### Groupoidification Made Easy

#### Posted by John Baez

Merry Christmas! It’s still Christmas here in California, despite what the time stamp on this blog may say. So, it’s not too late for one last present! Here’s one just for you, from Santa and his elves:

- John Baez, Alex Hoffnung and Christopher Walker, Groupoidification made easy.

You’ve probably heard me mutter and rhapsodize about groupoidification. I’ve been doing it for years — in fact, I have an entire webpage devoted to such rhapsodic mutterings.

But now, finally, you can learn the basic idea in an actual math paper that contains actual *proofs* of the most basic results! This is just the beginning of a much bigger story — but at least it’s a start.

Here’s the abstract:

Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of ‘degroupoidification’: a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present two applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter $q$ is a prime power. We illustrate this with the simplest nontrivial example, coming from the $A_2$ Dynkin diagram. In this example we show that the solution of the Yang–Baxter equation built into the $A_2$ Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field $\mathbb{F}_q$.

Be the first on your block to groupoidify something. Make a New Year’s resolution to do it! Ho-ho-ho!

## Re: Groupoidification Made Easy

Fascinating!

What happens with more general tensors than just vectors and linear operators, I wonder? Many-legged spans?

There doesn’t seem to be much to asymmetrically distinguish the two legs of a span in the case of linear operators to tell me which came from which half of $V \multimap V = V \otimes V^*$, but certainly tensor contraction seems to come out nicely as weak pullback for both the degroupoidification of matrix multiplication and the inner product.