\chapter{Systems}\label{systems}
\fixchapterheading
\section{Diagrams made with diagram.sty}
\captionstyleparagraph

%% The full documentation is in the file: diagram.sty

An example diagram appears below in Figure \ref{diagram.fig1}. This is
typical of what can made with the diagram package.

\begin{figure}[b]               % Place at bottom of this page
$$
\begin{diagram}
\node{U} \arrow{e,t}{i_1} \arrow{s}
\node{X} \arrow{s,r}{\pi} \\
\node{Y-\partial Q} \arrow{e,t}{j_1} \node{Y}
\end{diagram}
$$
\caption{Diagram example\label{diagram.fig1}}
\end{figure}

\section{Sample diagrams from diagram.tex}

Example diagrams reproduced here were taken from various sources.
Compare the three diagrams of increasing sizes in
Figure \ref{file.fig1}, Figure \ref{file.fig2}, Figure \ref{file.fig3}
with the three diagrams in Figure \ref{file.fig4}, Figure
\ref{file.fig5},
Figure \ref{file.fig6}.


\begin{figure}[b]
$$
\setlength{\dgARROWLENGTH}{3.0em}
\begin{diagram}[\strut A]
\node{A} \arrow{e} \arrow{s} \arrow{se} \node{B} \arrow{s} \\
\node{C} \arrow{e}                      \node{D}
\end{diagram}
$$
\caption{Base diagram, Arrowlength = 3.0em
\label{file.fig1}}
\end{figure}

\begin{figure}[p]
$$
\setlength{\dgARROWLENGTH}{6.0em}
\begin{diagram}[\strut A]
\node{A} \arrow{e} \arrow{s} \arrow{se} \node{B} \arrow{s} \\
\node{C} \arrow{e}                      \node{D}
\end{diagram}
$$
\caption{Same as Figure \protect\ref{file.fig1}, but Arrowlength = 6.0em
\label{file.fig2}}
\end{figure}

\begin{figure}[p]
$$
\setlength{\dgARROWLENGTH}{12.0em}
\begin{diagram}[\strut A]
\node{A} \arrow{e} \arrow{s} \arrow{se} \node{B} \arrow{s} \\
\node{C} \arrow{e}                      \node{D}
\end{diagram}
$$
\caption{Same as Figure \protect\ref{file.fig1}, but Arrowlength =
12.0em \label{file.fig3}}
\end{figure}

\begin{figure}[p]
$$
\setlength{\dgARROWLENGTH}{3.0em}
\begin{diagram}[\strut A]
\node{A} \arrow{e} \arrow{s} \arrow{se} \node{B} \arrow{s} \\
\node{C} \arrow{e}                      \node{D}
\end{diagram}
$$
\caption{Base figure, same as Figure \protect\ref{file.fig1}.
\label{file.fig4}}
\end{figure}
\clearpage % make page of floats

\begin{figure}[t]
$$
\setlength{\dgARROWLENGTH}{3.0em}
\begin{diagram}[\strut\hspace{6.0em}]
\node{A} \arrow{e} \arrow{s} \arrow{se} \node{B} \arrow{s} \\
\node{C} \arrow{e}                      \node{D}
\end{diagram}
$$
\caption{Same as Figure \protect\ref{file.fig4}, but Bignode = strut
hspace 6.0em. \label{file.fig5}}
\end{figure}

\begin{figure}[t]
$$
\setlength{\dgARROWLENGTH}{3.0em}
\begin{diagram}[\strut\hspace{12.0em}]
\node{A} \arrow{e} \arrow{s} \arrow{se} \node{B} \arrow{s} \\
\node{C} \arrow{e}                      \node{D}
\end{diagram}
$$
\caption{Same as Figure \protect\ref{file.fig1}, but Bignode = strut
hspace 12.0em \label{file.fig6}}
\end{figure}

Below we show diagrams from the manual with a few modifications. The
first in Figure \ref{file.fig7} is essentially as it appears in the
manual, whereas the second, Figure \ref{file.fig8} has been
rescaled to a larger size.
\begin{figure}[p]
$$
\begin{diagram}[B^*]
\node{A} \arrow{e,t}{a} \arrow{s,l}{c} \arrow{ese,b,1}{u}
   \node{B^*} \arrow{e,t}{b^*}
      \node{C} \arrow{s,r}{d} \arrow{wsw,b,1}{v} \\
\node{D} \arrow[2]{e,b}{e}
   \node[2]{E}
\end{diagram}
$$
\caption{First diagram from manual
\label{file.fig7}}
\end{figure}

\begin{figure}[p]
$$
\setlength{\dgARROWLENGTH}{.75em}
\begin{diagram}[B^*]
\node{A} \arrow[2]{e,t}{a} \arrow[2]{s,l}{c} \arrow[2]{ese,b,1}{u}
   \node[2]{B^*} \arrow[2]{e,t}{b^*}
      \node[2]{C} \arrow[2]{s,r}{d} \arrow{wsw,b,-}{v}
\\
        \node[3]{} \arrow{wsw}
\\
\node{D} \arrow[4]{e,b}{e}
   \node[4]{E}
\end{diagram}
$$
\caption{First diagram from manual, rescaled.\label{file.fig8}}
\end{figure}

Below are several diagrams created by Bill Richter. The first, Figure
\ref{file.fig9} is modified slightly to produce Figure
\ref{file.fig10}. Both use fractur fonts. The last one, Figure
\ref{file.fig11}, is a complicated example illustrating the limits of
what can be done with diagrams.

The diagram below in Figure \ref{file.fig12}, the last of our series of
illustrations, is by Anders Thorup (\verb"thorup@math.ku.dk"),
originally done with a package developed by himself and Steven Kleiman
(\verb"kleiman@math.mit.edu"):


%%%%
%%%% If you're missing fractur fonts, then comment out these next 5
%%%% lines and type instead;
%%%% \let\frak=\bf
%%%%
\font\tenfrak=eufm10 scaled \magstep1
\font\sevenfrak=eufm7 scaled \magstep1
\font\fivefrak=eufm5 scaled \magstep1
\newfam\frakfam \def\frak{\fam\frakfam\tenfrak} \textfont\frakfam=\tenfrak
\scriptfont\frakfam=\sevenfrak  \scriptscriptfont\frakfam=\fivefrak
%%%%
%%%%
\def\a{ \alpha }
\def\d{ \delta }
\def\s{ \sigma }
\def\l{ \lambda }
\def\p{ \partial }
\def\st{{\tilde\s}}
\def\O{ \Omega }
\def\S{\Sigma}
\def\Z{{   \Bbb Z }}
\def\@{ \otimes }
\def\^{ \wedge }
\def\({ \left( }
\def\){ \right) }
\def\K#1{{ K\(\Z/2,#1\) }}
\def\KZ#1{{K\(\Z/4,#1\) }}
\def\id{ \mathop{id}\nolimits }
\def\h{ {\frak h} }
\def\e{ {\frak e} }
\def\G{ G }
\def\pinch{{ \mathop{{\rm pinch}} }}
\def\tuber{{ \bar\tau }}
\begin{figure}[p]
$$
\setlength{\dgARROWLENGTH}{1.5em}
\begin{diagram}[ \KZ{8n-1}  ]
\node[4]{ \K{8n+1} } \\
\node[2]{ \KZ{8n-1}  } \arrow{e} \arrow{ene,t}{Sq^2}
   \node{E} \arrow{ne,b}{\Theta} \arrow{s,l}{\pi} \\
\node{ \S\O X \^ \O X  } \arrow{e,t}{H_\mu} \arrow{ne,t}{\s(\a\@\a)}
   \node{ \Sigma \O X } \arrow{e,t}{\sigma} \arrow{ne,t}{\st}
       \node{ X }  \arrow{e,t}{\a^2}
           \node{ \KZ{8n}. }
\end{diagram}
$$
\caption{Bill Richter, first diagram\label{file.fig9}}
\end{figure}

\begin{figure}[p]
$$
\setlength{\dgARROWLENGTH}{-2.75em}
\begin{diagram}[ \O^2 \( \S A \^ \S A \) ]
\node[3]{\O\S A} \arrow[2]{e,t}{\l_2}
  \node[2]{\O^2 \( \S A \^ \S A \)}
\\
\node[4]{\#}
\\
% Note: the next two lines are like
% \node{\O B}  \arrow[2]{e,t,1}{\d}     \arrow[2]{ne,t}{\O\(\p\)}
% but put a gap in first arrow to make room for crossing arrow
\node{\O B}  \arrow{e,t,-}{\d}  \arrow[2]{ne,t}{\O\(\p\)}
  \node{} \arrow{e}
    \node{F} \arrow[2]{e,t}{\h}  \arrow[2]{s,r}{\pi} \arrow[2]{n,r}{J}
      \node[2]{\O^2 \( B \^ \S A \)} \arrow[2]{n,r}{\O^2\(\p\^\id\)}
\\
\\
\node{A} \arrow[2]{ne,t}{\e} \arrow[2]{e,t}{f}  \arrow[2]{nne,t,1}{E}
  \node[2]{X}  \arrow[2]{e,t}{h}
    \node[2]{B.}
\end{diagram}
$$
\caption{Bill Richter, second diagram\label{file.fig10}}
\end{figure}
\clearpage % Expunge all figures


\begin{figure}[p]
$$
\setlength{\dgARROWLENGTH}{-3.9em}
\begin{diagram}[ J\(S^4\^S^4\) ]
\node[9]{\O S^5}
\\
\\
\\
\node[8]{\beth}
\\
\node{\O\( M^5_{2\i}\)} \arrow[4]{e,t}{\O\(\pinch\)}
        \node[4]{\O S^5} \arrow[2]{e,t,-}{\d}
                                \arrow[4]{ne,t}{\O\(2\i\)}
                \node[2]{} \arrow[2]{e}
                        \node[2]{\G}    \arrow[4]{e,t}{\h_2}
                                        \arrow[2]{s,r,-}{\pi} \arrow[4]{n,r}{J}
                                \node[4]{J\(S^4\^S^4\)}
\\
\\
\node[3]{J_2\( M^4_{2\i}\)} \arrow[3]{e,t,3,-}{\d_2}
                                \arrow[2]{ne,t}{\i}
        \node[3]{}      \arrow{e}
                \node{\G_2} \arrow[4]{e,t,3}{\h_2}
                                \arrow[2]{ne,t}{\i}
                        \node[2]{}      \arrow[2]{s}
                                \node[2]{S^8} \arrow[2]{ne,b}{E}
\\
\node[4]{\aleph}
\\
\node{M^{12}_{2\i}} \arrow[4]{e,t}{\tuber} \arrow[2]{ne,t}{\tau}
        \node[4]{S^4} \arrow[4]{e,t}{\i}
                                \arrow[2]{ne,t}{\e}
                                        \arrow[4]{nne,t,3}{E}
                \node[4]{M^5_{2\i}} \arrow[4]{e,t}{\pinch}
                        \node[4]{S^5}
\end{diagram}
$$
\caption{Bill Richter, third diagram\label{file.fig11}}
\end{figure}

\begin{figure}[p]
$$
\setlength{\dgARROWLENGTH}{-6em}
\begin{diagram}[H^k(B_G\times N;Q)=H^k_G(N;Q)]
\node{H^k(B_G\times N;Q)=H^k_G(N;Q)}
      \arrow[2]{e,t}{f^*_j} \arrow[2]{s,l}{p^*} \arrow{se,t}{\tilde f^*}
   \node[2]{H^k_G(F_j;Q)}
      \arrow[2]{s,r}{q^*_j} \\
\node[2]{H^k_G(M;Q)}
      \arrow{ne,t}{i^*_j} \arrow[2]{s,l,1}{i^*} \\
\node{H^k(N;Q)}
      \arrow{e,t,-}{\tilde f^*_j=f^*_j} \arrow{se,b}{\tilde f^*=f^*}
   \node{}
      \arrow{e}
   \node{H^k(F_j;Q)} \\
\node[2]{H^k(M;Q)}
      \arrow{ne,b}{i^*_j}
\end{diagram}
$$
\caption{Anders Thorup diagram\label{file.fig12}}
\end{figure}
\clearpage
