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 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.