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October 21, 2009

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?

Posted at October 21, 2009 5:41 PM UTC

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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.

Posted by: John Armstrong on October 21, 2009 6:39 PM | Permalink | Reply to this

Re: Aesthetics of Commutative Diagrams

I prefer the rectangular versions because I find them easier to read. It also makes it easier to recognise patterns of similar rows or columns.

Perhaps it’s also a question about the tools you use. In my experience using xypic I got the impression that in non-rectangular diagrams it happens all the time that objects and arrows end up in places and angles which aren’t quite what you had in mind and would be most clear. So you easily end up wasting a lot of time tweaking the diagram so it doesn’t look completely horrible.

Posted by: ssp on October 21, 2009 6:57 PM | Permalink | Reply to this

Re: Aesthetics of Commutative Diagrams

FWIW, the rectangular diagrams were made with xypic, but the non-rectangular ones were made with Ipe.

Posted by: Mike Shulman on October 21, 2009 8:57 PM | Permalink | PGP Sig | Reply to this

Re: Aesthetics of Commutative Diagrams

Of the A’s, I much prefer A3.

One principle to adopt for large commutative diagrams is: don’t frighten people. I think that means something like “try to persuade your reader that they could have come up with this themselves”. If the diagram has no apparent pattern to it, the reader might imagine that they would have needed hours to come up with it. Or perhaps more to the point, they might imagine that if they ever had to recreate the proof without your paper to hand, it would take them hours. So if there’s any kind of regularity in the diagram, you should make it stand out as much as possible. That’s why I like A3.

Posted by: Tom Leinster on October 21, 2009 8:42 PM | Permalink | Reply to this

Re: Aesthetics of Commutative Diagrams

So if there’s any kind of regularity in the diagram, you should make it stand out as much as possible. That’s why I like A3.

I like A3 for that reason as well. However, I wasn’t able to get the same sort of regularity to appear in B or C, and I don’t like B2 as much as B1. But another comment I received was that the style should be consistent in a single paper—either all rectangular or all globular. And I can see that too.

Posted by: Mike Shulman on October 21, 2009 9:04 PM | Permalink | PGP Sig | Reply to this

Re: Aesthetics of Commutative Diagrams

I like the last one of each. I'm not sure if I can describe general principles to back that up; although apparently I like rectangular diagrams, I didn't know that before I looked!

I will say that, while A1 is prettiest, A3 is most legible.

Posted by: Toby Bartels on October 22, 2009 12:53 AM | Permalink | Reply to this

Re: Aesthetics of Commutative Diagrams

I will say that, while A1 is prettiest, A3 is most legible.

What do you mean by “legible”? Do you mean, as Theo says, that it’s easier to tell “what’s going on”? What is “going on”?

Posted by: Mike Shulman on October 22, 2009 4:56 AM | Permalink | PGP Sig | Reply to this

Re: Aesthetics of Commutative Diagrams

What do you mean by “legible”? Do you mean, as Theo says, that it’s easier to tell “what’s going on”? What is “going on”?

I'd say that it's ‘easier to follow’.

To see ‘what's going on’, it might help to somehow stress the second line from the bottom; everything sort of flows out of that. But does that matter?

Posted by: Toby Bartels on October 22, 2009 9:48 PM | Permalink | Reply to this

Re: Aesthetics of Commutative Diagrams

I definitely don’t like most of the globular examples you posted — I have absolutely no idea at a glance what’s going on in them.

This is not to say that I think rectangles are the way to go. In particular, they give the strong impression that things in a give row or column are naturally related, and if this is not the case, then the rectangles are misleading.

Of the examples you’re posted, B2 and C3 are the most visually appealing. They look like if I tried to read them, I’d understand them.

One other comment. You talk about “zig-zagging”. But your diagrams have no zig-zags, only chained zigs and zags. I guess I would propose the following. I would draw a zig-zagging path along the top of a rectangle, and another zig-zag along the bottom, and I’d make the angles of the zig-zag mean something. I’d put very long equals signs along the lateral sides of the rectangle. Then I’d fill in the interior with the squares.

Actually, trying that on B still doesn’t look that good. But here’s another nifty fact. Each of your diagrams actually defines a poset on your vertices — there are no loops. I know you want to draw the diagrams on a plane, but perhaps drawing them on the outside of a bowl, with all arrows pointing up, might be better?

Posted by: Theo on October 22, 2009 2:26 AM | Permalink | Reply to this

Re: Aesthetics of Commutative Diagrams

[rectangles] give the strong impression that things in a given row or column are naturally related

Well, the opposite sides of a naturality square are of course related: they’re either two different components of the same natural transformation, or the image of the same map under two different functors. So in A3, for instance, all the parallel morphisms in one row or column are related in these ways. I’m not sure if that’s what you meant. This is less applicable to C2 and C4, though, where even an overall “rectangular” shape has to include various diagonals.

But your diagrams have no zig-zags, only chained zigs and zags.

I think it’s fairly common to refer to strings of composable forward-pointing arrows and backward-pointing weak equivalences as a “zigzag,” whether or not one draws them in a straight line. And I personally think it would look pretty ugly to actually draw them in what is colloquially called a zigzag.

I would draw a zig-zagging path along the top of a rectangle, and another zig-zag along the bottom, and I’d make the angles of the zig-zag mean something. I’d put very long equals signs along the lateral sides of the rectangle. Then I’d fill in the interior with the squares.

I don’t really know what you mean; can you do it instead of describing it?

I know you want to draw the diagrams on a plane, but perhaps drawing them on the outside of a bowl, with all arrows pointing up, might be better?

That might be a good idea if I could get bowl-shaped paper to print them on. (-:

Actually, I don’t like the idea of “all arrows pointing up,” because the backwardness of the weak equivalences is important to keep track of.

Posted by: Mike Shulman on October 22, 2009 4:34 AM | Permalink | PGP Sig | Reply to this

Re: Aesthetics of Commutative Diagrams

I’m finding most of these diagrams rather hard to read (not being familiar with “zigzags with backwards-pointing weak equivalences”), because the eye naturally wants to flow along the direction of the arrows to follow what’s going on. Having to stop and read some of the arrows backwards tremendously impedes my ability to form chains of connections from one position to another, and to visually compare them.

Would it be at all possible to introduce a new notation to replace these backward-pointing arrows? It should be something that is clearly distinct from the arrow, but gives the visual impression of something pointing in the “right” direction, so it can be easily seen to be part of a chain. Perhaps a circle-headed arrow? So instead of:

A <– B

with only a tilde to show that this arrow should be read backwards, we’d have

A –0 B

with a circle at the tail instead of an arrowhead.

Posted by: Stuart on October 22, 2009 6:20 AM | Permalink | Reply to this

Re: Aesthetics of Commutative Diagrams

Would it be at all possible to introduce a new notation to replace these backward-pointing arrows?

I think most of the intended audience of this paper will be familiar with zigzags in the way that I’ve used them, and would find it much more confusing if I tried to introduce something new. I certainly would if I were reading such a paper.

Posted by: Mike Shulman on October 22, 2009 3:52 PM | Permalink | PGP Sig | Reply to this

Re: Aesthetics of Commutative Diagrams

Yeah, don’t make up any new weirdness. If someone doesn’t know what a zigzag with backwards-pointing weak equivalences is, they probably have no business reading your paper.

Posted by: John Baez on October 22, 2009 9:21 PM | Permalink | Reply to this

Re: Aesthetics of Commutative Diagrams

Like others, I tend to prefer the rectangular diagrams. Part of it has to do with the layout used for the globular ones, however. The nice thing about the rectangular diagrams is that, because all the shapes are uniform, it’s easy to see what’s going on. With these globular examples, the shapes are all squashed and the lack of uniformity makes it hard to see what the point being demonstrated is.

However, for some diagrams it really makes sense to use shapes other than squares, provided that they’re fairly “regular” or symmetric shapes. The hexagons in the current version of your paper are a good example (though they could be made more uniform). And this paper has some rather nice diagrams with triangles, trapezoids, and parallelograms.

Posted by: wren ng thornton on October 22, 2009 9:37 AM | Permalink | Reply to this

Re: Aesthetics of Commutative Diagrams

I always try to remember to do two things when creating a very large diagram: make the source/target clear, and highlight the essential parts to make the reader feel like they know the key things to remember if they wanted to reconstruct it later for themselves. Having said that, I think symmetry is also important, and simple things like making all your squares look like squares in the same way helps, i.e., if some squares are actual squares but others are very strange regions that just happen to have 4 sides, that tends to be hard to read.

I came up with my opinions by first looking at the diagrams, then reading the descriptions of what I was supposed to be looking at, and then looking again. I think this does a good job of simulating how I might read a paper.

For the A’s, I strongly prefer A3. I knew what the source and target were before reading the description, and had decided I knew why about a third of the diagram commuted without having to be told.

For the B’s, I very slightly prefer B1. I could figure out its source and target without help, while for B2 I had a guess but was not sure. B1 was also visually divided in two: the “easy” parts on the bottom, and the less obvious parts on the top. But I will admit, that with some very simple instructions about how to read the diagrams, B2 might have been my favorite.

For the C’s, I find all of the diagrams intimidating, but I think C3 the least so. Long lines across any diagram tend to be harder to look at I think, with the possible exception of an exterior edge which is also a very simple morphism like the identity. Large shapes also make it hard to spot patterns, like a collection of naturality squares. I do think it can be useful to mark regions with symbols indicating why they commute as you did with these, especially if the different individual regions commute for lots of different reasons, instead of just naturality over and over again.

Posted by: Nick Gurski on October 22, 2009 12:30 PM | Permalink | Reply to this

Re: Aesthetics of Commutative Diagrams

Diagrams like A1, B1, C1 and C3 seem to have been scrawled out by an insane genius in a fit of fevered inspiration. As such, they seem intimidating. I think most people are more comfortable with rectangular shapes or highly symmetrical shapes. These diagrams seem disorganized and random.

What amazes me is that you had the energy to draw several versions of these diagrams!

Posted by: John Baez on October 22, 2009 9:13 PM | Permalink | Reply to this

Re: Aesthetics of Commutative Diagrams

I see Nick likes C3. This just proves that he’s an insane genius.

Posted by: John Baez on October 22, 2009 9:15 PM | Permalink | Reply to this

Re: Aesthetics of Commutative Diagrams

I am not sure about insane or genius, but I do feel a bit fearless when it comes to drawing large diagrams. Once you get the whole thing down, then you can try to make it readable.

One thing to remember about large diagrams: expect to draw them poorly a lot of times before you get something that looks good. The first time I draw any really big diagram, I try to do it on a chalk board. Since there is no way that diagram will make into a paper or even a preprint, I don’t feel any attachment to it. I think being able to step back helps me evaluate if the diagram is doing what I want more critically than if I start by TeXing it up.

Posted by: Nick Gurski on October 23, 2009 11:48 AM | Permalink | Reply to this

Re: Aesthetics of Commutative Diagrams

Diagrams like A1, B1, C1 and C3 seem to have been scrawled out by an insane genius in a fit of fevered inspiration. As such, they seem intimidating.

Interesting. To play devil’s advocate for a minute….

Diagrams A1, B1, and C1 are actually the first diagrams that I wrote down, on paper. Does that make me an insane genius too? (-: Whenever I have something like this to prove, I generally first write out what I’m given in a big empty “globular” shape and then just try to fill it in with things that I know commute—which usually means mostly naturality squares. So these diagrams kind of make me feel more comfortable, because they’re the sort of thing I would end up with if I sat down and tried to prove it myself. Whereas, seeing an orderly rectangular diagram makes me think there may be some deep structure which I need to understand in more detail to see what’s going on, rather than just blindly following my nose.

I see Nick likes C3. This just proves that he’s an insane genius.

Last Saturday I was discussing this with half a dozen people at Peter May’s birthday conference and passed around basically the same versions of exhibit C shown above. It turned out that C3 was the almost unanimous favorite. I guess we had a whole convention of insane geniuses! (-:

Posted by: Mike Shulman on October 22, 2009 10:58 PM | Permalink | Reply to this

Re: Aesthetics of Commutative Diagrams

Are these images still available somewhere? The PDF links are broken at the moment.

Posted by: Joshua Taylor on September 5, 2013 3:52 PM | Permalink | Reply to this

Re: Aesthetics of Commutative Diagrams

Sadly, I have no idea. I don’t know what happened to them or whether I kept copies anywhere else.

Posted by: Mike Shulman on September 6, 2013 10:20 PM | Permalink | Reply to this

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