The Unapologetic Mathematician

(Not) The Tensorator for Span 2-categories

Part of the disappointment I mentioned is that the road I was on just looked so pretty. I’ve said in various places that I agree with (what I understand to be) David Corfield’s view of mathematics as a process of telling good stories, and this was a great story, but unfortunately it just doesn’t quite ring true. Before I purge it, I want to show you the picture of the tensorator as I thought it would work.

Across the top are two tensor products of one span and one object each, and across the bottom are the other two, giving the compositions in both orders. The squares (that look like triangles) at the top and bottom are pullbacks, giving the actual composite spans. Then we can put the tensor product $F\otimes G$ in the middle, and get arrows up and down from the universal properties of the pullback squares. And it even looks like a big tensor product symbol!

But ultimately there’s no way to make this span we get always be unitary, or even invertible. And all the pretty pictures in the world can’t save a deeply flawed story. Just ask Michael Bay.