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\begin{document}
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All commercial rights reserved. May be freely distributed
and used with the following exceptions:\\
1. No commercial use without explicit permission.\\
2. It may not be distributed without this notice.
\def\\{{\tt \char 92}}
\section*{The diagram macros}
The file diagram includes a number of macros for drawing commutative
diagrams in special shapes as well as more flexible macros to paste
together more complicated diagrams.
In general, there are two kinds of macros. The first kind has the
following form
{\tt \\shape\-name[shape\_\-parameters]} and is intended to be used in
the following way:
{\verbatim
$$
\shapename[shape_parameters]
$$
|egroup
This will create a shape with the given parameters. It operates by
building its own picture environment.
The second kind of procedure must be used inside a picture environment
and allows the user to put a diagram of a given size and shape at a
given place. The usual way in which this is used is:
{\verbatim
$$
\bfig
\putshape1(x1,y1)[shape_parameters1]
|dots
\putshape(xn,yn)[shape_parametersn]
\efig
$$
|egroup
Here is an example of the first kind of macro. I will have to explain a
few details before giving examples of the second. The code
{\verbatim
$$
\square[A`B`C`D;f`g`h`k]
$$
|egroup
produces the diagram
$$
\square[A`B`C`D;f`g`h`k]
$$
and the code
{\verbatim
$$
\Atriangle[A`B`C;f`g`h]
$$
|egroup
produces the diagram
$$
\Atriangle[A`B`C;f`g`h]
$$
The reason for the shape name will be explained later.
If an arrow label extends especially high or low, the space
automatically expands to match. For example,
{\verbatim
$$
\square[A`B`C`D;\sum_{i=1}^{\infty}`g`h`\Psi^A_k]
$$
|egroup
will get the diagram
$$
\square[A`B`C`D;\sum_{i=1}^{\infty}`g`h`\Psi^A_k]
$$
and so on.
Before going on, I will explain about the parameters. Any shape will
have one or two parameters for height and/or length as well as one
parameter for each arrow in the diagram. These are in addition to the
parameters used in the actual procedures. For example, the procedure
{\verbatim
$$
\square<-1`0`2`-3;1000`700>[A`B`C`D;f`g`h`k]
$$
|egroup
gives the square:
$$
\square<-1`0`2`-3;1000`700>[A`B`C`D;f`g`h`k]
$$
The meaning is as follows. The first four parameters refer to four
arrows in linguistic order (top, left, right, bottom). A negative
number gives a backward arrow, while a zero causes it to be omitted. A
parameter with an absolute value of 1 is ordinary, while 2 gives an
arrow with a tail (monomorphism) and 3 a double headed arrow
(epimorphism). The last two parameters determine the width and height,
respectively, in units of 0.01 em (an em is the width of an M, the
widest letter in a font). It should be noted that it is not
possible to give only some of the optional parameters; either all
must be given or none is.
Here are the shapes. We have already seen the shape {\tt \\square}.
There are eight different triangles, all isoceles right triangles in
different orientations. The names all have the form {\tt
\\xtriangle}, where x is the letter that most closely resembles the
actual shape of the triangle.
Here are the names, followed by a sample of each one:
\begin{list}{}{}
\item{\tt \\btriangle}\samepage
$$
\btriangle[A`B`C;f`g`h]
$$
\pagebreak[0]
\item{\tt \\dtriangle}\samepage
$$
\dtriangle[A`B`C;f`g`h]
$$
\pagebreak[0]
\item{\tt \\ptriangle}\samepage
$$
\ptriangle[A`B`C;f`g`h]
$$
\pagebreak[0]
\item{\tt \\qtriangle}\samepage
$$
\qtriangle[A`B`C;f`g`h]
$$
\pagebreak[0]
\item{\tt \\Atriangle}\samepage
$$
\Atriangle[A`B`C;f`g`h]
$$
\pagebreak[0]
\item{\tt \\Vtriangle}\samepage
$$
\Vtriangle[A`B`C;f`g`h]
$$
\pagebreak[0]
\item{\tt \\Ctriangle}\samepage
$$
\Ctriangle[A`B`C;f`g`h]
$$
\pagebreak[0]
\item{\tt \\Dtriangle}\samepage
$$
\Dtriangle[A`B`C;f`g`h]
$$
\end{list}
In addition, there are two special diagrams that come up often enough to
be worth having a special macros for.
\begin{list}{}{}
\item{\tt \\Atrianglepair}\samepage
$$
\Atrianglepair[A`B`C`D;f`g`h`k`l]
$$
\item{\tt \\Vtrianglepair}\samepage
$$
\Vtrianglepair[A`B`C`D;f`g`h`k`l]
$$
\end{list}
Finally, there is one special shape that is probably not used by very
many mathematicians. Still I have it and the user might as well share
it.
{\verbatim
\recurse[A`B`R`C;s`f_0`f`t_0`t]
|egroup
$$
\recurse[A`B`R`C;s`f_0`f`t_0`t]
$$
Notice what happens if the first parameter is empty:
{\verbatim
\recurse[`B`R`C;s`f_0`f`t_0`t]
|egroup
$$
\recurse[`B`R`C;s`f_0`f`t_0`t]
$$
This takes care of the simple procedures. The remaining procedures are
there as building blocks for more complicated diagrams.
These take the form
\vskip0pt\noindent
{\tt \\putshape(xpos,ypos)[shape\_parameters]}
\vskip0pt\noindent
where
shape is one of the 11 shapes (excluding {\tt \\recurse}) described
above
and the parms are as described there. The parameters {\tt xpos} and
{\tt ypos} are the offsets from the lower left corner of the picture
measured in umits of .01 em in the usual \LaTeX\ fashion.
There is one
more shape which is simply an arrow with an attached label. This is
called {\tt \\putmorphism} and it is used with the syntax
\vskip0pt\noindent
{\tt
\\putmorphism(xpos,ypos)(run,rise)[node1`node2`label]\{dist\}\{type\}\{loc\}}
\vskip0pt\noindent
Here the parameters {\tt xpos} and
{\tt ypos} are as above. The slope is {\tt rise/run} where {\tt rise}
and {\tt run} are two numbers that give the slope in accordance with the
\LaTeX\ rules. That is rise and run must have no common divisor and
must not exceed 4 in absolute value. In addition, these procedures are
defined so that rise must not be positive and if it is zero, then run
must be positive. In other words, all arrows must go in the linguistic
direction, downwards or to the right. Arrows can be made to go in the
reverse direction as explained below.
The next three parameters are the node that appears first (in linguistic
order), the one that appears second and the arrow label. The next
paramater is the distance in the sense of \LaTeX\ between the centers of
the nodes. This means it is the horizontal component of the distance,
unless that is negative, in which case it is the vertical distance. The
second parameter is the code for the arrowtype, an integer between -3
and 3 with the same meaning as explained above. Using negative values
of this parameter allows one to draw an arrow that goes upwards or left.
Note, however, that node1 is always the node that is above or to the
left of node2. The last parameter should have the value a,b,l,r or m.
The values a and b are used only for horizontal arrows and direct the
arrow label to be placed above or below the arrow. The values l and r
are used for all other arrows and direct the label to be left or right
of the arrow. Finally, a vertical arrow only can be given the parameter
m, in which case the arrow will be gapped and the label placed in the
middle of the gap.
In general, the simple macros are designed to be used as indicated
either in the display
environment or in one of the others, usually
{\verbatim
\begin{equation}\begin{array}{c}
|dots
\end{array}\end{equation}|egroup
The \\putshape macros must be used inside a picture environment that is
normally placed inside a display or similar environment. The
reference point for the positioning parameters is determined as the
lower left corner of the smallest rectangle with sides parallel to the
coordinate axes that includes the center points of all nodes on its
border. This rectangle will be degenerate in the case of a horizontal
or vertical morphism. The reference point may either be on or outside
the actual figure. For example, in the case of the btriangle, it is the
center of the lower left node, while for a qtriangle it is outside the
triangle itself being the fourth corner of the enclosing square.
The macros are made so that they fit together well. That is why all
distances are from node centers to node centers. If a vertex is part of
two shapes, it is probably best not to repeat it, since it is
conceivable that round-off errors will cause its two appearances to be
slightly offset. It can either be omitted or, since the width is used
to determine the amount to shorten horizontal arrows, replaced by {\tt
\\phantom} versions. Both methods are illustrated in the example given
at the end.
A somewhat baroque (but taken from an actual text) example illustrates
most of these points.
{\verbatim
$$
\bfig
\putmorphism(0,2100)(0,-1)[``T\eta'T]{1400}1l
\putmorphism(0,2100)(1,0)[TT`T`\mu]{700}1a
\putmorphism(0,2100)(1,-1)[`TTT'`TT\eta']{700}1l
\putmorphism(700,2100)(1,-1)[`TT'`T\eta]{700}1r
\put(700,1750){\makebox(0,0){1}}
\putmorphism(700,1420)(1,0)[\phantom{TTT'}`\phantom{TT'}`\mu
T']{700}1a
\putmorphism(700,1380)(1,0)[\phantom{TTT'}`%
\phantom{TT'}`T\sigma]{700}1b
\putsquare<0`1`1`1;700`700>(700,700)[TTT'`TT'`TT'TT'`TT'T';`T\eta'TT'``]
\putmorphism(700,700)(1,0)[\phantom{TT'TT'}`%
\phantom{TT'T'}`TT'\sigma]{700}1a
\put(300,1400){\makebox(0,0){2}}
\put(950,1050){\makebox(0,0){3}}
\putbtriangle<0`1`0;700>(1400,700)[``TT';T\eta'T'`id`]
\putmorphism(1400,700)(1,0)[\phantom{TT'T'}`%
\phantom{TT'}`T\mu']{700}1a
\put(1600,1050){\makebox(0,0){6}}
\putsquare<1`1`0`1;700`700>(0,0)[TT'T`\phantom{TT'TT'}`T'T`T'TT';%
TT'T\eta'`\sigma T``T'T\eta']
\putmorphism(700,0)(1,0)[\phantom{T'TT'}`%
\phantom{T'T'}`T'\sigma]{700}1b
\putsquare<0`0`1`1;700`700>(1400,0)[``T'T'`T';``\sigma`\mu']
\putmorphism(700,700)(0,-1)[``\sigma TT']{700}1m
\putmorphism(1400,700)(0,-1)[``\sigma T']{700}1m
\put(300,350){\makebox(0,0){4}}
\put(1050,350){\makebox(0,0){5}}
\put(1750,350){\makebox(0,0){7}}
\efig
$$
|egroup
which produces
$$
\bfig
\putmorphism(0,2100)(0,-1)[``T\eta'T]{1400}1l
\putmorphism(0,2100)(1,0)[TT`T`\mu]{700}1a
\putmorphism(0,2100)(1,-1)[`TTT'`TT\eta']{700}1l
\putmorphism(700,2100)(1,-1)[`TT'`T\eta]{700}1r
\put(700,1750){\makebox(0,0){1}}
\putmorphism(700,1420)(1,0)[\phantom{TTT'}`\phantom{TT'}`\mu
T']{700}1a
\putmorphism(700,1380)(1,0)[\phantom{TTT'}`%
\phantom{TT'}`T\sigma]{700}1b
\putsquare<0`1`1`1;700`700>(700,700)[TTT'`TT'`TT'TT'`TT'T';`T\eta'TT'``]
\putmorphism(700,700)(1,0)[\phantom{TT'TT'}`%
\phantom{TT'T'}`TT'\sigma]{700}1a
\put(300,1400){\makebox(0,0){2}}
\put(950,1050){\makebox(0,0){3}}
\putbtriangle<0`1`0;700>(1400,700)[``TT';T\eta'T'`id`]
\putmorphism(1400,700)(1,0)[\phantom{TT'T'}`%
\phantom{TT'}`T\mu']{700}1a
\put(1600,1050){\makebox(0,0){6}}
\putsquare<1`1`0`1;700`700>(0,0)[TT'T`\phantom{TT'TT'}`T'T`T'TT';%
TT'T\eta'`\sigma T``T'T\eta']
\putmorphism(700,0)(1,0)[\phantom{T'TT'}`%
\phantom{T'T'}`T'\sigma]{700}1b
\putsquare<0`0`1`1;700`700>(1400,0)[``T'T'`T';``\sigma`\mu']
\putmorphism(700,700)(0,-1)[``\sigma TT']{700}1m
\putmorphism(1400,700)(0,-1)[``\sigma T']{700}1m
\put(300,350){\makebox(0,0){4}}
\put(1050,350){\makebox(0,0){5}}
\put(1750,350){\makebox(0,0){7}}
\efig
$$
Here is a page of samples of the results from the various
{\tt\\putshapes}. The code
{\verbatim
$$
\bfig
\putsquare(0,0)[A`B`C`D;f`g`h`k]
\putbtriangle(0,1500)[A`B`C;f`g`h]
\putdtriangle(2200,1500)[A`B`C;f`g`h]
\putptriangle(0,3000)[A`B`C;f`g`h]
\putqtriangle(1000,500)[A`B`C;f`g`h]
\putCtriangle(1500,750)[A`B`C;f`g`h]
\putDtriangle(700,1000)[A`B`C;f`g`h]
\putAtriangle(2000,2200)[A`B`C;f`g`h]
\putAtrianglepair(0,2500)[A`B`C`D;f`g`h`k`l]
\putVtriangle(2000,3000)[A`B`C;f`g`h]
\putVtrianglepair(1000,2500)[A`B`C`D;f`g`h`k`l]
\efig
$$
|egroup
produces the diagram:
$$
\bfig
\putsquare(0,0)[A`B`C`D;f`g`h`k]
\putbtriangle(0,1500)[A`B`C;f`g`h]
\putdtriangle(2200,1500)[A`B`C;f`g`h]
\putptriangle(0,3000)[A`B`C;f`g`h]
\putqtriangle(1000,500)[A`B`C;f`g`h]
\putCtriangle(1500,750)[A`B`C;f`g`h]
\putDtriangle(700,1000)[A`B`C;f`g`h]
\putAtriangle(2000,2200)[A`B`C;f`g`h]
\putAtrianglepair(0,2500)[A`B`C`D;f`g`h`k`l]
\putVtriangle(2000,3000)[A`B`C;f`g`h]
\putVtrianglepair(1000,2500)[A`B`C`D;f`g`h`k`l]
\efig
$$
Added: For horizontal and vertical lines only, arrow parameters of
$\pm4$, $\pm5$ and $\pm6$ will give dashed lines.
Here is a double cube:
{\verbatim
$$
\bfig
\putsquare(0,750)[A`B`C`D;```]
\putsquare(0,250)[\phantom{C}`\phantom{D}`E`F;```]
\putsquare(250,500)[A'`B'`C'`D';```]
\putsquare(250,0)[\phantom{C'}`\phantom{D'}`E'`F';```]
\putmorphism(0,1250)(1,-1)[```]{250}1a
\putmorphism(500,1250)(1,-1)[```]{250}1a
\putmorphism(0,750)(1,-1)[```]{250}1a
\putmorphism(500,750)(1,-1)[```]{250}1a
\putmorphism(0,250)(1,-1)[```]{250}1a
\putmorphism(500,250)(1,-1)[```]{250}1a
\efig
$$
|egroup
$$
\bfig
\putsquare(0,750)[A`B`C`D;```]
\putsquare(0,250)[\phantom{C}`\phantom{D}`E`F;```]
\putsquare(250,500)[A'`B'`C'`D';```]
\putsquare(250,0)[\phantom{C'}`\phantom{D'}`E'`F';```]
\putmorphism(0,1250)(1,-1)[```]{250}1a
\putmorphism(500,1250)(1,-1)[```]{250}1a
\putmorphism(0,750)(1,-1)[```]{250}1a
\putmorphism(500,750)(1,-1)[```]{250}1a
\putmorphism(0,250)(1,-1)[```]{250}1a
\putmorphism(500,250)(1,-1)[```]{250}1a
\efig
$$
\end{document}