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When doing $1$-category theory, there are often no tangible advantages/disadvantages to drawing diagrams in one shape or another; for example, choosing to draw a naturality diagram for a natural transformation as a square as opposed to a trapezoid, or parallelogram, or circle, etc.

In $2$-dimensional category theory, the choices seem to become less arbitrary; there are diagrams which are 'easy to read' when written in some geometric configurations, and 'difficult to read' in others.

In $3$-dimensional category theory, diagram geometry becomes crucial for clear communication of ideas using diagrams; here the vertices are objects, edges are $1$-cells, faces are $2$-cells, and enclosed $3$-d regions are $3$-cells. (Unless you're working with string diagrams, then reverse the ordering.)

Has there been any work done on a 'systematic' way to 'correctly' write out diagrams in higher category theory?

It's usually possible to figure out a reasonably nice way to write a diagram through trial and error, but the larger the diagram in question (and the more dimensions in play) the longer this trial and error process typically takes. It would be nice if there was some systematic formula one could follow when drawing out higher dimensional diagrams, maximizing the chances of 'readability'. Any pointers are appreciated.

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How to draw diagrams is largely artistic choice.

The one mathematically based rule I can think of is that pullbacks should always be parallelograms.

The point is that pullbacks capture substitution, change of base, etc., so that they translate one part of a diagram into another.

The word translate there has a deliberately double meaning: the linguistic one and (so long as you use parallelograms) the geometrical one.

On second thoughts, this argument applies to naturality squares too.

Other than that, I try always to draw adjunctions vertically, so that the left adjoint is on the left and the right on the right.

OK, so maybe there is more than one rule.

Happy commuting!

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  • $\begingroup$ I agree that diagram drawing is an art form as it stands, and a piece of my soul would wither if there was some algorithm that produced better results than my artistic inclinations, but... change is the only constant, and this seems to be the way things are going (AI art, AI music, etc.). But I like your rules as a fellow human artist ;^). $\endgroup$
    – Alec Rhea
    Commented Aug 4 at 17:38

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