### 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($Set_*$).

Some earlier comments:

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.

## Re: Klein 2-Geometry VIIIS

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.