### Higher-Dimensional Algebra VII: Groupoidification

#### Posted by John Baez

Check this out:

- John Baez, Alex Hoffnung and Christopher Walker, Higher-dimensional algebra VII: groupoidification — latest version with all known corrections, arXiv version or published version.

This is a somewhat expanded and improved version of our paper Groupoidification made easy, more suitable for publication. It now includes material on Hall algebras — or, very loosely speaking, quantum groups — and various different recipes for turning spans of groupoids into linear operators between vector spaces.

Comments and corrections are most welcome!

I still haven’t satisfies Urs’ request to include material about groupoidified traces. I’m not quite sure where to put it.

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 three 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 $\mathbb{F}_q$. The third application is to Hall algebras. We explain how the standard construction of the Hall algebra from the category of $\mathbb{F}_q$ representations of a simply-laced quiver can be seen as an example of degroupoidification. This in turn provides a new way to categorify—or more precisely, groupoidify—the positive part of the quantum group associated to the quiver.

Alex Hoffnung will go a lot further with Hecke algebras in his thesis, and Christopher Walker will dig into Hall algebras in his.

## Re: Higher-Dimensional Algebra VII: Groupoidification

Am I right in thinking that the proposition on the bottom of p9 can be broken into

1) a statement about taking the tensor product of two vectors in different vector spaces

2) a groupoidification of “R tensor V is isomorphic to V”

that would make your “here’s how to multiply a vector by a scalar” into “here’s how to tensor two vectors, and if one of them is a scalar, then you can think of the tensor product as living in the original vector space”

?