path: root/Master/texmf-dist/doc/eplain/base/arrow.texi

@c arrow.texi - documentation for Eplain's commutative diagrams.
@c Copyright (C) 1991, 1992 Steven Smith.
@c This is part of the Eplain manual.
@c
@c This file is free software; you can redistribute it and/or modify
@c it under the terms of the GNU General Public License as published by
@c the Free Software Foundation; either version 2, or (at your option)
@c any later version.
@c
@c This file is distributed in the hope that it will be useful,
@c but WITHOUT ANY WARRANTY; without even the implied warranty of
@c MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
@c GNU General Public License for more details.
@c
@c You should have received a copy of the GNU General Public License
@c along with this file; if not, write to the Free Software
@c Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
@c 02110-1301, USA.

@comment \input texinfo
@comment setfilename arrow.info
@comment settitle Arrow Theoretic Diagrams

@c Input arrow macros without altering texinfo's \catcodes.
@c Use the arrow macros in an `@iftex @tex ... @end iftex' environment.

@iftex
@catcode`@$=3 @catcode`@%=14 @catcode`@&=4 @catcode`@#=6
@catcode`@^=7 @catcode`@_=8
@catcode`@"=@other @catcode`@<=@other @catcode`@>=@other
@catcode`@\=0
\catcode`\@=\other
\input arrow
\catcode`\@=0
@catcode`@\=@active
@catcode`@$=@other @catcode`@%=@other @catcode`@&=@other @catcode`@#=@other
@catcode`@^=@active @catcode`@_=@active
@catcode`@"=@active @catcode`@<=@active @catcode`@>=@active
@end iftex


@node Arrow theoretic diagrams
@chapter Arrow theoretic diagrams

This chapter describes definitions for producing commutative diagrams.

Steven Smith wrote this documentation (and the macros).

@menu
* Slanted lines and vectors::   
* Commutative diagrams::        
@end menu

@node Slanted lines and vectors
@section Slanted lines and vectors

The macros @code{\drawline}
@findex drawline
and @code{\drawvector}
@findex drawvector
provide the capability found in @LaTeX{}'s
@cindex @LaTeX{}
@cindex picture mode
picture mode to draw slanted lines and vectors of certain directions.
Both of these macros take three arguments: two integer arguments to
specify the direction of the line or vector, and one argument to specify
its length.  For example, @samp{\drawvector(-4,1)@{60pt@}} produces the
vector
@iftex
@tex
$$\vbox{\hbox{\drawvector(-4,1){60pt}}\smallskip
  \hbox{$\mathop{\hbox to60pt{\leftarrowfill\hskip-5pt\rightarrowfill}}
  \limits_{{\fam0 60\,pt}}$}}$$
@end tex
@end iftex
@ifinfo

@center (A vector in the 2d quadrant of length 60 pt appears here.)

@end ifinfo
@cindex lines
@cindex vectors
which lies in the 2d quadrant, has a slope of minus 1/4, and a width of
60 pt.

Note that if an @code{\hbox} is placed around @code{\drawline} or
@code{\drawvector}, then the width of the @code{\hbox} will be the
positive dimension specified in the third argument, except when a
vertical line or vector is specified, e.g.,
@code{\drawline(0,1)@{1in@}}, which has zero width.  If the specified
direction lies in the 1st or 2d quadrant (e.g., @code{(1,1)} or
@code{(-2,3)}), then the @code{\hbox} will have positive height and zero
depth.  Conversely, if the specified direction lies in the 3d or 4th
quadrant (e.g., @code{(-1,-1)} or @code{(2,-3)}), then the @code{\hbox}
will have positive depth and zero height.

There are a finite number of directions that can be specified.  For
@code{\drawline}, the absolute value of each integer defining the
direction must be less than or equal to six, i.e., @code{(7,-1)} is
incorrect, but @code{(6,-1)} is acceptable. For @code{\drawvector}, the
absolute value of each integer must be less than or equal to four.
Furthermore, the two integers cannot have common divisors; therefore, if
a line with slope 2 is desired, say @code{(2,1)} instead of
@code{(4,2)}.  Also, specify @code{(1,0)} instead of, say, @code{(3,0)}
for horizontal lines and likewise for vertical lines.

Finally, these macros depend upon the @LaTeX{} font @code{line10}.  If
your site doesn't have this font, ask your system administrator to get
it.  Future enhancements will include macros to draw dotted lines and
dotted vectors of various directions.


@node Commutative diagrams
@section Commutative diagrams

@cindex commutative diagrams

The primitive commands @code{\drawline} and @code{\drawvector} can be
used to typeset arrow theoretic diagrams.  This section describes (1)
macros to facilitate typesetting arrows and morphisms, and (2) macros to
facilitate the construction of commutative diagrams.  All macros
described in this section must be used in math mode.

@menu
* Arrows and morphisms::        
* Construction of commutative diagrams::  
* Commutative diagram parameters::  
@end menu

@node Arrows and morphisms
@subsection Arrows and morphisms
@cindex arrows
@cindex morphisms

The macros @code{\mapright} and @code{\mapleft} produce right and left
@findex mapright
@findex mapleft
pointing arrows, respectively.  Use superscript (@code{^}) to place a
morphism above the arrow, e.g., @samp{\mapright^\alpha}; use subscript
(@code{_}) to place a morphism below the arrow, e.g.,
@samp{\mapright_@{\tilde l@}}.  Superscripts and subscripts may be used
simulataneously, e.g., @samp{\mapright^\pi_@{\rm epimor.@}}.

Similarly, the macros @code{\mapup} and @code{\mapdown} produce up and
@findex mapup
@findex mapdown
down pointing arrows, respectively.  Use @code{\rt}
@findex rt
to place a morphism to the right of the arrow, e.g., @samp{\mapup\rt@{\rm
id@}}; use @code{\lft}
@findex lft
to place a morphism to the left of the arrow, e.g., @samp{\mapup\lft\omega}.
@code{\lft} and @code{\rt} may be used simultaneously, e.g.,
@samp{\mapdown\lft\pi\rt@{\rm monomor.@}}.

Slanted arrows are produced by the macro @code{\arrow}, which takes
@findex arrow
a direction argument (e.g., @samp{\arrow(3,-4)}).  Use @code{\rt} and
@code{\lft} to place morphisms to the right and left, respectively, of
the arrow.  A slanted line (no arrowhead) is produced with the macro
@code{\sline},
@findex sline
whose syntax is identical to that of @code{\arrow}.

The length of these macros is predefined by the default @TeX{}
dimensions @code{\harrowlength},
@findex harrowlength
for horizontal arrows (or lines),
@code{\varrowlength},
@findex varrowlength
for vertical arrows (or lines), and
@code{\sarrowlength},
@findex sarrowlength
for slanted arrows (or lines).  To change any of these dimensions, say,
e.g., @samp{\harrowlength=40pt}.  As with all other @TeX{} dimensions,
the change may be as global or as local as you like.  Furthermore, the
placement of morphisms on the arrows is controlled by the dimensions
@code{\hmorphposn},
@findex hmorphposn
@code{\vmorphposn},
@findex vmorphposn
and @code{\morphdist}.
@findex morphdist
The first two dimensions control the horizontal and vertical position of
the morphism from its default position; the latter dimension controls
the distance of the morphism from the arrow.  If you have more than one
morphism per arrow (i.e., a @code{^}/@code{_} or @code{\lft}/@code{\rt}
construction), use the parameters
@code{\hmorphposnup},
@findex hmorphposnup
@code{\hmorphposndn},
@findex hmorphposndn
@code{\vmorphposnup},
@findex vmorphposnup
@code{\vmorphposndn},
@findex vmorphposndn
@code{\hmorphposnrt},
@findex hmorphposnrt
@code{\hmorphposnlft},
@findex hmorphposnlft
@code{\vmorphposnrt},
@findex vmorphposnrt
and @code{\vmorphposnlft}.
@findex vmorphposnlft
The default values of all these dimensions are provided in the section
on parameters that follows below.

There is a family of macros to produce horizontal lines, arrows, and
adjoint arrows.  The following macros produce horizontal maps and have
the same syntax as @code{\mapright}:

@table @code

@item \mapright
@findex mapright
@code{$X\mapright Y$}
@iftex
@tex
$\equiv$ $\harrowlength=20ptX\mapright Y$.
@end tex
@end iftex
@ifinfo
= (a right arrow).
@end ifinfo

@item \mapleft
@findex mapleft
@code{$X\mapleft Y$}
@iftex
@tex
$\equiv$ $\harrowlength=20ptX\mapleft Y$.
@end tex
@end iftex
@ifinfo
= (a left arrow).
@end ifinfo

@item \hline
@findex hline
@code{$X\hline Y$}
@iftex
@tex
$\equiv$ $\harrowlength=20ptX\hline Y$.
@end tex
@end iftex
@ifinfo
= (horizontal line)
@end ifinfo

@ignore
@item \dothline
@findex dothline
(dotted horizontal line)	{@bf Unimplemented.}
@end ignore

@item \bimapright
@findex bimapright
@code{$X\bimapright Y$}
@iftex
@tex
$\equiv$ $\harrowlength=20ptX\bimapright Y$.
@end tex
@end iftex
@ifinfo
= (two right arrows).
@end ifinfo

@item \bimapleft
@findex bimapleft
@code{$X\bimapleft Y$}
@iftex
@tex
$\equiv$ $\harrowlength=20ptX\bimapleft Y$.
@end tex
@end iftex
@ifinfo
= (two left arrows)
@end ifinfo

@item \adjmapright
@findex adjmapright
@code{$X\adjmapright Y$}
@iftex
@tex
$\equiv$ $\harrowlength=20ptX\adjmapright Y$.
@end tex
@end iftex
@ifinfo
= (two adjoint arrows; left over right)
@end ifinfo

@item \adjmapleft
@findex adjmapleft
@code{$X\adjmapleft Y$}
@iftex
@tex
$\equiv$ $\harrowlength=20ptX\adjmapleft Y$.
@end tex
@end iftex
@ifinfo
= (two adjoint arrows; right over left)
@end ifinfo

@item \bihline
@findex bihline
@code{$X\bihline Y$}
@iftex
@tex
$\equiv$ $\harrowlength=20ptX\bihline Y$.
@end tex
@end iftex
@ifinfo
= (two horizontal lines)
@end ifinfo
@end table

There is also a family of macros to produce vertical lines, arrows, and
adjoint arrows.  The following macros produce vertical maps and have
the same syntax as @code{\mapdown}:

@table @code

@item \mapdown
@findex mapdown
(a down arrow)

@item \mapup
@findex mapup
(an up arrow)

@item \vline
@findex vline
(vertical line)

@ignore
@item \dotvline
@findex dotvline
(dotted vertical line) {@bf Unimplemented.}
@end ignore

@item \bimapdown
@findex bimapdown
(two down arrows)

@item \bimapup
@findex bimapup
(two up arrows)

@item \adjmapdown
@findex adjmapdown
(two adjoint arrows; down then up)

@item \adjmapup
@findex adjmapup
(two adjoint arrows; up then down)

@item \bivline
@findex bivline
(two vertical lines)
@end table

Finally, there is a family of macros to produce slanted lines, arrows,
and adjoint arrows.  The following macros produce slanted maps and have
the same syntax as @code{\arrow}:

@table @code

@item \arrow
@findex arrow
(a slanted arrow)

@item \sline
@findex sline
(a slanted line)

@item \biarrow
@findex biarrow
(two straight arrows)

@item \adjarrow
@findex adjarrow
(two adjoint arrows)

@item \bisline
@findex bisline
(two straight lines)

@end table

The width between double arrows is controlled by the parameter
@code{\channelwidth}.
@findex channelwidth
The parameters @code{\hchannel} and @code{\vchannel}, if nonzero,
override @code{\channelwidth} by controlling the horizontal and vertical
shifting from the first arrow to the second.

There are no adornments on these arrows to distinguish inclusions from
epimorphisms from monomorphisms.  Many texts, such as Lang's book
@cite{Algebra}, use as a tasteful alternative the symbol `inc' (in roman) next
to an arrow to denote inclusion.
@cindex Lang, Serge

Future enhancements will include a mechanism to draw curved arrows
found in, e.g., the Snake Lemma, by employing a version of the
@code{\path} macros of Appendix D of @cite{The @TeX{}book}.
@cindex Snake Lemma


@node Construction of commutative diagrams
@subsection Construction of commutative diagrams

There are two approaches to the construction of commutative diagrams
described here.  The first approach, and the simplest, treats
commutative diagrams like fancy matrices, as Knuth does in Exercise
18.46 of @cite{The @TeX{}book}.  This case is covered by the macro
@code{\commdiag},
@findex commdiag
which is an altered version of the Plain @TeX{} macro @code{\matrix}.
@findex matrix
An example suffices to demonstrate this macro.  The following
commutative diagram (illustrating the covering homotopy property; Bott
and Tu, @cite{Differential Forms in Algebraic Topology})
@cindex Bott, Raoul
@cindex Tu, Loring W.
@cindex covering homotopy property
@iftex
@tex
$$\commdiag{Y&\mapright^f&E\cr \mapdown&\arrow(3,2)\lft{f_t}&\mapdown\cr
Y\times I&\mapright^{\bar f_t}&X}$$
@end tex
@end iftex
@ifinfo

@center (A commutative diagram appears here in the printed output.)

@end ifinfo
is produced with the code

@example
$$\commdiag@{Y&\mapright^f&E\cr \mapdown&\arrow(3,2)\lft@{f_t@}&\mapdown\cr
Y\times I&\mapright^@{\bar f_t@}&X@}$$
@end example

Of course, the parameters may be changed to produce a different effect.
The following commutative diagram (illustrating the universal mapping
property; Warner, @cite{Foundations of Differentiable Manifolds and Lie
Groups})
@cindex Warner, Frank W.
@cindex universal mapping property
@iftex
@tex
$$\varrowlength=20pt
\commdiag{V\otimes W\cr \mapup\lft\phi&\arrow(3,-1)\rt{\tilde l}\cr
V\times W&\mapright^l&U\cr}$$
@end tex
@end iftex
@ifinfo

@center (A commutative diagram appears here in the printed output.)

@end ifinfo
is produced with the code

@example
$$\varrowlength=20pt
\commdiag@{V\otimes W\cr \mapup\lft\phi&\arrow(3,-1)\rt@{\tilde l@}\cr
V\times W&\mapright^l&U\cr@}$$
@end example

A diagram containing isosceles triangles is achieved by placing the apex
of the triangle in the center column, as shown in the example
(illustrating all constant minimal realizations of a linear system;
Brockett, @cite{Finite Dimensional Linear Systems})
@cindex Brockett, Roger W.
@cindex minimal realizations
@cindex linear systems theory
@iftex
@tex
$$\sarrowlength=.42\harrowlength
\commdiag{&R^m\cr &\arrow(-1,-1)\lft{\fam6 B}\quad \arrow(1,-1)\rt{\fam6 G}\cr
R^n&\mapright^{\fam6 P}&R^n\cr
\mapdown\lft{e^{{\fam6 A}t}}&&\mapdown\rt{e^{{\fam6 F}t}}\cr
R^n&\mapright^{\fam6 P}&R^n\cr
&\arrow(1,-1)\lft{\fam6 C}\quad \arrow(-1,-1)\rt{\fam6 H}\cr
&R^q\cr}$$
@end tex
@end iftex
@ifinfo

@center (A commutative diagram appears here in the printed output.)

@end ifinfo
which is produced with the code

@example
$$\sarrowlength=.42\harrowlength
\commdiag@{&R^m\cr &\arrow(-1,-1)\lft@{\bf B@}\quad \arrow(1,-1)\rt@{\bf G@}\cr
R^n&\mapright^@{\bf P@}&R^n\cr
\mapdown\lft@{e^@{@{\bf A@}t@}@}&&\mapdown\rt@{e^@{@{\bf F@}t@}@}\cr
R^n&\mapright^@{\bf P@}&R^n\cr
&\arrow(1,-1)\lft@{\bf C@}\quad \arrow(-1,-1)\rt@{\bf H@}\cr
&R^q\cr@}$$
@end example

Other commutative diagram examples appear in the file
@code{commdiags.tex}, which is distributed with this package.

In these examples the arrow lengths and line slopes were carefully
chosen to blend with each other.  In the first example, the default
settings for the arrow lengths are used, but a direction for the arrow
must be chosen.  The ratio of the default horizontal and vertical arrow
lengths is approximately the golden mean
@cindex golden mean
@iftex
@tex
$\gamma=1.618\ldots$;
@end tex
@end iftex
@ifinfo
gamma=1.618...;
@end ifinfo
@cindex golden mean
the arrow direction closest to this mean is @code{(3,2)}. In the second
example, a slope of
@iftex
@tex
$-1/3$
@end tex
@end iftex
@ifinfo
-1/3
@end ifinfo
is desired and the default horizontal arrow length is 60 pt; therefore,
choose a vertical arrow length of 20 pt.  You may affect the interline
glue settings of @code{\commdiag} by redefining the macro
@code{\commdiagbaselines}.
@findex commdiagbaselines
(cf@. Exercise 18.46 of @cite{The @TeX{}book} and the section on
parameters below.)

The width, height, and depth of all morphisms are hidden so that the
morphisms' size do not affect arrow positions.  This can cause a large
morphism at the top or bottom of a diagram to impinge upon the text
surrounding the diagram.  To overcome this problem, use @TeX{}'s
@code{\noalign} primitive to insert a @code{\vskip} immediately above or
below the offending line, e.g.,
@samp{$$\commdiag@{\noalign@{\vskip6pt@}X&\mapright^\int&Y\cr ...@}}.

The macro @code{\commdiag} is too simple to be used for more complicated
diagrams, which may have intersecting or overlapping arrows.  A second
approach, borrowed from Francis Borceux's @cite{Diagram} macros for
@LaTeX{}, treats the commutative diagram like a grid of identically
shaped boxes.  To compose the commutative diagram, first draw an equally
spaced grid, e.g.,
@cindex grid
@cindex Borceux, Francis
@cindex Diagram, macros for LaTeX
@iftex
@tex
$$\def\grid{\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\cr}
\matrix{\grid\grid\grid\grid}$$
@end tex
@end iftex
@ifinfo

@center                    . . . . . .
@center                    . . . . . .
@center                    . . . . . .
@center                    . . . . . .

@end ifinfo
on a piece of scratch paper.  Then draw each element (vertices and
arrows) of the commutative diagram on this grid, centered at each
grid point.  Finally, use the macro @code{\gridcommdiag}
@findex gridcommdiag
to implement your design as a @TeX{} alignment.  For example, the cubic
diagram
@cindex cube
@iftex
@tex
$$\harrowlength=48pt \varrowlength=48pt \sarrowlength=20pt
\def\cross#1#2{\setbox0=\hbox{$#1$}%
  \hbox to\wd0{\hss\hbox{$#2$}\hss}\llap{\unhbox0}}
\gridcommdiag{&&B&&\mapright^b&&D\cr
&\arrow(1,1)\lft a&&&&\arrow(1,1)\lft d\cr
A&&\cross{\hmorphposn=12pt\mapright^c}{\vmorphposn=-12pt\mapdown\lft f}
&&C&&\mapdown\rt h\cr\cr
\mapdown\lft e&&F&&\cross{\hmorphposn=-12pt\mapright_j}
{\vmorphposn=12pt\mapdown\rt g}&&H\cr
&\arrow(1,1)\lft i&&&&\arrow(1,1)\rt l\cr
E&&\mapright_k&&G\cr}$$
@end tex
@end iftex
@ifinfo

@center (A commutative diagram appears here.)

@end ifinfo
that appears in Francis Borceux's documentation can be implemented on
a 7 by 7 grid, and is achieved with the code

@example
$$\harrowlength=48pt \varrowlength=48pt \sarrowlength=20pt
\def\cross#1#2@{\setbox0=\hbox@{$#1$@}%
  \hbox to\wd0@{\hss\hbox@{$#2$@}\hss@}\llap@{\unhbox0@}@}
\gridcommdiag@{&&B&&\mapright^b&&D\cr
&\arrow(1,1)\lft a&&&&\arrow(1,1)\lft d\cr
A&&\cross@{\hmorphposn=12pt\mapright^c@}@{\vmorphposn=-12pt\mapdown\lft f@}
&&C&&\mapdown\rt h\cr\cr
\mapdown\lft e&&F&&\cross@{\hmorphposn=-12pt\mapright_j@}
@{\vmorphposn=12pt\mapdown\rt g@}&&H\cr
&\arrow(1,1)\lft i&&&&\arrow(1,1)\rt l\cr
E&&\mapright_k&&G\cr@}$$
@end example

The dimensions @code{\hgrid} and @code{\vgrid}
@findex hgrid
@findex vgrid
control the horizontal and vertical spacing of the grid used by
@code{\gridcommdiag}. The default setting for both of these dimensions
is 15 pt.  Note that in the example of the cube the arrow lengths must
be adjusted so that the arrows overlap into neighboring boxes by the
desired amount.  Hence, the @code{\gridcommdiag} method, albeit more
powerful, is less automatic than the simpler @code{\commdiag} method.
Furthermore, the ad hoc macro @code{\cross} is introduced to allow the
effect of overlapping arrows.  Finally, note that the positions of four
of the morphisms are adjusted by setting @code{\hmorphposn} and
@code{\vmorphposn}.

One is not restricted to a square grid.  For example, the proof of
Zassenhaus's Butterfly Lemma can be illustrated by the diagram (appearing
in Lang's book @cite{Algebra})
@cindex Zassenhaus, Hans
@cindex Lang, Serge
@cindex Butterfly Lemma
@iftex
@tex
$$\hgrid=16pt \vgrid=8pt \sarrowlength=32pt
\def\cross#1#2{\setbox0=\hbox{$#1$}%
  \hbox to\wd0{\hss\hbox{$#2$}\hss}\llap{\unhbox0}}
\def\l#1{\llap{$#1$\hskip.5em}}
\def\r#1{\rlap{\hskip.5em$#1$}}
\gridcommdiag{&&U&&&&V\cr &&\bullet&&&&\bullet\cr
&&\sarrowlength=16pt\sline(0,1)&&&&\sarrowlength=16pt\sline(0,1)\cr
&&\l{u(U\cap V)}\bullet&&&&\bullet\r{(U\cap V)v}\cr
&&&\sline(2,-1)&&\sline(2,1)\cr
&&\cross{=}{\sline(0,1)}&&\bullet&&\cross{=}{\sline(0,1)}\cr\cr
&&\l{^{\textstyle u(U\cap v)}}\bullet&&\cross{=}{\sline(0,1)}&&
 \bullet\r{^{\textstyle(u\cap V)v}}\cr
&\sline(2,1)&&\sline(2,-1)&&\sline(2,1)&&\sline(2,-1)\cr
\l{u}\bullet&&&&\bullet&&&&\bullet\r{v}\cr
&\sline(2,-1)&&\sline(2,1)&&\sline(2,-1)&&\sline(2,1)\cr
&&\bullet&&&&\bullet\cr &&u\cap V&&&&U\cap v\cr}$$
@end tex
@end iftex
@ifinfo

@center (A commutative diagram appears here.)

@end ifinfo
This diagram may be implemented on a 9 by 12 grid with an aspect ratio
of 1/2, and is set with the code

@example
$$\hgrid=16pt \vgrid=8pt \sarrowlength=32pt
\def\cross#1#2@{\setbox0=\hbox@{$#1$@}%
  \hbox to\wd0@{\hss\hbox@{$#2$@}\hss@}\llap@{\unhbox0@}@}
\def\l#1@{\llap@{$#1$\hskip.5em@}@}
\def\r#1@{\rlap@{\hskip.5em$#1$@}@}
\gridcommdiag@{&&U&&&&V\cr &&\bullet&&&&\bullet\cr
&&\sarrowlength=16pt\sline(0,1)&&&&\sarrowlength=16pt\sline(0,1)\cr
&&\l@{u(U\cap V)@}\bullet&&&&\bullet\r@{(U\cap V)v@}\cr
&&&\sline(2,-1)&&\sline(2,1)\cr
&&\cross@{=@}@{\sline(0,1)@}&&\bullet&&\cross@{=@}@{\sline(0,1)@}\cr\cr
&&\l@{^@{\textstyle u(U\cap v)@}@}\bullet&&\cross@{=@}@{\sline(0,1)@}&&
 \bullet\r@{^@{\textstyle(u\cap V)v@}@}\cr
&\sline(2,1)&&\sline(2,-1)&&\sline(2,1)&&\sline(2,-1)\cr
\l@{u@}\bullet&&&&\bullet&&&&\bullet\r@{v@}\cr
&\sline(2,-1)&&\sline(2,1)&&\sline(2,-1)&&\sline(2,1)\cr
&&\bullet&&&&\bullet\cr &&u\cap V&&&&U\cap v\cr@}$$
@end example

Again, the construction of this diagram requires careful choices for the
arrow lengths and is facilitated by the introduction of the ad hoc
macros @code{\cross}, @code{\r}, and @code{\l}.  Note also that
superscripts were used to adjust the position of the vertices
@iftex
@tex
$u(U\cap v)$ and $(u\cap V)v$.
@end tex
@end iftex
@ifinfo
u(U intersection v) and (u intersection V)v.
@end ifinfo
Many diagrams may be typeset with the predefined macros that appear
here; however, ingenuity is often required to handle special cases.

@node Commutative diagram parameters
@subsection Commutative diagram parameters

The following is a list describing the parameters used in the
commutative diagram macros.  These dimensions may be changed globally or
locally.

@table @code
@item \harrowlength
@findex harrowlength
(Default: 60 pt) The length of right or left arrows.

@item \varrowlength
@findex varrowlength
(Default: 0.618@code{\harrowlength}) The length of up or down
arrows.

@item \sarrowlength
@findex sarrowlength
(Default: 60 pt) The horizontal length of slanted arrows.

@item \hmorphposn
@findex hmorphposn
(Default: 0 pt) The horizontal position of the morphism with
respect to its default position.  There are also the dimensions
@code{\hmorphposnup},
@findex hmorphposnup
@code{\hmorphposndn},
@findex hmorphposndn
@code{\hmorphposnrt},
@findex hmorphposnrt
and @code{\hmorphposnlft}
@findex hmorphposnlft
for @code{^}/@code{_} or @code{\lft}/@code{\rt} constructions.

@item \vmorphposn
@findex vmorphposn
(Default: 0 pt) The vertical position of the morphism with
respect to its default position. There are also the dimensions
@code{\vmorphposnup},
@findex vmorphposnup
@code{\vmorphposndn},
@findex vmorphposndn
@code{\vmorphposnrt},
@findex vmorphposnrt
and @code{\vmorphposnlft}
@findex vmorphposnlft
for @code{^}/@code{_} or @code{\lft}/@code{\rt} constructions.

@item \morphdist
@findex morphdist
(Default: 4 pt) The distance of morphisms from slanted lines
or arrows.

@item \channelwidth
@findex channelwidth
(Default: 3 pt) The distance between double lines or arrows.

@item \hchannel, \vchannel
@findex hchannel
@findex vchannel
(Defaults: 0 pt) Overrides @code{\channelwidth}. The
horizontal and vertical shifts between double lines or arrows.

@item \commdiagbaselines
@findex commdiagbaselines
(Default: @code{\baselineskip=15pt
\lineskip=3pt
\lineskiplimit=3pt })
The parameters used by @code{\commdiag} for setting interline glue.

@item \hgrid
@findex hgrid
(Default: 15 pt) The horizontal spacing of the grid used by
@code{\gridcommdiag}.

@item \vgrid
@findex vgrid
(Default: 15 pt) The vertical spacing of the grid used by
@code{\gridcommdiag}.

@end table

@comment bye