-LaTeX file
-
-%%% Copyright (c) 2000 Philip S. Hirschhorn
-%
-% This document may be distributed and/or modified under the
-% conditions of the LaTeX Project Public License, either version 1.2
-% 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.
-%
-
-
-%%% Philip Hirschhorn
-%%% Department of Mathematics
-%%% Wellesley College
-%%% Wellesley, MA 02481
-%%% psh@poincare.wellesley.edu
-%%% psh@math.mit.edu
-
-
-%%% This is an attempt to explain how to get up and running with
-%%% AmS-LaTeX, if you've already got some familiarity with either
-%%% TeX or AMS-TeX or LaTeX.
-
-
-
-
-\newcommand{\filedate}{August 10, 2000}
-\newcommand{\fileversion}{Version 2.0}
-
-%---------------------------------------------------------------------
-\documentclass[12pt]{amsart}
-%\documentclass{amsart}
-\usepackage{amscd}
-
-
-\newcommand{\tensor}{\otimes}
-\newcommand{\homotopic}{\simeq}
-\newcommand{\homeq}{\cong}
-\newcommand{\iso}{\approx}
-\newcommand{\ho}{\operatorname{Ho}}
-
-
-
-
-\newcommand{\C}{{\cal C}}
-\newcommand{\M}{{\cal M}}
-\newcommand{\W}{{\cal W}}
-
-
-\newenvironment{technical}{\begingroup \small}{\endgroup}
-
-
-
-%---------------------------------------------------------------------
-%---------------------------------------------------------------------
-
-\numberwithin{equation}{section}
-
-
-% Theorem environments
-
-\theoremstyle{plain} %% This is the default, anyway
-\newtheorem{thm}[equation]{Theorem}
-\newtheorem{cor}[equation]{Corollary}
-\newtheorem{lem}[equation]{Lemma}
-\newtheorem{prop}[equation]{Proposition}
-
-
-\theoremstyle{definition}
-\newtheorem{defn}[equation]{Definition}
-
-\theoremstyle{remark}
-\newtheorem{rem}[equation]{Remark}
-\newtheorem{ex}[equation]{Example}
-\newtheorem{notation}[equation]{Notation}
-\newtheorem{terminology}[equation]{Terminology}
-
-
-
-
-%---------------------------------------------------------------------
-%---------------------------------------------------------------------
-%---------------------------------------------------------------------
-%---------------------------------------------------------------------
-
-\begin{document}
-
-
-\title[Running \AmS-\LaTeX]
-{Getting up and running\\
-with \AmS-\LaTeX}
-
-\author{Philip S. Hirschhorn}
-
-\address{Department of Mathematics\\
- Wellesley College\\
- Wellesley, Massachusetts 02481}
-
-\email{psh@math.mit.edu}
-
-
-\date{\filedate, \fileversion}
-
-
-\begin{abstract}
- 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.
-\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.
-
-
-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
-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
-confuses you.
-
-In case you haven't guessed, these instructions were printed using
-\AmS-\LaTeX, so you can get some idea what it all looks like.
-
-
-
-%---------------------------------------------------------------------
-%---------------------------------------------------------------------
-\section{Basic \LaTeX{} stuff}
-\label{sec:basicstuff}
-
-In this section, we'll describe the three commands that have to be
-part of any \LaTeX{} document: \verb"\documentclass",
-\verb"\begin{document}", and \verb"\end{document}". The complete
-explanation of these can be found in the \LaTeX{} User's
-Guide~\cite{latex} or in \emph{The not so short introduction to
- \LaTeXe}~\cite{NotShort}. We'll also explain how to begin a new
-section or subsection of the paper, and how \LaTeX{} manages to get
-the cross-references right (which is also the explanation of why you
-need to run a file through \LaTeX{} \emph{twice} to be sure that all
-the cross-references are correct).
-
-
-\subsection{The \texttt{documentclass} command}
-\label{sec:DocClsCom}
-
-Before you type anything that actually appears in the paper, you must
-include a \verb"\documentclass" command. It's easiest to just put the
-\verb"\documentclass" command at the very beginning of the file,
-possibly with a few lines of comments before it.
-
-It's actually the choice of document class that determines whether
-you're using \AmS-\LaTeX{} or just plain old \LaTeX. There are 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
-\begin{center}
-\verb"\documentclass{amsart}"
-\end{center}
-This will give you the default type size, which is 10~point type. If
-you'd like to use 12~point type, then you should include the optional
-argument \verb"[12pt]"; this makes the command
-\begin{center}
-\verb"\documentclass[12pt]{amsart}"
-\end{center}
-
-
-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
-\begin{center}
- \begin{tabular}{l}
- \verb"\documentclass{amsart}"\\
- \verb"\usepackage{amscd}"
- \end{tabular}
-\end{center}
-If you want 12~point type and you want to use the commutative
-diagram macros, then the commands are
-\begin{center}
- \begin{tabular}{l}
- \verb"\documentclass[12pt]{amsart}"\\
- \verb"\usepackage{amscd}"
- \end{tabular}
-\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
-\begin{center}
- \begin{tabular}{l}
- \verb"\documentclass{amsart}"\\
- \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
-\begin{center}
- \begin{tabular}{l}
- \verb"\documentclass[12pt]{amsart}"\\
- \verb"\usepackage{amssymb}"
- \end{tabular}
-\end{center}
-This document uses both of these packages, and so we used the commands
-\begin{center}
- \begin{tabular}{l}
- \verb"\documentclass[12pt]{amsart}"\\
- \verb"\usepackage{amscd}"\\
- \verb"\usepackage{amssymb}"
- \end{tabular}
-\end{center}
-
-%---------------------------------------------------------------------
-
-\subsection{\texttt{begin\{document\}} and \texttt{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
-\verb"\begin{document}", and that's actually a good place for them,
- but that's about all.
-
-%---------------------------------------------------------------------
-\subsection{Sections and subsections}
-\label{sec:sections}
-
-To begin a new section, you give the command
-\begin{center}
-\verb"\section{Section name}"
-\end{center}
-To begin the present section, I gave the command
-\begin{center}
-\verb"\section{Basic \LaTeX{} stuff}"
-\end{center}
-A section number is supplied automatically. If you want to be able to
-make reference to that section, then you need to \emph{label} it.
-Since I wanted to be able to demonstrate the cross-reference commands,
-I actually began this section with the lines
-\begin{center}
- \begin{tabular}{l}
- \verb"\section{Basic \LaTeX{} stuff}"\\
- \verb"\label{sec:basicstuff}"
- \end{tabular}
-\end{center}
-This allows me to say ``\verb"Section~\ref{sec:basicstuff}"'' and have
-it printed as Section~\ref{sec:basicstuff}.
-
-To begin a new subsection, you give the command
-\begin{center}
-\verb"\subsection{Subsection name}"
-\end{center}
-To begin the present subsection, I gave the command
-\begin{center}
-\verb"\subsection{Sections and subsections}"
-\end{center}
-A subsection number is supplied automatically. If you want to be able
-to make reference to that subsection, then you need to \emph{label}
-it. This subsection was begun with the lines
-\begin{center}
- \begin{tabular}{l}
- \verb"\subsection{Sections and subsections}"\\
- \verb"\label{sec:sections}"
- \end{tabular}
-\end{center}
-so if we say ``\verb"Section~\ref{sec:sections}",'' it is printed as
-Section~\ref{sec:sections}.
-
-Labels always take the number of the smallest enclosing structure.
-Thus, a \verb"\label" command that's inside a section but \emph{not}
-inside a subsection or Theorem or anything else will take the value of
-the section counter, while a \verb"\label" command that's inside the
-statement of a Theorem will take the value of that Theorem number.
-For more information on this, see Section~\ref{sec:xreferences}.
-
-%--------------------------------------------------------------------
-\subsubsection{Yes, there are subsubsections too}
-
-I began this subsubsection with the command
-\begin{center}
-\verb"\subsubsection{Yes, there are subsubsections too}"
-\end{center}
-I refuse to even experiment to see if there are subsubsubsections.
-
-
-
-%--------------------------------------------------------------------
-\subsubsection*{Sections without numbers}
-
-I began this subsubsection with the command
-\begin{center}
-\verb"\subsubsection*{Sections without numbers}"
-\end{center}
-and got a subsubsection that wasn't numbered. If you give the command
-\begin{center}
-\verb"\section*{A Section Title}"
-\end{center}
-then you'll begin a new section that will not have a number.
-
-
-
-%---------------------------------------------------------------------
-\subsection{Italics \emph{for emphasis}}
-
-If you want to use italics to emphasize a word or two, the \LaTeX{}
-convention is not to switch explicitly to italics, but rather to use
-the command \verb"\emph" (which means \emph{emphasize}). This command
-works just like a font change command, except that it switches you
-\emph{into} italics if the current font is upright, and switches you
-\emph{out of} italics if the current font is italics.
-
-For example, if you type
-\begin{center}
-\verb"The whole is \emph{more} than the sum of its parts."
-\end{center}
-you'll get
-\begin{center}
-The whole is \emph{more} than the sum of its parts.
-\end{center}
-but if you type
-\begin{verbatim}
-\begin{thm}
- The whole is \emph{more} than the sum of its parts.
-\end{thm}
-\end{verbatim}
-you'll get
-\begin{thm}
- The whole is \emph{more} than the sum of its parts.
-\end{thm}
-
-
-\subsubsection*{Note}
-
-The \verb"\emph" command is a recent addition to \LaTeX, and it has
-the feature that it automatically inserts an italic correction where
-needed. If you don't know what an italic correction is, you can
-safely ignore this paragraph (but I will at least mention that all
-those ``\verb"\/"'' commands frequently seen in \TeX{} (and older
-\LaTeX) documents are all inserting italic corrections; the point of
-this paragraph is that, with the current version of \LaTeX, you don't
-have to do that anymore).
-
-
-
-%---------------------------------------------------------------------
-\subsection{Once is never enough}
-
-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.
-
-
-\subsubsection*{Cross-References}
-
-Every time \LaTeX{} processes your file, it writes an \emph{auxiliary}
-file. Since the file containing these instructions is called
-\verb"amshelp.tex", the auxiliary file is called \verb"amshelp.aux".
-The auxiliary file contains the definitions of all the keys used for
-cross-references. When \LaTeX{} begins to process your file, it first
-looks for an \verb".aux" file, and reads it in if it exists. Of
-course, this is the \verb".aux" file that was produced the \emph{last}
-time that your file was processed, so the Theorem numbers, Section
-numbers, etc., are all the ones from the last time the file was
-processed.
-
-The very first time that \LaTeX{} processes your file, there is no
-\verb".aux" file, and so \LaTeX{} gives \emph{lots} of warning
-messages about undefined labels, or whatever. Ignore all of this.
-The \emph{next} time that you run \LaTeX, there \emph{will} be an
-\verb".aux" file, and all the references will be filled in. (Yes, it
-is possible, at least in theory, for some page number to change every
-time you run \LaTeX{} on your file, even without any changes in the
-source file, but this isn't very likely.)
-
-\subsubsection*{The Table of Contents}
-
-If you give the command \verb"\tableofcontents", then \LaTeX{} will
-try to write a table of contents, including the page numbers of the
-sections. Obviously, \LaTeX{} can't know those page numbers or
-section titles yet, so as \LaTeX{} processes your file, it writes a
-\verb".toc" file containing the information it needs. (The
-\verb".toc" file for these instructions is \verb"amshelp.toc".) Once
-again, \LaTeX{} is always using the information from the \emph{last}
-time that it processed your file.
-
-If you \emph{do} include a table of contents in your document, and if
-the table of contents takes up at least a page or so of space, then
-you might have to run \LaTeX{} \emph{three} times in order to get all
-of the cross-references right. The reason for this is that the first
-time you run \LaTeX{} there isn't any \verb".toc" file listing the
-section titles, and so the table of contents has nothing in it. The
-second time you run \LaTeX{} you'll get a table of contents that lists
-the page numbers for the sections from the last time you ran \LaTeX,
-when the table of contents took up no space at all. Unfortunately,
-during this second run, the table of contents will be created, and
-will take up enough space to change the page numbers of the sections
-from what they were during the first run. Only during the
-\emph{third} run will the correct page numbers be written into the
-table of contents. Since this doesn't change the amount of space that
-the table of contents occupies, this version will be correct.
-
-\subsubsection*{How do I know when everything is correct?}
-
-After processing your file, \LaTeX{} checks whether all the
-cross-reference numbers that it read from the \verb".aux" file are
-correct. If any of them are incorrect, it prints a warning on the
-screen at the very end of the run advising you that labels may have
-changed, and that you should run \LaTeX{} again to get the
-cross-references right. Unfortunately, \LaTeX{} doesn't seem to check
-that the table of contents entries are correct, so if you change the
-name of a section in a way that doesn't make any page references
-incorrect, you won't be warned to run \LaTeX{} again.
-
-
-
-
-
-%---------------------------------------------------------------------
-%---------------------------------------------------------------------
-\section{Title, Author, and the \texttt{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.
-
-%---------------------------------------------------------------------
-
-\subsection{The title}
-You announce the title with the
-command
-\begin{center}
-\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}
-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
-\begin{center}
- \begin{tabular}{l}
- \verb"\title{Getting up and running\\"\\
- \verb"with \AmS-\LaTeX}"
- \end{tabular}
-\end{center}
-
-
-
-%---------------------------------------------------------------------
-\subsection{The author, and the author's address}
-
-
-The author is specified with an \verb"author" command:
-\begin{center}
-\verb"\author{Author's name}"
-\end{center}
-These directions used the command
-\verb"\author{Philip S. Hirschhorn}".
-The author's address is given in an address command, with double
-backslashes to indicate line breaks. These instructions used the
-command
-\begin{center}
- \begin{tabular}{l}
- \verb"\address{Department of Mathematics\\"\\
- \verb"Wellesley College\\"\\
- \verb"Wellesley, Massachusetts 02481}"
- \end{tabular}
-\end{center}
-If the author's current address is different from the address at which
-the research was carried out, then you can specify the current address
-with the command \verb"\curraddr". For example, you might say
-\begin{center}
- \begin{tabular}{l}
- \verb"\curraddr{Department of Mechanics\\"\\
- \verb"Brake and Wheel Bearing Division\\"\\
- \verb"Serene Service Center\\"\\
- \verb"Salem, Massachusetts 02139}"
- \end{tabular}
-\end{center}
-You can also include an email address, with the \verb"\email"
-command. These instructions used the command
-\begin{center}
- \verb"\email{psh@math.mit.edu}"
-\end{center}
-To acknowledge support, use the command \verb"\thanks", e.g.,
-\begin{center}
- \verb"\thanks{Supported in part by NSF grant 3.14159}"
-\end{center}
-This will be printed as a footnote on the first page.
-
-
-
-
-
-\subsubsection*{Multiple authors}
-
-If there are several authors, then each one should have a separate
-\verb"\author" command, with each individual's address (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 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.)
-
-
-%---------------------------------------------------------------------
-
-\subsection{The date}
-
-This is pretty straightforward:
-\begin{center}
-\verb"\date{Whatever date you please}"
-\end{center}
-To have the date of processing used, use the command
-\verb"\date{\today}".
-
-%---------------------------------------------------------------------
-\subsection{\texttt{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.
-
-
-
-%---------------------------------------------------------------------
-%---------------------------------------------------------------------
-\section{Theorems, Propositions, Lemmas, etc.}
-
-The instructions in this section assume that you're using the
-\verb"newtheorem" commands that I put in the file
-\verb"template.tex".
-
-\subsection{Stating theorems, propositions, etc.}
-
-To state a theorem, you do the following:
-\begin{verbatim}
-\begin{thm}
-The square of the hypotenuse of a right triangle
-is equal to the sum of the squares of the two
-adjacent sides.
-\end{thm}
-\end{verbatim}
-If you do that, you'll get the following:
-\begin{thm}
-\label{pythagthm}
-The square of the hypotenuse of a right triangle
-is equal to the sum of the squares of the two
-adjacent sides.
-\end{thm}
-If you thought that it was only a proposition, you'd use
-\begin{verbatim}
-\begin{prop}
-The square of the hypotenuse of a right triangle
-is equal to the sum of the squares of the two
-adjacent sides.
-\end{prop}
-\end{verbatim}
-and you'd get
-\begin{prop}
-The square of the hypotenuse of a right triangle
-is equal to the sum of the squares of the two
-adjacent sides.
-\end{prop}
-
-If you think it's a theorem again, but you'd like to make reference to
-it in some other part of the paper, you have to choose a \emph{key}
-with which you'll refer to it, and then \emph{label} the theorem. If
-you want to use the key \emph{pythagthm}, then it would look like the
-following:
-\begin{verbatim}
-\begin{thm}
-\label{pythagthm}
-The square of the hypotenuse of a right triangle
-is equal to the sum of the squares of the two
-adjacent sides.
-\end{thm}
-\end{verbatim}
-If you later give the command \verb"\ref{pythagthm}", then that
-command will expand to the \emph{number} that was assigned to that
-theorem (in this case, \ref{pythagthm}). For more explanation of
-cross-references, see Section~\ref{sec:xreferences}.
-
-If you'd like to state a theorem and give a \emph{name} to it, then
-you can add an optional argument to the \verb"\begin{thm}" command.
-If you type
-\begin{verbatim}
-\begin{thm}[Pythagorus]
-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.
-\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.
-
-$$
-\begin{tabular}{c@{\hspace{4em}}l@{\hspace{4em}}c}
-
-\multicolumn{3}{c}{Theorem Environments}\\*[8pt]
-
-\hspace{1em}Name& Printed Form& Body font\\*[6pt]
-
-\texttt{thm}& \textbf{Theorem}& Italic\\
-\texttt{cor}& \textbf{Corollary}& Italic\\
-\texttt{lem}& \textbf{Lemma}& Italic\\
-\texttt{prop}& \textbf{Proposition}& Italic\\
-\texttt{defn}& \textbf{Definition}& Normal\\
-\texttt{rem}& \textit{Remark}& Normal\\
-\texttt{ex}& \textit{Example}& Normal\\
-\texttt{notation}& \textit{Notation}& Normal\\
-\texttt{terminology}& \textit{Terminology}& Normal\\*[3pt]
-\end{tabular}
-$$
-
-For full details, see the beginning of the template file (reproduced
-here in Section~\ref{sec:template}), after the comment ``The Theorem
-Environments.''
-
-
-
-%---------------------------------------------------------------------
-\subsection{Proofs}
-
-To give a proof, you do the following:
-\begin{verbatim}
-\begin{proof}
-As any fool can plainly see, it's true!
-\end{proof}
-\end{verbatim}
-and you'll get the following:
-\begin{proof}
-As any fool can plainly see, it's true!
-\end{proof}
-If the theorem said that a condition was both necessary and sufficient
-for something, and you want to prove each part separately,
-you can do the following:
-\begin{verbatim}
-\begin{proof}[Proof (sufficiency)]
-Well, it's \emph{obviously} sufficient!
-\end{proof}
-\end{verbatim}
-and you'll get
-\begin{proof}[Proof (sufficiency)]
-Well, it's \emph{obviously} sufficient!
-\end{proof}
-that is, the \verb"proof" environment allows you to use an optional
-second argument that will appear in place of the word \verb"Proof".
-
-If the proof of Theorem~\ref{pythagthm} does not appear immediately
-after its statement, you might use the following:
-\begin{verbatim}
-\begin{proof}[Proof of Theorem~\ref{pythagthm}]
-As any fool can plainly see, it's true!
-\end{proof}
-\end{verbatim}
-and you'd get
-\begin{proof}[Proof of Theorem~\ref{pythagthm}]
-As any fool can plainly see, it's true!
-\end{proof}
-
-
-
-%---------------------------------------------------------------------
-%---------------------------------------------------------------------
-\section{Cross-References}
-\label{sec:xreferences}
-
-This section explains how to make reference to numbered sections,
-theorems, equations, and bibliography items, with the correct
-reference numbers filled in automatically by \LaTeX.
-
-%---------------------------------------------------------------------
-\subsection{References to sections, theorems and equations}
-\label{sec:thmrefs}
-
-For each structure in the manuscript to which you'll be making
-reference, you must assign a \emph{key} that you'll use to refer to
-that structure. For sections, theorems 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}".
-
-
-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 $f_* = g_* \colon \mathrm{H}_*X \to \mathrm{H}_*Y$.
-\end{thm}
-
-We typed this 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
-$f_* = g_* \colon \mathrm{H}_*X \to \mathrm{H}_*Y$.
-\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
-\begin{center}
- \begin{tabular}{l}
- \verb"\section{Cross-References}"\\
- \verb"\label{sec:xreferences}"
- \end{tabular}
-\end{center}
-and we began this subsection by typing
-\begin{center}
- \begin{tabular}{l}
- \verb"\subsection{References to sections, theorems and equations}"\\
- \verb"\label{sec:thmrefs}"
- \end{tabular}
-\end{center}
-The phrase ``\verb"See Section~\ref{sec:xreferences}"'' is printed as
-``See Section~\ref{sec:xreferences},'' while the phrase
-``\verb"See Section~\ref{sec:thmrefs}"'' is printed as ``See
-Section~\ref{sec:thmrefs}.''
-
-
-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}.
-
-
-\subsubsection*{References to equations}
-
-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
-\begin{verbatim}
-\begin{equation}
-\label{additivity}
-\mathrm{H}_* \bigvee_{\alpha\in A} X_\alpha \iso
- \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
-\end{equation}
-If we now say
-\begin{verbatim}
-\begin{thm}
-Equation~\eqref{additivity} is true for all sorts of
-functors $\mathrm{H}$.
-\end{thm}
-\end{verbatim}
-then we'll get
-\begin{thm}
-Equation~\eqref{additivity} is true for all sorts of
-functors $\mathrm{H}$.
-\end{thm}
-Notice the parentheses around the equation number, and the fact that
-even though the theorem is set in slanted type, the equation
-number is set in an upright font. This is the advantage of using
-\verb"\eqref" instead of \verb"\ref"; the command \verb"\eqref"
-arranges it so that the number and surrounding parentheses are in an
-upright font no matter what the surrounding font, and supplies an
-italic correction if it's needed.
-
-
-%---------------------------------------------------------------------
-\subsection{References to page numbers}
-
-If you want to make reference to the \emph{page} that contains a
-label, rather than to the structure that is labeled, use the command
-\verb"\pageref{key}". For example, if you type
-\begin{verbatim}
-Look at page~\pageref{homotopy} to find
-Theorem~\ref{homotopy}.
-\end{verbatim}
-you'll get ``Look at page~\pageref{homotopy} to find
-Theorem~\ref{homotopy}.''
-
-
-
-
-
-
-
-%---------------------------------------------------------------------
-\subsection{Bibliographic references}
-\label{sec:bibreferences}
-
-
-Bibliography entries receive a \emph{key} as part of their basic
-structure. Each item in the bibliography is entered as
-\begin{verbatim}
-\bibitem{key} The actual bibliography item goes here.
-\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
-%
-\begin{verbatim}
-\bibitem{HA}
-D. G. Quillen, \emph{Homotopical Algebra,} Lecture Notes in
-Mathematics number 43, Springer-Verlag, Berlin, 1967.
-\end{verbatim}
-%
-If we say ``\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
-``\verb"see~\cite[Chapter I]{HA}"'', then it will be printed as
-``see~\cite[Chapter I]{HA}''.
-
-
-
-
-
-%---------------------------------------------------------------------
-%---------------------------------------------------------------------
-\section{Mathematics}
-
-
-\subsection{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
-\begin{center}
-\verb"Let $f\colon X \to Y$ be a continuous function."
-\end{center}
-you'll get
-\begin{center}
-Let $f\colon X \to Y$ be a continuous function.
-\end{center}
-
-The only novelty that \LaTeX{} introduces is that, instead of using
-a dollar sign to toggle math mode on and off, you can use
-`\verb"\("' to \emph{begin} math mode, and `\verb"\)"' to \emph{end}
-math mode. Thus, the example above is equivalent to typing
-\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.
-
-
-%---------------------------------------------------------------------
-\subsection{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"$$"), it will be interpreted in math mode and displayed.
-Thus, if you've previously given the command
-\begin{center}
- \verb"\newcommand{\iso}{\approx}"
-\end{center}
-(see Section~\ref{sec:definitions}), and you type
-\begin{verbatim}
-$$
-\pi_1(X \vee Y) \iso \pi_1X * \pi_1Y
-$$
-\end{verbatim}
-you'll get
-$$
-\pi_1(X \vee Y) \iso \pi_1X * \pi_1Y
-$$
-The new choices are that exactly the same thing will be obtained by
-either
-\begin{verbatim}
-\[
-\pi_1(X \vee Y) \iso \pi_1X * \pi_1Y
-\]
-\end{verbatim}
-or by
-\begin{verbatim}
-\begin{displaymath}
-\pi_1(X \vee Y) \iso \pi_1X * \pi_1Y
-\end{displaymath}
-\end{verbatim}
-or by
-\begin{verbatim}
-\begin{equation*}
-\pi_1(X \vee Y) \iso \pi_1X * \pi_1Y
-\end{equation*}
-\end{verbatim}
-
-
-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
-\begin{verbatim}
-\begin{equation}
-\pi_1(X \vee Y) \iso \pi_1X * \pi_1Y
-\end{equation}
-\end{verbatim}
-you'll get
-\begin{equation}
-\label{pi1eqn}
-\pi_1(X \vee Y) \iso \pi_1X * \pi_1Y
-\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:
-\begin{verbatim}
-\begin{equation}
-\label{pi1eqn}
-\pi_1(X \vee Y) \iso \pi_1X * \pi_1Y
-\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}.
-
-
-
-%---------------------------------------------------------------------
-\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}
-
-To produce the diagram
-$$
-\begin{CD}
-A @>>> B @= B\\
-@AAA @| @VVV\\
-X @<<< B @>>> Y
-\end{CD}
-$$
-you type
-\begin{verbatim}
-$$
-\begin{CD}
-A @>>> B @= B\\
-@AAA @| @VVV\\
-X @<<< B @>>> Y
-\end{CD}
-$$
-\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
-\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}
-\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}
-
-
-\subsubsection*{Leaving out part of the rectangle}
-
-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
-\begin{verbatim}
-\begin{displaymath}
-\begin{CD}
-X\\
-@VfVV\\
-Y @= Y\\
-@. @VVgV\\
- @. Z
-\end{CD}
-\end{displaymath}
-\end{verbatim}
-you'll get
-\begin{displaymath}
-\begin{CD}
-X\\
-@VfVV\\
-Y @= Y\\
-@. @VVgV\\
- @. Z
-\end{CD}
-\end{displaymath}
-
-
-
-
-
-%---------------------------------------------------------------------
-%---------------------------------------------------------------------
-\section{Macro definitions, a.k.a.\ \texttt{newcommand}}
-\label{sec:definitions}
-
-
-\LaTeX{} allows you to use the same \verb"\def" command that you use
-in plain \TeX, but it's considered bad style. Instead, \LaTeX{} has
-the \verb"\newcommand" and \verb"\renewcommand" commands, which do a
-little error checking for you. In plain \TeX, you might use the
-command
-\begin{center}
-\verb"\def\tensor{\otimes}"
-\end{center}
-but in \LaTeX, the preferred form is
-\begin{center}
-\verb"\newcommand{\tensor}{\otimes}"
-\end{center}
-The advantage of this is that \LaTeX{} will check to see if there
-already is a command with the name \verb"\tensor", and give you an
-error message if there is. If you know that there is a previous
-definition of \verb"\tensor" but you \emph{want} to override it,
-then you use the command
-\begin{center}
-\verb"\renewcommand{\tensor}{\otimes}"
-\end{center}
-
-
-If you want to use macros with replaceable parameters, the
-\verb"newcommand" command allows this. For the equivalent of the
-plain \TeX{} command
-\begin{center}
-\verb"\def\pushout#1#2#3{#1\cup_{#2}#3}"
-\end{center}
-you use the \LaTeX{} command
-\begin{center}
-\verb"\newcommand{\pushout}[3]{#1\cup_{#2}#3}"
-\end{center}
-i.e., the command name is enclosed in braces, and the number of
-parameters is enclosed in square brackets.
-
-
-
-%---------------------------------------------------------------------
-%---------------------------------------------------------------------
-\section{Lists: \texttt{itemize, enumerate, and description}}
-
-There are three list making environments: \texttt{itemize},
-\texttt{enumerate}, and \texttt{description}. The \texttt{itemize}
-environment just lists the items with a marker in front of each one.
-If you type
-%
-\begin{verbatim}
-\begin{itemize}
-\item
-This is the first item in the list, which runs on long enough to
-spill over onto a second line.
-\item
-This is the second item in the list, which is a bit shorter.
-\item
-This is the last item.
-\end{itemize}
-\end{verbatim}
-%
-then you'll get
-%
-\begin{itemize}
-\item
-This is the first item in the list, which runs on long enough to
-spill over onto a second line.
-\item
-This is the second item in the list, which is a bit shorter.
-\item
-This is the last item.
-\end{itemize}
-
-
-
-The \texttt{enumerate} environment looks the same, except that the items
-in the list are numbered. If you type
-\begin{verbatim}
-\begin{enumerate}
-\item
-This is the first item in the list, which runs on long enough to
-spill over onto a second line.
-\item
-This is the second item in the list, which is a bit shorter.
-\item
-This is the last item.
-\end{enumerate}
-\end{verbatim}
-then you'll get
-\begin{enumerate}
-\item
-This is the first item in the list, which runs on long enough to
-spill over onto a second line.
-\item
-This is the second item in the list, which is a bit shorter.
-\item
-This is the last item.
-\end{enumerate}
-
-The \texttt{description} environment requires an extra argument for each
-\verb"\item" command, which will be printed at the beginning of the
-item. If you type
-\begin{verbatim}
-\begin{description}
-\item[sedge]
-A green plant, found in both wetlands and uplands.
-Sedges are often confused with grasses and rushes.
-\item[grass]
-A green plant, found in both wetlands and uplands.
-Grasses are often confused with sedges and rushes.
-\item[rush]
-A green plant, found in both wetlands and uplands.
-Rushes are often confused with sedges and grasses
-\end{description}
-\end{verbatim}
-you'll get
-\begin{description}
-\item[sedge]
-A green plant, found in both wetlands and uplands. Sedges are often
-confused with grasses and rushes.
-\item[grass]
-A green plant, found in both wetlands and uplands. Grasses are often
-confused with sedges and rushes.
-\item[rush]
-A green plant, found in both wetlands and uplands. Rushes are often
-confused with sedges and grasses
-\end{description}
-
-
-These environments can be inserted within each other, and the
-\verb"enumerate" environment keeps track of what level it's at, and
-numbers its items accordingly. If you type
-\begin{verbatim}
-\begin{enumerate}
-\item I went to the dry cleaners.
-\item I went to the supermarket. I bought
-\begin{enumerate}
-\item Bread,
-\item cheese, and
-\item Tabasco sauce.
-\end{enumerate}
-\item I went to the bank.
-\end{enumerate}
-\end{verbatim}
-%
-you'll get
-%
-\begin{enumerate}
-\item I went to the dry cleaners.
-\item I went to the supermarket. I bought
-\begin{enumerate}
-\item Bread,
-\item cheese, and
-\item Tabasco sauce.
-\end{enumerate}
-\item I went to the bank.
-\end{enumerate}
-
-
-
-
-
-
-
-
-%---------------------------------------------------------------------
-%---------------------------------------------------------------------
-\section{The bibliography}
-\label{sec:bibliography}
-
-\subsection{\texttt{begin\{thebibliography\}} and
- \texttt{end\{thebibliography\}}}
-
-
-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}".
-
-After listing each item in the bibliography, you end the bibliography
-with the \verb"\end{thebibliography}" command.
-
-
-\subsection{Bibliography items}
-
-Each item is begun with a \verb"\bibitem" command. The format is
-\begin{center}
-\verb"\bibitem{key for cross-references}Item entry"
-\end{center}
-For example, the bibliography in these instructions contains 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 allows you to say
-\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 this, see Section~\ref{sec:bibreferences}.
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-%---------------------------------------------------------------------
-%---------------------------------------------------------------------
-\section{The template file}
-\label{sec:template}
-
-The following is the text of the file \verb"template.tex".
-
-\begin{verbatim}
-%%% template.tex
-%%% This is a template for making up an AMS-LaTeX file
-%%% Version of August 10, 2000
-%%%---------------------------------------------------------
-%%% 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}
-\usepackage{amssymb}
-
-
-
-
-
-%%% 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.
-
-\newcommand{\tensor}{\otimes}
-\newcommand{\homotopic}{\simeq}
-\newcommand{\homeq}{\cong}
-\newcommand{\iso}{\approx}
-
-\DeclareMathOperator{\ho}{Ho}
-\DeclareMathOperator*{\colim}{colim}
-
-
-
-\newcommand{\R}{\mathbb{R}}
-\newcommand{\C}{\mathbb{C}}
-\newcommand{\Z}{\mathbb{Z}}
-
-\newcommand{\M}{\mathcal{M}}
-\newcommand{\W}{\mathcal{W}}
-
-
-
-\newcommand{\itilde}{\tilde{\imath}}
-\newcommand{\jtilde}{\tilde{\jmath}}
-\newcommand{\ihat}{\hat{\imath}}
-\newcommand{\jhat}{\hat{\jmath}}
-
-
-
-%%%-------------------------------------------------------------------
-%%%-------------------------------------------------------------------
-%%% The Theorem environments:
-%%%
-%%%
-%%% The following commands set it up so that:
-%%%
-%%% All Theorems, Corollaries, Lemmas, Propositions, Definitions,
-%%% Remarks, Examples, Notations, and Terminologies will be numbered
-%%% in a single sequence, and the numbering will be within each
-%%% section. Displayed equations will be numbered in the same
-%%% sequence.
-%%%
-%%%
-%%% Theorems, Propositions, Lemmas, and Corollaries will have the most
-%%% formal typesetting.
-%%%
-%%% Definitions will have the next level of formality.
-%%%
-%%% Remarks, Examples, Notations, and Terminologies will be the least
-%%% formal.
-%%%
-%%% Theorem:
-%%% \begin{thm}
-%%%
-%%% \end{thm}
-%%%
-%%% Corollary:
-%%% \begin{cor}
-%%%
-%%% \end{cor}
-%%%
-%%% Lemma:
-%%% \begin{lem}
-%%%
-%%% \end{lem}
-%%%
-%%% Proposition:
-%%% \begin{prop}
-%%%
-%%% \end{prop}
-%%%
-%%% Definition:
-%%% \begin{defn}
-%%%
-%%% \end{defn}
-%%%
-%%% Remark:
-%%% \begin{rem}
-%%%
-%%% \end{rem}
-%%%
-%%% Example:
-%%% \begin{ex}
-%%%
-%%% \end{ex}
-%%%
-%%% Notation:
-%%% \begin{notation}
-%%%
-%%% \end{notation}
-%%%
-%%% Terminology:
-%%% \begin{terminology}
-%%%
-%%% \end{terminology}
-%%%
-%%% Theorem environments
-
-% The following causes equations to be numbered within sections
-\numberwithin{equation}{section}
-
-% We'll use the equation counter for all our theorem environments, so
-% that everything will be numbered in the same sequence.
-
-% Theorem environments
-
-\theoremstyle{plain} %% This is the default, anyway
-\newtheorem{thm}[equation]{Theorem}
-\newtheorem{cor}[equation]{Corollary}
-\newtheorem{lem}[equation]{Lemma}
-\newtheorem{prop}[equation]{Proposition}
-
-
-\theoremstyle{definition}
-\newtheorem{defn}[equation]{Definition}
-
-\theoremstyle{remark}
-\newtheorem{rem}[equation]{Remark}
-\newtheorem{ex}[equation]{Example}
-\newtheorem{notation}[equation]{Notation}
-\newtheorem{terminology}[equation]{Terminology}
-
-
-%%%-------------------------------------------------------------------
-%%%-------------------------------------------------------------------
-%%%-------------------------------------------------------------------
-%%%-------------------------------------------------------------------
-%%%-------------------------------------------------------------------
-%%%-------------------------------------------------------------------
-%%%-------------------------------------------------------------------
-\begin{document}
-
-%%% In the title, use a double backslash "\\" to show a linebreak:
-%%% Use one of the following two forms:
-%%% \title{Text of the title}
-%%% or
-%%% \title[Short form for the running head]{Text of the title}
-\title
-
-
-
-
-\author{}
-
-
-
-%%% In the address, show linebreaks with double backslashes:
-\address{}
-
-
-
-
-%%% Email address is optional.
-\email{}
-
-%%% To have the current date inserted, use \date{\today}:
-\date{}
-
-
-\maketitle
-
-%%% To include a table of contents, uncomment the next line:
-% \tableofcontents
-%%%-------------------------------------------------------------------
-%%%-------------------------------------------------------------------
-%%% Start the body of the paper here! E.G., maybe use:
-%%% \section{Introduction}
-%%% \label{sec:intro}
-
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-
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-%%%-------------------------------------------------------------------
-%%%-------------------------------------------------------------------
-%%% 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}
-
-
-
-
-%%% 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.
-
-
-
-
-
-
-
-
-\end{thebibliography}
-\end{document}
-\end{verbatim}
-
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-
-%---------------------------------------------------------------------
-%---------------------------------------------------------------------
-
-\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".
-
-\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}
-