September 4, 2006

Klein 2-Geometry V

Posted by David Corfield

I had hoped to mark my first appearance in the Café with a striking new contribution to our Klein 2-geometry project. The project began on my old blog back in May, and you can follow it through its twists and turns over the next 3 monthly instalments. I have enjoyed both participating in a mathematical dialogue and, as a philosopher, thinking about what such participation has to do with a theory of enquiry. The obvious comparison for me is with the fictional dialogue Proofs and Refutations written by the philosopher Imre Lakatos in the early 1960s. The clearest difference between these two dialogues is that Lakatos takes the engine of conceptual development to be a process of

conjectured result (perhaps imprecisely worded) - proposed (sketched) proof - suggested counterexample - analysis of proof for hidden assumptions - revised definitions, conjecture, and improved proof,

whereas John, I and other contributors look largely to other considerations to get the concepts ‘right’. For instance, it is clear that one cannot get very far without a heavy dose of analogical reasoning, something Lakatos ought to have learned more about from Polya, both in person and through his books.

So where’s the next great step on the 2-Kleinian project, I hear you ask impatiently. Well, I have to confess that the Aquitainian sunshine, the etangs, the aroma of ripe plums, and, no doubt most significantly, the demands of my children, were not conducive to mathematical research, and so I come here largely empty-handed. I was thinking, however, that it is key to understand how vector 2-subspaces work, e.g., how to sum them, what are complements, etc.

We seemed to have established that 2-vector spaces could be characterised by 2 natural numbers b0 and b1, its Betti numbers. All indications were that a sub-2-vector space would have to have first Betti no greater than b1. Several indications suggested the same holds for the zeroth Betti number.

Two thoughts, then. First, if we require sub 2-vector spaces to have complements, and we knew what happens to Betti numbers when 2-vector spaces are added, this would set us the right way concerning the issue in the previous paragraph. Second, as a sub 2-group we should see what happens when we perform our quotienting operation of a sub-2-vector space acting on the 2-vector space.

Did we ever decide that we must write - sub 2-vector space, rather than 2-vector subspace? Not to speak of pushing the ‘2’ in front of the ‘space’.

Posted at September 4, 2006 10:21 AM UTC

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Re: Klein 2-Geometry V

[see improved version below]

Posted by: David Roberts on September 5, 2006 2:33 AM | Permalink | Reply to this

Re: Klein 2-Geometry V

[A reposting - despite all my previous usage of this interface, I forgot the itex to MathML option. Thanks for the nudge, John]

I posed this question over at David’s blog on what was probably the day he went away, but here it is again for discussion:

In a category, or at least in a topos, a subobject is an equivalence class of monic arrows $A \to B$. How then do we reconcile this with the consideration of sub-2-groups as essentially injective (or whatever other adjective was decided on) arrows $H_2 \to G_2$?

If we want universal properties to work out (a normal sub-2-group being a kernel of a regular epimorphism, say), a sub-2-group should only be defined up to a unique equivalence of categories anyway.

Help!

Posted by: David Roberts on September 6, 2006 1:29 AM | Permalink | Reply to this

Re: Klein 2-Geometry V

Dave Roberts writes:

In a category, or at least in a topos, a subobject is an equivalence class of monic arrows $A \to B$.

This is certainly what people do, and it’s certainly a good idea in many ways, but personally I think of this as a case of decategorification. For any object $B$ in a category, there really is an obvious category of monic arrows into $B$, where the morphisms are commutative triangles. When we decategorify this category - by which I mean take its set of isomorphism class of objects - we get the usual set of subobjects of $B$.

I believe this category of monics into $B$ is always a “preorder”, meaning a category equivalent to a partially ordered set. This makes sense: we have not just a set of all subsets of some set, but a set partially ordered by inclusion. So, there’s some juicy information that fades from view when we decategorify.

Of course the toposophers know how to get this information back in other ways, so I’m not really criticizing them. But, I just like the category of monics, and it’s probably worth keeping it in mind when we categorify everything in sight. How does it work for an object $B$ in an arbitrary 2-category? I don’t know, but maybe it’s a “2-poset” in the sense explained by Mike Shulman here, in the subsection on “Posets, Fibers and Topoi”.

But anyway, you’re right.

If we want universal properties to work out (a normal sub-2-group being a kernel of a regular epimorphism, say), a sub-2-group should only be defined up to a unique equivalence of categories anyway.

You’re right. I tried to factor in what you wrote here when composing my summary of where we stand now.

Posted by: John Baez on September 6, 2006 6:14 AM | Permalink | Reply to this

Re: Klein 2-Geometry V

Let me try to summarize our work on this Klein 2-geometry business. I wouldn’t want to deny our new customers the delight of reading our May, June, July and August conversations on this subject. But, some of them may have jobs - so they may not have time to read all that stuff! And, starting a new group blog is a good moment to take stock of what we’ve done.

Klein 2-geometry is an attempt to categorify Klein’s Erlangen program by replacing groups with 2-groups, which are like groups but categories instead of sets.

The idea of Klein’s original Erlangen program is to study geometries that have some group of symmetries, $G$. If this group acts transitively on some kind of geometrical figure - e.g. points, lines, etc. - the space of figures of this kind is $G/H$, where $H \subseteq G$ is the stabilizer of such a figure: that is, the subgroup of symmetries that preserve it.

To categorify this we’d like to replace $G$ with a 2-group, and replace $H$ with a “sub-2-group”. We then need to define the suitable analogue of the quotient $G/H$, and see in what sense $G$ acts transitively on this.

How do we define a “sub-2-group”?

For groups, we can think of a subgroup of $G$ as an equivalence class of one-to-one homomorphisms

$i: H \to G$

So, let’s try the same for 2-groups! We know what a homomorphism of 2-groups is, but we need to know the right analogue of “one-to-one”. Since 2-groups are categories, and their homomorphisms are functors, one natural guess is to look at homomorphisms

$i: H \to G$

that are one-to-one both on objects and morphisms. Alas, this definition is evil: it is not respected by equivalence of 2-groups.

After pondering this for a while, we realized there are four basic concepts relating to “one-to-one-ness” and “ontoness” for homomorphisms of 2-groups. The trick is to think of a 2-group as a connected pointed space with only $\pi_1$ and $\pi_2$ nontrivial. If this sounds scary, just take your 2-group $G$ and make these definitions:

• $\pi_1(G)$ is the group of isomorphism class of objects of $G$.
• $\pi_2(G)$ is the group of automorphisms of the unit object $1 \in G$.

The result is the same. The point is, just as homotopy groups are preserved by homotopy equivalence, these concepts are preserved by equivalence of 2-groups!

Any homomorphism

$f : H \to G$

between 2-groups induces homomorphisms

$\pi_1(f) : \pi_1(H) \to \pi_1(G)$

and

$\pi_1(f) : \pi_1(H) \to \pi_1(G)$

This lets us write down a quartet of definitions that generalize the twin concepts of “one-to-one” and “onto” from groups to 2-groups:

• $f$ is essentially surjective iff $\pi_1(f)$ is onto.
• $f$ is essentially injective iff $\pi_1(f)$ is one-to-one.
• $f$ is full on automorphisms if $\pi_2(f)$ is onto.
• $f$ is faithful iff $\pi_2(f)$ is one-to-one.

Some secondary definitions are also in common use:

• $f$ is full iff $\pi_1(f)$ is one-to-one and $\pi_2(f)$ is onto (it’s full on automorphisms and essentially injective).
• $f$ is an equivalence iff both $\pi_1(f)$ and $\pi_2(f)$ are one-to-one and onto.

There seems to be some evidence that a sub-2-group should be an equivalence class of homomorphisms

$i : H \to G$

that are faithful ($\pi_2(i)$ one-to-one). Namely, if we let $G$ act on some category, and look at the appropriately defined stabilizer of an object, that’s the kind of sub-2-group the stabilizer is!

On the other hand, we also noticed some evidence saying we want $i$ to be faithful and essentially injective ($\pi_1(i)$ and $\pi_2(i)$ one-to-one). Namely, if we categorify the concept of vector space, we get a concept of “vector 2-space”, which is a special kind of 2-group, just as a vector space is a kind of group. And, if we look at “sub-2-spaces” of a vector 2-space, we see they’re only “smaller” if their inclusion homomorphism is both faithful and essentially injective!

The issue of vector 2-spaces and their sub-2-spaces is important when we categorify projective geometry - a famous special case of Klein’s approach to geometry. So, 2-groups show up in two very different ways in our work, and don’t seem to want the same concept of “sub-2-group”. This got us scared, but I now think it’s okay. We should expect some things become a bit more complex when we categorify them, and this may be one. The point is, we’re no longer living in a world where “one-to-one” and “onto” rule the roost! These two concepts have split into four, so life is bound to be a bit more interesting.

One thing I’d enjoy doing is marching ahead and working out the theory of projective 2-geometry. We need some concrete examples under our belt. Anyone interested?

Posted by: John Baez on September 5, 2006 12:45 PM | Permalink | Reply to this

Re: Klein 2-Geometry V

One thing I’d enjoy doing is marching ahead and working out the theory of projective 2-geometry. We need some concrete examples under our belt. Anyone interested?

“Of course we are!”, there came the resounding reply.

What do we need to find out first? What do the Betti numbers tell us about what happens when we form the projective 2-space from a 2-vector space?

Posted by: David Corfield on September 5, 2006 2:05 PM | Permalink | Reply to this

Re: Klein 2-Geometry V

David almost wrote:

What do we need to find out first? What do the Betti numbers tell us about what happens when we form the projective 2-space from a vector 2-space?

Exactly. A vector 2-space $V$ is secretly the same as a 2-term chain complex of vector spaces, so it’s characterized up to equivalence by two numbers: the dimensions of its 0th and 1st homology groups, otherwise known as the Betti numbers $b_0(V)$ and $b_1(V)$.

I’m pretty sure that if we have vector 2-space and natural numbers $b_0' \le b_0(V)$ and $b_1' \le b_1(V)$, we can find a vector 2-subspace in the strong sense (inclusion injective on both homology groups) whose Betti numbers are $b_0'$ and $b_1'$. This allows us to ponder the 2-Grassmannian of such 2-subspaces.

The simplest case of a Grassmannian is a projective space, so we can start by categorifying that case ($b_0' = 1$, $b_1' = 0$), trying to really see what it looks like.

So, we should get a 2-parameter family of projective 2-spaces, depending on natural numbers $b_0$ and $b_1$.

Posted by: John Baez on September 5, 2006 4:22 PM | Permalink | Reply to this

Re: Klein 2-Geometry V

So for a nice concrete example, we might look at the projective 2-space of (1,0) sub-2-spaces of, say, (3,1). Does this gives us a 2d projective space of objects, each with a 1d space of automorphisms?

Posted by: David Corfield on September 5, 2006 5:25 PM | Permalink | Reply to this

Re: Klein 2-Geometry V

David writes:

So for a nice concrete example, we might look at the projective 2-space of (1,0) sub-2-spaces of, say, (3,1). Does this gives us a 2d projective space of objects, each with a 1d space of automorphisms?

Great! We’re getting in gear now…

But, if you don’t mind me switching into reverse for a moment, let me just check for backwards-compatibility: always a good thing when you’re categorifying. What about the dimension (1,0) sub-2-spaces of $k^{(n,0)}$?

Nota Bene: Some new notation is cropping up here. We say a 2-vector space has dimension $(p,q)$ if its 0th Betti number is $p$ and its 1st Betti number is $q$. The simplest 2-vector space of this dimension looks as follows when viewed as a chain complex:

$k^{p} \stackrel{0}{\leftarrow} k^{q}$

where $k$ is our ground field, and we call this guy $k^{p,q}$. Every 2-vector space of this dimension is equivalent, though not isomorphic, to this one.

The vector 2-space $k^{n,0}$ is just a fancy new way of talking about the vector space $k^n$: since it only has identity morphisms, it’s a category that might as well be a set!

A subspace of dimension $(p,0)$ sitting inside $k^{n,0}$ is just the same as a $p$-dimensional subspace of $k^n$. Moreover these subspaces have no interesting automorphisms (I bet).

So, the 2-Grassmannian of $(p,0)$-dimensional subspaces of $k^{n,0}$ is just the usual Grassmannian of $p$-dimensional subspaces of $k^n$, decked out in fancy new clothes!

Good. Okay, now let’s switch back to first gear and move forwards slowly.

What’s a $(1,0)$-dimensional subspace of $k^{3,1}$? I’m going to jump the gun a bit and guess it’s any sub-chain-complex of $k^{3,1}$ (viewed as a chain complex) whose dimension is $(1,0)$.

This is just the same as a 1-dimensional subspace of $k^3$. In other words, it’s a point in the projective plane $kP^2$.

These guys are the objects of the projective 2-space David wants us to look at. We also need to figure out the morphisms. For those, I should probably not have jumped the gun quite so fast: I should have described a $(1,0)$-dimensional sub-2-space of $k^{3,1}$ in terms of a chain map

$i: k^{1,0} \to k^{3,1},$

namely an “inclusion”. This lets me study chain homotopies between such inclusions. The chain homotopies between $i$ and itself should (probably) be the automorphisms of the given point in our projective 2-space.

Okay - if this is what we should do, it’s easy to see what we get: chain homotopies are simple when the differentials in our 2-term chain complexes vanish! For one thing, a chain homotopy can only go from from a chain map to itself. For another thing, a chain homotopy is just any map from 0-chains of the source to 1-chains in the target.

So, a chain homotopy from $i$ to itself boils down to any map

$h: k^1 \to k^1$

from 0-chains in $k^{1,0}$ to 1-chains in $k^{3,1}$.

So, we get a line’s worth of automorphisms of any point in our projective 2-space.

Verdict: when we form the projective 2-space associated to $k^{3,1}$, its objects form the projective plane $kP^2$, and each object has a line’s worth of automorphisms. There are no morphisms between different objects: we’re working with a “skeletal” version of our projective 2-space.

So, David is right - with much less work than it took me! Perhaps some clever person can guess the precise description of the projective space associated to $k^{p,q}$.

Posted by: John Baez on September 6, 2006 5:43 AM | Permalink | Reply to this

Re: Klein 2-Geometry V

So, David is right - with much less work than it took me!

That’s because I cunningly used the other formulation you gave us $(V - \{0\})//Disc(k^*)$ way back when, where $(V - \{0\})$ has the component connected to 0 removed from $V$.

That presumably also helps us describe the projective space associated to $k^{p,q}$, as $kP^{p-1}$ worth of objects each with $k^q$ worth of automorphisms.

When we move on to subspaces of the form (2, 0) and higher, is there an alternative formulation like the above, or must we go through the chain map route?

Then we can move on to (0,1) sub-2-spaces of (0, $n$) 2-spaces.

Posted by: David Corfield on September 7, 2006 8:41 AM | Permalink | Reply to this

Re: Klein 2-Geometry V

John wrote:

So, David is right - with much less work than it took me! David wrote:

That’s because I cunningly used the other formulation you gave us $(V - \{0\})//Disc(k^*)$ way back when, where $(V - \{0\})$ has the component connected to 0 removed from $V$.

Good! I completely forgot that other description of projective 2-spaces, which is more like “a vector space minus the origin, mod scalar multiplication”. The one I was using now is like “the space of all lines”. It’s nice that they agree.

That presumably also helps us describe the projective space associated to $k^{p,q}$, as $kP^{p-1}$ worth of objects each with $k^q$ worth of automorphisms.

Right! The space of morphisms in our projective 2-space form a vector bundle over the space of objects, which is a projective space. Indeed, any vector bundle gives a 2-space in this way! Unfortunately for the connossieur of vector bundles, this particular one is a trivial bundle.

When we move on to subspaces of the form (2, 0) and higher, is there an alternative formulation like the above, or must we go through the chain map route?

I think we need the chain map route. But, that method actually isn’t much work. I was mainly just kidding about how I did a calculation, while you simply stated the answer in oracular fashion.

Okay - so for my next trick, I should describe all the categorified Grassmanians. Then we can consider incidence relations, like “a point lies on a line”.

But, right now I need to do some actual work - turns out I’d latexed up a paper for a conference proceedings using the wrong stylefile; have to do it again.

Posted by: John Baez on September 7, 2006 9:08 AM | Permalink | Reply to this

quasi isomorphisms

Above, the idea is that every $n$-term chain complex is equivalent to its cohomology chain complex (all whose morphisms are the zero morphism).

How does this relate to the crucial fact in the context of the derived category of chain complexes, that there exist quasi-isomorphisms, namely morphisms of chain complexes which are not isomorphisms, but do become isomorphisms when sent to cohomology?

Is there some interesting relation between quasi-isomorphisms and those equivalences that are not isomorphisms?

Posted by: urs on October 1, 2006 10:34 PM | Permalink | Reply to this

Re: Klein 2-Geometry V

Unfortunately for the connoisseur of vector bundles, this particular one is a trivial bundle.

There’s been a principal bundle and gauge theory feel to our discussions all along. How will we feel if all this Kleinian 2-geometry turns out to be some known chapter of differential geometry?

A while ago we saw that categorified 2d Euclidean space was $E(2) \leftarrow 1//P \leftarrow 1$ in the notation we used ($P$ being the stabilizer of a point). Presumably real projective 2-space is $PGL(3,R) \leftarrow 1//Q \leftarrow 1$, $Q$ the stabilizer of a point. Presumably things get more interesting when we replace the 1 by something more exciting. There ought to be such a replacement which corresponds to the projective space associated with $R^{3,q}$.

Posted by: David Corfield on September 7, 2006 9:40 AM | Permalink | Reply to this

Re: Klein 2-Geometry V

David wrote:

How will we feel if all this Kleinian 2-geometry turns out to be some known chapter of differential geometry?

There’s no way that all this stuff can turn out to be “just” something already known - the outlook is too new. If it’s related to a bunch of stuff in differential geometry, that’s good: more reason to study it! If some parts wind up reducing to known stuff, we turn to other parts. If somehow miraculously it all winds up seeming too easy, we just turn up the dial a notch and look at Klein 3-geometry!

It’s interesting to hear you worrying a bit like this. I guess I can understand your worries, but it’s way too late for me to share them. About 5 years ago I told Dan Freed I was working on 2-bundles, a kind of a generalization of gerbes where the structure 2-group wasn’t abelian. He asked me for “examples occuring in nature” - meaning within known math - and I admitted I didn’t have them yet. He told me that it was probably not a good idea then. I got worried, but I stubbornly persisted. Now it’s clear that 2-bundles and 3-bundles are lurking all over string theory and M-theory.

Note how easy it is to get discouraged if you take the wrong attitude: either your ideas are “just something known already” or else they’re “not found in nature” - either way you’re in trouble!

It’s certainly easy to have dumb ideas. But when an idea has a solid foundation, it’s almost bound to lead somewhere unless you really bungle it. Klein geometry: good. Categorification: good. Groups naturally categorify to 2-groups, with lots of nice examples: good. Klein geometry is about differing degrees of structure, and categorifying structure gives stuff: good. So how could we possibly not do well by categorifying Klein geometry? It would be like failing to find sand in the Sahara!

I’m not saying we’ve found the really interesting examples yet. This projective 2-geometry stuff may be too wimpy to be deeply exciting… we’ll see. But sometimes it takes a while to find the really good examples. People were looking for nice noncommutative, non-cocommutative Hopf algebras for decades before they found quantum groups.

The other reason for my deep lack of worry is that categorification is a very broad project, like a huge tidal wave hitting the whole length of the shoreline. If certain parts don’t advance as fast as others, it’s really no big deal.

Posted by: John Baez on September 7, 2006 1:50 PM | Permalink | Reply to this

Re: Klein 2-Geometry V

The other reason for my deep lack of worry is that categorification is a very broad project, like a huge tidal wave hitting the whole length of the shoreline.

And The n-Category Café is the Tsunami Warning System?

Posted by: David Corfield on September 7, 2006 2:01 PM | Permalink | Reply to this

Re: Klein 2-Geometry V

And The n-Category Café is the Tsunami Warning System?

I thought what we are trying to do is this.

Posted by: urs on September 7, 2006 2:16 PM | Permalink | Reply to this

Re: Klein 2-Geometry V

I’m reassigning this comment to the main flow of the river rather than as part of a tributary.

John wrote:

I think we need the chain map route. But, that method actually isn’t much work.

Oh yes, I see. It looks like chain homotopies again correspond to any map between 0-chains of the source and 1-chains of the target. So the 2-space of ($n$, 0) sub-2-spaces of $k^{p,q}$ has the grassmanian $G_{n,p}$ as objects and linear maps from $k^n$ to $k^q$ as automorphisms.

While we’re at it, isn’t then the set of $(a,b)$ sub 2-spaces of $k^{c,d}$ just $G_{a,c} \times G_{b,d}$ of objects each with linear maps $k^a$ to $k^d$ worth of automorphisms?

Posted by: David Corfield on September 8, 2006 11:41 AM | Permalink | Reply to this

Re: Klein 2-Geometry V

Here’s something I’d like to check before proceeding. John wrote:

I should have described a $(1,0)$-dimensional sub-2-space of $k^{3,1}$ in terms of a chain map $i: k^{1,0} \to k^{3,1},$ namely an “inclusion”.

Does this take into account that a scalar multiple of an inclusion of $k^p$ into $k^q$ corresponds to the same subspace of $k^q$? We seem to be implicitly factoring out such equivalence of maps when we look at 2-Grassmanians, but have we then been careful enough with those chain maps going from 0-chains of the source to 1-chains of the target? Are those which differ by a scalar factor different?

John also wrote:

The space of morphisms in our projective 2-space form a vector bundle over the space of objects, which is a projective space. Indeed, any vector bundle gives a 2-space in this way! Unfortunately for the connoisseur of vector bundles, this particular one is a trivial bundle.

If we try to find the symmetry 2-group of this bundle, we have to respect the projective-ness of the space of objects. What kind of map must a map of fibres be?

Posted by: David Corfield on September 10, 2006 10:18 AM | Permalink | Reply to this

Re: Klein 2-Geometry V

In The heritage of S.Lie and F.Klein: Geometry via transformation groups, E. Ortacgil asks:

What happened to Erlanger Programm? The main reason for this question is that the total space $P$ of the principal bundle $P \rightarrow M$ does not have any group-like structure and therefore does not act on the base manifold $M$. Thus $P$ emerges in this framework as a new entity whose relation to the geometry of $M$ may not be immediate and we must deal with $P$ now seperately. Consequently, it seems that the most essential feature of Klein’s realization of geometry is given up by this approach. Some mathematicians already expressed their dissatisfaction of this state of affairs in literature with varying tones…

Posted by: David Corfield on September 20, 2006 9:59 AM | Permalink | Reply to this

Re: Klein 2-Geometry V

David mentioned:

The heritage of S.Lie and F.Klein: Geometry via transformation groups

I wasn’t aware before that people wished to phrase all of geometry in terms of fixed points of group actions. I also wasn’t aware that, as the authors of the above article say, principal bundles were invented with the desire to unify Riemannian and Kleinian geometry.

At first sight, the “unpleasant question” the authors mention on the top of p.2 seems to indicate that the wish to describe everything in terms of the Erlangen program is simply misguided.

Can you give some indications why one would expect that this is not so?

Posted by: urs on September 20, 2006 10:13 PM | Permalink | Reply to this

Re: Klein 2-Geometry V

Thinking of principal bundles as extensions of transitive groupoids probably helps here:

$P \times_G G \to P \times_G P \to M \times M$ (this is just the exponentiated Atiyah sequence)

One `problem’ in dealing with groups and groupoids is that people expect groups, thought of as groupoids, to act like groups. This is certainly not the case if we compare a nonabelian group $G$ and the groupoid $G \to *$ : the latter is not a 2-group (this is where Giraud tripped up). This doesn’t mean we can’t use $G \to *$ as we would - it just means we might as well use general groupoids.

Posted by: David Roberts on September 21, 2006 2:37 AM | Permalink | Reply to this

exponentiated Atiyah sequence

David Roberts wrote:

(1)$P \times_G G \to P \times_G P \to M \times M$

(this is just the exponentiated Atiyah sequence)

Hey, that’s what I said here - without having (ever bothered to work out the computation necessary for) the proof. (See the item list after “I believe it is known…”)

Do you have a reference for this?

Posted by: urs on September 21, 2006 12:29 PM | Permalink | Reply to this

Re: exponentiated Atiyah sequence

urs schreibt:

Do you have a reference for this?

I don’t have a reference, but it is an old result of Ehresmann - try the complete works ;)

Posted by: David Roberts on October 3, 2006 8:31 AM | Permalink | Reply to this

Re: Klein 2-Geometry V

David quoted Ortacgil as saying:

What happened to Erlanger Programm? The main reason for this question is that the total space $P$ of the principal bundle $P \to M$ does not have any group-like structure and therefore does not act on the base manifold $M$. […] Consequently, it seems that the most essential feature of Klein’s realization of geometry is given up by this approach. Some mathematicians already expressed their dissatisfaction of this state of affairs […]

The best attempt to tackle this problem is very old and sadly unappreciated: it’s called “Cartan geometry”, and it was actually developed by Cartan before Ehresmann invented the principal bundle formalism that Ortacgil is worrying about. Cartan geometry is really a very subtle way of stretching the Klein geometry idea to let it handle geometries that are “bumpy”, like those in general relativity, rather than perfectly “homogeneous” like those in special relativity. Ehresmann’s formalism of principal bundles, while utterly wonderful in its own way, is in another sense a watered-down version of Cartan’s work.

My student Derek Wise is doing his thesis on Cartan geometry and quantum gravity, and he’s been talking to James Dolan on a weekly basis for almost a year now. So, we’ve made a lot of progress in understanding Cartan geometry, and in a few months Derek should come out with a paper about this.

I don’t know if Derek will dare say this in his paper, but just to whet your appetite: Cartan geometry is about torsoroids. Alas, these “torsoroids” are not as to groupoids as torsors are to groups - they’re not a result of systematic process of oidization - they’re just Jim’s silly name for some gadget that has a bit less structure than a torsor.

Posted by: John Baez on September 23, 2006 4:40 PM | Permalink | Reply to this

Re: Klein 2-Geometry V

So after Klein 2-geometry, if we ever get there (insert appropriate emoticon), we should then do Cartan 2-geometry. By the way, I’d love to hear if James has any thought on what we’ve been doing here.

Posted by: David Corfield on September 24, 2006 12:42 PM | Permalink | Reply to this
Read the post Klein 2-Geometry VI
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Excerpt: Resuming the categorification of Kleinian Geometry
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