The n-Category Café

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.

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.

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.

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.

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?

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.

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.

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.

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!

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