The n-Category Café

Urs would no doubt look to $\mathrm{BiMod}(\mathrm{Set}_*)$.

I would indeed, at least if I were not fully absorbed by thinking about Baez-Crans $n$-vector spaces at the moment!

(Namely we are finally making some progress in understanding higher morphisms of Lie $n$-algebras, it seems. As always, once you see the solution it appears so very obvious…)

But back when I was still in quantization mode (currently I am in write-up-classical-mode), I did indeed think about $\mathrm{Set}-\mathrm{Mod}$ a bit in the context of the canonical quantization of the 1-particle.

Recall, there the idea was to replace numbers by sets in order to be able to write the path integral as a colimit.

And just as we have a canonical inclusion $$\mathrm{BiMod}(\mathrm{Vect}) \hookrightarrow \mathrm{Vect}-\mathrm{Mod}$$ we have a canonical inclusion $$\mathrm{BiMod}(\mathrm{Set}) \hookrightarrow \mathrm{Set}-\mathrm{Mod}$$ unless I am mixed up.

(I hope I find the time to get back into quantization mode soon. That “canonical quantization” thing was really thrilling. Not sure how anyone else felt about this (was probably unreadable, I know), but when I got that relation between the Leinster-measure of the binary tree and the exponentiated Laplace operator obtained from pushing forward I felt there was something going on there…)

Anyway, that’s why I was interested in $\mathrm{BiMod}(\mathrm{Set})$, or variants of that.

When I saw you guys discussing modules for the field with one element recently I got the vague impression that this might be relevant to these quantumly considerations from last winter (maybe that would help get a better understanding of what it means to regard functions as “bundles of numbers”), but I am not sure yet.

P.S.

By the way, recently it seemed that I made some progress with persuading some experts to attack the first $n$-Café millenium prize.

David wrote:

Now it might be worth considering what is a $2$-vector space over $F_1$. That would depend on which version of $2$-vector space one chose. Baez-Crans would presumably look at $2$-term chain complexes. These would be functions between pointed sets preserving the point.

Wouldn’t that be an arbitrary category with a distinguished object? (The distinguished morphism would be its identity. That would follow from the fact that the source, target and identity functions all have to preserve the distinguished point.)

I guess we might also have taken a look at orbifolds as spaces whose symmetries form 2-groups. A paper by Eugene Lerman defines things in such a way that the relevant 2-groups are strict.

That oughtn’t to be too hard to think up figures to preserve in a space such as a cone, and construct the space of such figures as a 2-group quotient.

David wrote:

Perhaps the simplest type of non-trivial, non-discrete category have a finite number of objects, and a copy of an identical group, G, of arrows at each object, and no other arrows. The 2-group of symmetries for this is $disc(S_n) \cdot (AUT(G))^n$, where $AUT(G)$ will have as 2-arrows the natural transformations which correspond to relevant conjugacy relations, and ‘$\cdot$’ is a kind of (semi-direct) product.

I sense a mild discomfort in this sentence. I know the feeling. It takes time to learn to love wreath products, which is what we’re seeing here.

The idea is simple: if someone hands you $n$ identical regular tetrahedra, the symmetry group of this collection is the wreath product of the permutation group $S_n$ and the symmetry group of the tetrahedron.

In general, if you have a group $G$ and a subgroup $H \subseteq S_n$, the wreath product $H \wr G$ is just the semidirect product of $H$ and $G^n$, where $H$ acts on $G^n$ by permuting the factors.

Here you are inventing a wreath product of 2-groups — or more precisely, of the group $S_n$ and the 2-group $\AUT(G)$.

It would be fun, if possible, to invent a more categorified wreathe product where instead of $S_n$ or any subgroup of the permutations of a set, we used any sub-2-group of the automorphism 2-group of some category!

Lots more to say, but I gotta run to my weekly meeting with Derek and Jeff. Their thesis clock is ticking loudly.