## May 7, 2007

### Klein 2-Geometry VIIIS

#### Posted by David Corfield

Some odd remarks about Klein 2-geometry which occurred to me in slack moments in Amsterdam. (The ‘S’ of the title represents a half.)

2-groups measure the symmetry of a category. 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.

We can also restrict the symmetries of the objects. We might have as objects the vertices of a cube. We might also restrict the symmetry of the arrows. Rather like the symmetry of a tangent bundle is determined by symmetries of the base, whereas this is not the case for a trivial bundle, we might consider three elements fibred above each vertex of a cube whose behaviour corresponds to the way the three adjacent faces of a vertex transform under motions of the cube.

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. Chain maps would follow easily. As for chain homotopies without subtraction available, I could imagine they might exist between identical chain maps, using the designated point as a kind of zero vector.

Kapranov-Voevodsky might look to $(Set_*)^n$. Urs would no doubt look to BiMod($Set_*$).

A vector bundle can be considered a category. Each point of the base is an object, and the elements of a fibre are its arrows with addition as composition.

Now, take a bundle such as the trivial bundle over the sphere with fibre equal to $\mathbb{R}^2$. Consider the cornucopia of figures we could look to preserve: a point on the sphere; a point in the fibre above a particular point; a subspace of the fibre above a particular point; a great circle; a section of the bundle restricted to a circle; a sub-bundle of the bundle restricted to the circle, such as an infinitely wide Möbius strip winding about over the equator; etc.

Presumably each has as stabilizer a sub-2-group of the 2-group of automorphism of the bundle. And presumably the respective quotients give the space (2-space?) of figures of that type. And double quotienty things give incidence relations.

Now, take the category whose objects are points of the Euclidean plane, and each vertex having the group $S_2$ worth of arrows. Then the 2-group of (Euclidean) symmetries will have $E(2)$ worth of objects, and 2 arrows at each object. It’s rather like a double plane, whose symmetries only differ from a single plane by the ability to exchange copies. Then there are different figures to preserve. Not only a point on a plane, but a set of twin points, not only a line, but a set of twin lines, etc. A double quotient of the 2-group by sub-2-groups fixing a point and a line will now not just correspond to the distance between point and line, but also include a binary answer to the question of coplanarity.

Posted at May 7, 2007 1:50 PM UTC

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### Re: Klein 2-Geometry VIIIS

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.

Posted by: urs on May 7, 2007 9:04 PM | Permalink | Reply to this

### Re: Klein 2-Geometry VIIIS

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.)

Posted by: Tim Silverman on May 7, 2007 10:04 PM | Permalink | Reply to this

### Re: Klein 2-Geometry VIIIS

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.

Posted by: David Corfield on May 8, 2007 2:23 PM | Permalink | Reply to this

### Re: Klein 2-Geometry VIIIS

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.

Posted by: John Baez on May 9, 2007 5:31 PM | Permalink | Reply to this

### Re: Klein 2-Geometry VIIIS

Ah, I felt this must be familiar.

Is there a Cayley representation theorem for 2-groups? So that all 2-groups are sub-2-groups of some kind of permutation 2-group?

I suppose a 2-group $G$ is a sub-2-group of the 2-group AUT($U(G)$), where $U(G)$ is the underlying category of $G$.

So a permutation 2-group has the form AUT($C$) for some category $C$?

And if the Yoneda Lemma is a ‘vast generalisation of Cayley’s theorem’, then would the ‘categorified Yoneda Lemma’ mentioned here be a vast generalisation of the construction above?

If all this works, could you have for $H$ a sub-2-group of AUT($C$), and the category Fun($C, A$), where $A$ is a 2-group, a categorified wreath product?

Posted by: David Corfield on May 9, 2007 9:50 PM | Permalink | Reply to this

### Re: Klein 2-Geometry VIIIS

The operation of wreath product corresponds to species substitution (or partitional composition) of species.

Mightn’t we expect the operation of a categorified wreath product to correspond to partition composition of stuff types?

Posted by: David Corfield on May 10, 2007 1:42 PM | Permalink | Reply to this
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