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Is there a software for writing 3D commutative diagrams?
Homotopy.io is all about higher dimensional diagrams although more surface diagrams, which may not be what you mean. Asymptote is a programming language for vector graphics that has the best capabilities of anything for mixing LaTeX and 3D. But you have to learn a whole programming language. (It’s pretty similar syntax to C.)
What do you mean by "3D commutative diagram"?
I mean writing a 1-categorical diagram but with the shape of a cube for instance.
You can draw such diagrams in quiver, but you will need to add crossing over
by hand.
You can also draw them in @Ambroise's editor — which doesn't appear to have any official name yet — by manually adjusting the ‘z’ level of arrows.