(eplain.info)Construction of commutative diagrams
6.2.2 Construction of commutative diagrams
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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')
(A commutative diagram appears here in the printed output.)
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')
(A commutative diagram appears here in the printed output.)
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')
(A commutative diagram appears here in the printed output.)
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 gamma=1.618...; the arrow
direction closest to this mean is '(3,2)'. In the second example, a
slope of -1/3 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., '$$\commdiag{\noalign{\vskip6pt}X&\mapright^\int&Y\cr ...}'.
The macro '\commdiag' is too simple to be used for more complicated
diagrams, which may have intersecting or overlapping arrows. A second
approach, borrowed from Francis Borceux's 'Diagram' macros for LaTeX,
treats the commutative diagram like a grid of identically shaped boxes.
To compose the commutative diagram, first draw an equally spaced grid,
e.g.,
. . . . . .
. . . . . .
. . . . . .
. . . . . .
on a piece of scratch paper. Then draw each element (vertices and
arrows) of the commutative diagram on this grid, centered at each grid
point. Finally, use the macro '\gridcommdiag' to implement your design
as a TeX alignment. For example, the cubic diagram
(A commutative diagram appears here.)
that appears in Francis Borceux's documentation can be implemented on
a 7 by 7 grid, and is achieved with the code
$$\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}$$
The dimensions '\hgrid' and '\vgrid' control the horizontal and
vertical spacing of the grid used by '\gridcommdiag'. The default
setting for both of these dimensions is 15 pt. Note that in the example
of the cube the arrow lengths must be adjusted so that the arrows
overlap into neighboring boxes by the desired amount. Hence, the
'\gridcommdiag' method, albeit more powerful, is less automatic than the
simpler '\commdiag' method. Furthermore, the ad hoc macro '\cross' is
introduced to allow the effect of overlapping arrows. Finally, note that
the positions of four of the morphisms are adjusted by setting
'\hmorphposn' and '\vmorphposn'.
One is not restricted to a square grid. For example, the proof of
Zassenhaus's Butterfly Lemma can be illustrated by the diagram
(appearing in Lang's book 'Algebra')
(A commutative diagram appears here.)
This diagram may be implemented on a 9 by 12 grid with an aspect
ratio of 1/2, and is set with the code
$$\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 u(U intersection v) and (u
intersection V)v. Many diagrams may be typeset with the predefined
macros that appear here; however, ingenuity is often required to handle
special cases.
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