The n-Category Café

The conjectures concern 2-rigs over a field k of characteristic zero. Here they are:

Conjecture 8.6. $\mathsf{Rep}(\text{M}(n,k))$ is the free 2-rig on an object of bosonic subdimension n.

Conjecture 8.7. $\mathsf{Rep}(\text{GL}(n,k))$ is the free 2-rig on an object of bosonic dimension n.

Conjecture 8.8. If an object has bosonic dimension n, then it also has bosonic subdimension n. If an object has fermionic dimension n, then it also has fermionic subdimension n.

However, Coulembier says he needed to fix our definition of ‘bosonic dimension’ to prove some of these conjectures. We had said

• a 2-rig over a field k is a Cauchy complete k-linear symmetric monoidal category

and an object x in a 2-rig

• is a line object if there’s an object y with $x \otimes y \cong I$.

• is a bosonic line object if it is a line object and the symmetry $x \otimes x \to x \otimes x$ is the identity morphism.

• is a fermionic line object if it is a line object and the symmetry $x \otimes x \to x \otimes x$ is minus the identity morphism.

• has bosonic dimension n if its nth exterior power is a bosonic line object.

• has fermionic dimension n if its nth symmetric power is a fermionic line object.

• has bosonic subdimension n if its (n+1)st exterior power vanishes.

• has fermionic subdimension n if its (n+1)st symmetric power vanishes.

However, Coulembier noted that with these definitions, we get some very strange objects of bosonic dimension n. Namely, any fermionic line object has bosonic dimension n for every even natural number n. The reason is that its second tensor power, and thus all its even tensor powers, are bosonic line objects. However, a fermionic line object does not have bosonic subdimension n for any natural number n, since its exterior powers are the same as its tensor powers, and none of these vanish.

So, Coulembier rules out this case: he defines an object to have bosonic dimenion n if its (n+1)st exterior power vanishes and it is not a fermionic line object.

If one does this, one should similarly change the definition of an object with ‘fermionic dimension n’ to rule out bosonic line objects.

It’s great to see some interest in 2-rig theory! Anyone who wants more conjectures to tackle can try the big conjecture in the introduction to our paper, or Conjectures 34–37 in Tannaka reconstruction and the monoid of matrices.

By the way, Todd and I have already proved Conjecture 8.6 in that paper. But we used very different techniques than Coulembier’s, so it’s interesting to see his new proof.