Concerning the idea of groupoidification, which you say involves replacing vector spaces by groupoids and linear maps by spans of groupoids:

spans in some category (like $\mathrm{Set}$) form themselves a weak 2-category.

Spans in some 2-category (like $\mathrm{Grpd}$) should form themselves even a weak 3-category.

For instance, monads in spans in groupoids are nothing but *double groupoids*, I think.

So it would seem that replacing $\mathrm{Vect}$ by $\mathrm{Span}(\mathrm{Grpd})$ is actually an act of categorifying *twice* – unless you want to divide out isomorphisms.

Actually, I am not sure yet that I understand what the special role of groupoids is in this program: why not consider replacing linear maps by spans in arbitrary (finite, probably) categories?

You motivate groupoids as the categorification of non-negative rational numbers, like sets categorify natural numbers.

But we have learned from Tom Leinster that general finite categories generalize this even to possibly negative rational numbers. So if I wanted to categorify a matrix with arbitrary rational entries, I would think of considering spans in finite categories, instead of just in groupoids.

Or not? Why not?

By the way, what I find very inspiring here is this:

over at our discussion of the canonical 1-particle I complained about how it is not clear to me precisely what 2-category should replace $\mathrm{Vect}$ when we think of the 1-particle as coupled to a vector bundle with connection, but keeping in mind that the corresponding parallel transport should be a *pseudo*functor, such that its sections are functors, such that they are subject to Tom’s theorem.

I was thinking about replacing $\mathrm{Vect}_\mathbb{C} = \mathbb{C}-\mathrm{Mod}$ with the 2-category $\mathbb{C}\mathrm{Set}-\mathrm{Mod}$. But maybe I should think of replacing it just with $\mathrm{Span}(\mathbb{C}\mathrm{Set})$.

That would actually nicely harmonize with the fact that the quantum theory obtained from that also involves spans for expressing its linear maps.

That’s something to think about.

## Re: This Week’s Finds in Mathematical Physics (Week 247)

I am really excited by the “Tale of Groupoidification”. Can’t wait for the next TWF!