### Aesthetics of Commutative Diagrams

#### Posted by Mike Shulman

I’ve recently run into the question of how best to lay out a fairly large commutative diagram. Some diagrams have a “natural” shape such as a cube or a simplex, but as far as I can tell that is not the case for the diagrams in question. They aren’t complicated, mostly just a bunch of naturality squares stuck together. But different people seem to have different aesthetic viewpoints on what makes the layout of a diagram “look good.” So I thought I’d share my data so far, and see whether anyone here has additional insights.

I’ll give links to the diagrams in a moment, but first let me say a few words about the context. These diagrams are from a new revision in progress of this paper, which you don’t need to read or know anything about. But it will probably help to know that the diagrams live in some category with weak equivalences, but they contain zigzags with backwards-pointing weak equivalences (generally labeled with a $\sim$) that can only be composed in the homotopy category. Thus, the questions about commutativity only happen in the homotopy category.

**Exhibit A.** In this diagram, we are given two zigzags and want to show that they are equal in the homotopy category. Here are three possibilities:

In the first two versions, the two zigzags we’re comparing should hopefully be evident: one goes along the top and the other along the bottom. In the third, the beginning and end of the two zigzags are marked with the derived functors they represent; the two zigzags in question are just the two ways to get around the outside from one to the other.

**Exhibit B.** In this diagram, we are given *one* zigzag and want to show that it is equal to the identity in the homotopy category.

In the first version, the given zigzag is along the top, and the bottom is a zigzag representing the identity. In the second version, the start and end are marked with boxes, the given zigzag is in solid arrows, the zigzag representing the identity is in dashed arrows, and all the other arrows are dotted.

**Exhibit C.** In this diagram, we are given one zigzag and want to show that it represents an isomorphism in the homotopy category. We do this by showing it is equal to another zigzag which is composed entirely of weak equivalences.

In the first version, the two zigzags in question are along the top and bottom. In the second, one goes along the top-right and the other along the bottom-left. The third and fourth are similar to the first and second, except that we break off part of the diagram into a secondary smaller one. Each of these has a square marked ⊛ which is special (the others are naturality squares; this one is a definition of one of its sides in terms of the other three).

**Conclusions so far.** Some people seem to prefer diagrams with rectangular shapes, like A3, B2, C2, and C4, saying that they’re more orderly and less frightening. Others like the “globular” shapes A2, B1, C1, and C3, saying that they make it more clear what the two composites are that are being compared. (In A3 and B2 I experimented with a couple of different methods to try to make it clear what the two composites are in a rectangular version; in C2 and C4 I was able to put the ultimate source and target at the corners.) Some people really object to the long arrows and long thin squares in A2 and C1; others don’t seem to mind them.

What do you prefer, and why? What general principles do you adhere to when organizing a diagram? How would you lay out these diagrams differently?

## Re: Aesthetics of Commutative Diagrams

I don’t mind the “globular” diagrams. Indeed, I rather like diagram A2 for its lenticular shape, which is very suggestive of the idea of filling in a 2-morphism between two 1-morphisms. The only qualm I have is that they tend to look somewhat haphazard and difficult to read unless you already understand all the terms.