The n-Category Café

Picturing Morphisms of 3-Functors

Posted by Urs Schreiber

I knew this would happen one day - sooner or later: I would be in need of differentiating a smooth 3-morphisms of a smooth 3-functor.

Now, with realizing non-fake flat 2-transport in terms of 3-functors ($\to$), that day has finally come.

(And I always thought I wouldn’t need that until seriously tackling Chern-Simons theory 3-functorially (cf. $\to$)).

Certainly, somewhere out there is a text which has everything about 1-, 2- and 3-morphisms of 3-functors in it that I would ever need. But there is nothing like doing it yourself. Plus, I need LaTeX templates for these 3-dimensional diagrams for my own use. Last not least, I might want to refer to this in some of my future entries on $n$-transport ($\to$) - which I am sure you are all looking forward to.

So I spent the better part of this evening with drawing 3D-diagrams. Or rather, programming LaTeX such as to draw these diagrams.

This takes a while, and of course I ran out of time before drawing the last diagram, that for 3-morphisms of 3-functors.

But anyway, I thought before going into the weekend I’d share some of the pictures.

Find seven pages of higher-order gauge transformations here:

$\;\;\;\;$ Morphisms of 3-Functors

Physicists are invited to think of everything 2-dimensional in there as an image of internal degrees of freedom of a charged string, and of everything 3-dimensional in there as an image of internal degrees of freedom of a charged membrane.

Comments are welcome. (I hope I don’t have any too obvious blunders in there, since at the end I was a little bit under time pressure.)