This chapter describes definitions for producing commutative diagrams.
Steven Smith wrote this documentation (and the macros).
The macros \drawline
and \drawvector
provide the capability found in LaTeX's
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, `\drawvector(-4,1){60pt}' produces the
vector
which lies in the 2d quadrant, has a slope of minus 1/4, and a width of
60 pt.
Note that if an \hbox
is placed around \drawline
or
\drawvector
, then the width of the \hbox
will be the
positive dimension specified in the third argument, except when a
vertical line or vector is specified, e.g.,
\drawline(0,1){1in}
, which has zero width. If the specified
direction lies in the 1st or 2d quadrant (e.g., (1,1)
or
(-2,3)
), then the \hbox
will have positive height and zero
depth. Conversely, if the specified direction lies in the 3d or 4th
quadrant (e.g., (-1,-1)
or (2,-3)
), then the \hbox
will have positive depth and zero height.
There are a finite number of directions that can be specified. For
\drawline
, the absolute value of each integer defining the
direction must be less than or equal to six, i.e., (7,-1)
is
incorrect, but (6,-1)
is acceptable. For \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 (2,1)
instead of
(4,2)
. Also, specify (1,0)
instead of, say, (3,0)
for horizontal lines and likewise for vertical lines.
Finally, these macros depend upon the LaTeX font 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.
The primitive commands \drawline
and \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.
The macros \mapright
and \mapleft
produce right and left
pointing arrows, respectively. Use superscript (^
) to place a
morphism above the arrow, e.g., `\mapright^\alpha'; use subscript
(_
) to place a morphism below the arrow, e.g.,
`\mapright_{\tilde l}'. Superscripts and subscripts may be used
simulataneously, e.g., `\mapright^\pi_{\rm epimor.}'.
Similarly, the macros \mapup
and \mapdown
produce up and
down pointing arrows, respectively. Use \rt
to place a morphism to the right of the arrow, e.g., `\mapup\rt{\rm
id}'; use \lft
to place a morphism to the left of the arrow, e.g., `\mapup\lft\omega'.
\lft
and \rt
may be used simultaneously, e.g.,
`\mapdown\lft\pi\rt{\rm monomor.}'.
Slanted arrows are produced by the macro \arrow
, which takes
a direction argument (e.g., `\arrow(3,-4)'). Use \rt
and
\lft
to place morphisms to the right and left, respectively, of
the arrow. A slanted line (no arrowhead) is produced with the macro
\sline
,
whose syntax is identical to that of \arrow
.
The length of these macros is predefined by the default TeX
dimensions \harrowlength
,
for horizontal arrows (or lines),
\varrowlength
,
for vertical arrows (or lines), and
\sarrowlength
,
for slanted arrows (or lines). To change any of these dimensions, say,
e.g., `\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
\hmorphposn
,
\vmorphposn
,
and \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 ^
/_
or \lft
/\rt
construction), use the parameters
\hmorphposnup
,
\hmorphposndn
,
\vmorphposnup
,
\vmorphposndn
,
\hmorphposnrt
,
\hmorphposnlft
,
\vmorphposnrt
,
and \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 \mapright
:
\mapright
$X\mapright Y$
\mapleft
$X\mapleft Y$
\hline
$X\hline Y$
\bimapright
$X\bimapright Y$
\bimapleft
$X\bimapleft Y$
\adjmapright
$X\adjmapright Y$
\adjmapleft
$X\adjmapleft Y$
\bihline
$X\bihline Y$
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 \mapdown
:
\mapdown
\mapup
\vline
\bimapdown
\bimapup
\adjmapdown
\adjmapup
\bivline
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 \arrow
:
\arrow
\sline
\biarrow
\adjarrow
\bisline
The width between double arrows is controlled by the parameter
\channelwidth
.
The parameters \hchannel
and \vchannel
, if nonzero,
override \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 Algebra, use as a tasteful alternative the symbol `inc' (in roman) next to an arrow to denote inclusion.
Future enhancements will include a mechanism to draw curved arrows
found in, e.g., the Snake Lemma, by employing a version of the
\path
macros of Appendix D of The TeXbook.
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 The TeXbook. This case is covered by the macro
\commdiag
,
which is an altered version of the Plain TeX macro \matrix
.
An example suffices to demonstrate this macro. The following
commutative diagram (illustrating the covering homotopy property; Bott
and Tu, Differential Forms in Algebraic Topology)
is produced with the code
$$\commdiag{Y&\mapright^f&E\cr \mapdown&\arrow(3,2)\lft{f_t}&\mapdown\cr Y\times I&\mapright^{\bar f_t}&X}$$
Of course, the parameters may be changed to produce a different effect. The following commutative diagram (illustrating the universal mapping property; Warner, Foundations of Differentiable Manifolds and Lie Groups) is produced with the code
$$\varrowlength=20pt \commdiag{V\otimes W\cr \mapup\lft\phi&\arrow(3,-1)\rt{\tilde l}\cr V\times W&\mapright^l&U\cr}$$
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, Finite Dimensional Linear Systems) which is produced with the code
$$\sarrowlength=.42\harrowlength \commdiag{&R^m\cr &\arrow(-1,-1)\lft{\bf B}\quad \arrow(1,-1)\rt{\bf G}\cr R^n&\mapright^{\bf P}&R^n\cr \mapdown\lft{e^{{\bf A}t}}&&\mapdown\rt{e^{{\bf F}t}}\cr R^n&\mapright^{\bf P}&R^n\cr &\arrow(1,-1)\lft{\bf C}\quad \arrow(-1,-1)\rt{\bf H}\cr &R^q\cr}$$
Other commutative diagram examples appear in the file
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
the arrow direction closest to this mean is (3,2)
. In the second
example, a slope of
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 \commdiag
by redefining the macro
\commdiagbaselines
.
(cf. Exercise 18.46 of The TeXbook 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
\noalign
primitive to insert a \vskip
immediately above or
below the offending line, e.g.,
`$$\commdiag{\noalign{\vskip6pt}X&\mapright^\int&Y\cr ...}'.
The macro \commdiag
is too simple to be used for more complicated
diagrams, which may have intersecting or overlapping arrows. A second
approach, borrowed from Francis Borceux's Diagram macros for
LaTeX, treats the commutative diagram like a grid of identically
shaped boxes. To compose the commutative diagram, first draw an equally
spaced grid, e.g.,
on a piece of scratch paper. Then draw each element (vertices and
arrows) of the commutative diagram on this grid, centered at each
grid point. Finally, use the macro \gridcommdiag
to implement your design as a TeX alignment. For example, the cubic
diagram
that appears in Francis Borceux's documentation can be implemented on
a 7 by 7 grid, and is achieved with the code
$$\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}$$
The dimensions \hgrid
and \vgrid
control the horizontal and vertical spacing of the grid used by
\gridcommdiag
. The default setting for both of these dimensions
is 15 pt. Note that in the example of the cube the arrow lengths must
be adjusted so that the arrows overlap into neighboring boxes by the
desired amount. Hence, the \gridcommdiag
method, albeit more
powerful, is less automatic than the simpler \commdiag
method.
Furthermore, the ad hoc macro \cross
is introduced to allow the
effect of overlapping arrows. Finally, note that the positions of four
of the morphisms are adjusted by setting \hmorphposn
and
\vmorphposn
.
One is not restricted to a square grid. For example, the proof of Zassenhaus's Butterfly Lemma can be illustrated by the diagram (appearing in Lang's book Algebra) This diagram may be implemented on a 9 by 12 grid with an aspect ratio of 1/2, and is set with the code
$$\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}$$
Again, the construction of this diagram requires careful choices for the
arrow lengths and is facilitated by the introduction of the ad hoc
macros \cross
, \r
, and \l
. Note also that
superscripts were used to adjust the position of the vertices
Many diagrams may be typeset with the predefined macros that appear
here; however, ingenuity is often required to handle special cases.
The following is a list describing the parameters used in the commutative diagram macros. These dimensions may be changed globally or locally.
\harrowlength
\varrowlength
\harrowlength
) The length of up or down
arrows.
\sarrowlength
\hmorphposn
\hmorphposnup
,
\hmorphposndn
,
\hmorphposnrt
,
and \hmorphposnlft
for ^
/_
or \lft
/\rt
constructions.
\vmorphposn
\vmorphposnup
,
\vmorphposndn
,
\vmorphposnrt
,
and \vmorphposnlft
for ^
/_
or \lft
/\rt
constructions.
\morphdist
\channelwidth
\hchannel, \vchannel
\channelwidth
. The
horizontal and vertical shifts between double lines or arrows.
\commdiagbaselines
\baselineskip=15pt
\lineskip=3pt
\lineskiplimit=3pt
)
The parameters used by \commdiag
for setting interline glue.
\hgrid
\gridcommdiag
.
\vgrid
\gridcommdiag
.