Klein 2-geometry
As a small experiment in collective, public thinking, I'm going to devote a post to the attempt to categorify Kleinian geometry, and update the date so it doesn't slip off the radar of 'Previous Posts'.
Update: I'm very happy for it to be more collective. Your comments are welcome.
So let's see what we have so far:
DC: Here's a thought: gaining new insights by categorifying the very simplest entities seems a good way to bring in the punters. You've treated various kinds of number, natural, rational, etc. [E.g., From finite sets to Feynman diagrams] Now, if you ask anyone what else they first learned at school, they'll say 2-dimensional Euclidean geometry (if they're old enough to have been taught properly). Where, they may ask, are categorified lines and circles? If all interesting equations are lies, what of Pythagoras' theorem? Or, when we say the intersection of any pair of altitudes of a triangle is the *same* point as that of any other, is there room for weakening?
I suppose you might give two responses:
1) Euclidean geometry although it came first is actually a complicated affair. First you need to categorify a stripped down 'geometry' such as differential topology.
2) OK. 2d-Euclidean space is a homogeneous space, the points corresponding to cosets of the quotient of the Lie group of Euclidean transformations by the stabilizer of a point. All we need is a Lie 2-group version.
Either way what prevents a Erlangen program for 2-groups?
JB (John Baez): Hi -
> Where, they may ask, are categorified lines and
> circles?
Interesting idea; one could take it in various directions, I suppose.
> 1) Euclidean geometry although it came first is actually a
> complicated affair. First you need to categorify a stripped
> down 'geometry' such as differential topology.
Mainly you need to see where the categories are, so you can see if there are interesting n-categories lurking beneath them.
> 2) OK. 2d-Euclidean space is a homogeneous space, the
> points corresponding to cosets of the quotient of the Lie
> group of Euclidean transformations by the stabilizer of a
> point. All we need is a Lie 2-group version.
> Either way what prevents a Erlangen program for 2-groups?
Nothing! Especially since the Erlangen program is just the flip side of Galois theory: (see especially the slide about the icosahedron), and Galois theory has already been n-categorified to powerful and still growing effect.
But you're right - nobody seems to have thought hard about Klein geometry with Lie 2-groups (or higher) replacing Lie groups. Somehow people have skipped straight to categorifying principal bundles, even though principal bundles are a stripped-down way of thinking about Cartan geometries, which generalize Klein geometries! Sometimes ontogeny fails to recapitulate phylogeny. So, maybe the "punters" should be handed a nice specific Lie 2-group, some 2-spaces on which it acts, and be asked to study the "incidence relations" between these figures. Incidence geometry could be given a whole new lease on life!
Best, jb
DC:
In a pleasanter world I'd be funded to think longer about such things. For those with the leisure time, you can read about incidence geometry at TWF 178, which treats incidence relations in projective geometry in terms of the Dynkin diagrams An. Perhaps we should start with projective rather than Euclidean geometry. Is there an obvious candiate for a Lie 2-group one step up from SL(n, C)? What then is projective 2-geometry? What are Dynkin 2-diagrams?
Or doing things axiomatically, perhaps we can categorify the axioms of projective geometry, such as those for the projective plane, taken from week 145:
A) Given two distinct points, there exists a unique line that both points lie on.
B) Given two distinct lines, there exists a unique point that lies on both lines.
C) There exist four points, no three of which lie on the same line.
D) There exist four lines, no three of which have the same point lying on them.
TL (Tom Leinster): David asks (or rather, has a hypothetical character ask) what categorified lines and circles are. John points out that this could be taken in various directions. Here's a possible beginning of an answer.
We have to decide which aspects of lines and circles we're interested in. Let's treat them as metric spaces. Then the "categorified circle" should be a categorified metric space. OK, so what's a categorified metric space?
Here we can follow Lawvere, who's done a lot to develop the thesis that everything is a category. (I'm exaggerating, but see the first page proper of his metric spaces paper.) If a thing can be regarded some kind of category, that increases our chances of being able to perform some useful sort of categorification. This is ironic, as Lawvere seems to disapprove of categorification...
Anyway, Lawvere interprets metric spaces as being categories enriched in R, the poset of non-negative reals ordered by >=, made monoidal by its additive structure. So it looks as if our task is to categorify R - for then a categorified metric space could be defined as a category (weakly) enriched in the categorified R.
As far as I know, there's no really compelling answer yet to "what is the categorification of the reals?"
DC: Could you also recapitulate Descartes' coordinate-based approach to geometry? With categorified reals you could carve out subcategories of R^2 in terms of isomorphisms such as X^2 + Y^2 is isomorphic to 1. Seems like you wouldn't be too far from Joyal's species.
JB:
> Is there an obvious candidate
>for a nice specific Lie 2-group?
I list a bunch of nice 2-groups in HDA5, and if I were going to categorify Klein geometry I would I would look at a bunch in parallel. The fun of course is seeing the effect and significance of the 2-morphisms (sorry, now I'm thinking of a 2-group as a 1-object 2-groupoid). We've got the ordinary groups, with no 2-morphisms to speak of; then we've got the 2-groups with only one morphism and an abelian group of 2-morphisms.In between these extremes, and more interesting, are groups built from a group G acting on an abelian group A; a great example is the "Euclidean 2-group" where G = SO(n), A = R^n. Or, it might be nice to let G be the whole Euclidean group and A some abelian group on which it acts: this would decategorify to the Euclidean group.
DC:
> I list a bunch of nice 2-groups in HDA5, and if I were
> going to categorify Klein geometry I would I would look
> at a bunch in parallel. The fun of course is seeing the effect and
> significance of the 2-morphisms (sorry, now I'm thinking
> of a 2-group as a 1-object 2-groupoid).
I misread this when I first read it, but even the misreading raised a question. I thought you were talking about 2-morphisms BETWEEN 2-groups (should be 3-morphisms I suppose). Back on the level of Kleinian 1-geometry, what role could group homomorphisms play in the Erlangen program? I guess what's already treated is group inclusions, e.g., projective transformations within affine transformations within Euclidean transformations. But is there scope for more interesting homomorphisms?
JB: Good point. Sure! Each group determines, or we could say "is", a Klein geometry, but groups form a category - in fact a 2-category, since groups are categories - so we get a 2-category of Klein geometries.
Ignoring the 2-morphisms for a moment, though they're interesting and important, let's think about the morphisms: group homomorphisms, viewed as morphisms between Klein geometries.
The examples you mention are already very interesting, because they show how geometry is a unified subject, not just a bunch of isolated "geometries". They show that including a little group in a big one can be seen as making a geometry "more flexible", by adding new transformations.
But, there's another way inclusions of groups show up: in Klein geometry, a "figure" is more or less given by the subgroup that stabilizes it. But, a subgroup is an inclusion of groups! So, the examples you listed of group inclusions can also be seen as "figures"! The most famous example is getting affine geometry by taking projective geometry and restricting to the transformations that stabilize a "point at infinity", or "line at infinity", or...
I hadn't noticed this dual viewpoint, for some reason.
Anyway, some other examples come from outer automorphisms of groups: the outer automorphism of SL(n) is the "duality" that switches points and lines, and for Spin(8) one has 3! acting as outer automorphisms, called "triality". For simple Lie groups one can read these off from the Dynkin diagramn
symmetries.
There are also inner automorphisms, which are just "changes of reference frame". The difference between inner and outer automorphisms is nicely handled by the 2-categorical structure of Grp. (Ever figured out what a natural transformation between a functor betwen groups is?)
This leaves the epimorphisms of groups: quotient maps. When you mod out the Euclidean group by the translation subgroup, you get the rotation group. What does this short exact sequence mean for the corresponding 3 Klein geometries?
This is a lot of fun, and it will be even more fun to categorify.
JB:
If you want to keep talking about this, maybe you should pick a 2-group, and I'll tell you about it, and then we can start trying to develop the corresponding "Klein 2-geometry". Or, we could work in abstract generality. Or both: lots of generalities, but with a key example or two to light the way. That's usually what I do.
Of course the fun will start as soon as we try to generalize the concept of "figures" familiar from ordinary Klein geometry. Ordinarily, a "type of figure" is a subgroup H of our symmetry group G, and the space of figures of that type is G/H. So, the concepts of "sub-2-group" and "categorified quotient space" will soon need some clarification. And, we'll soon start wondering if in addition to "figures" there are some new "2-figures", or "morphisms between figures", or something....
For this it'll be good to describe the theory of ordinary Klein geometry quite clearly using category theory, so we can see how it categorifies.
DC: The "Euclidean 2-group" where G = SO(n), A = R^n looks like a good candidate. So, we need to find sub-2-groups and work out what their quotient 2-spaces are. A couple of obvious cases are G = SO(n), A = {0}, and G = {id}, A = R^n. And then there's G=SO(n-1), A = R^n. Hmm, have you worked out the theory of 2-cosets yet? Or is that co-2-sets, which I suppose ought to be cocategories, but is already taken.
Update: I'm very happy for it to be more collective. Your comments are welcome.
So let's see what we have so far:
DC: Here's a thought: gaining new insights by categorifying the very simplest entities seems a good way to bring in the punters. You've treated various kinds of number, natural, rational, etc. [E.g., From finite sets to Feynman diagrams] Now, if you ask anyone what else they first learned at school, they'll say 2-dimensional Euclidean geometry (if they're old enough to have been taught properly). Where, they may ask, are categorified lines and circles? If all interesting equations are lies, what of Pythagoras' theorem? Or, when we say the intersection of any pair of altitudes of a triangle is the *same* point as that of any other, is there room for weakening?
I suppose you might give two responses:
1) Euclidean geometry although it came first is actually a complicated affair. First you need to categorify a stripped down 'geometry' such as differential topology.
2) OK. 2d-Euclidean space is a homogeneous space, the points corresponding to cosets of the quotient of the Lie group of Euclidean transformations by the stabilizer of a point. All we need is a Lie 2-group version.
Either way what prevents a Erlangen program for 2-groups?
JB (John Baez): Hi -
> Where, they may ask, are categorified lines and
> circles?
Interesting idea; one could take it in various directions, I suppose.
> 1) Euclidean geometry although it came first is actually a
> complicated affair. First you need to categorify a stripped
> down 'geometry' such as differential topology.
Mainly you need to see where the categories are, so you can see if there are interesting n-categories lurking beneath them.
> 2) OK. 2d-Euclidean space is a homogeneous space, the
> points corresponding to cosets of the quotient of the Lie
> group of Euclidean transformations by the stabilizer of a
> point. All we need is a Lie 2-group version.
> Either way what prevents a Erlangen program for 2-groups?
Nothing! Especially since the Erlangen program is just the flip side of Galois theory: (see especially the slide about the icosahedron), and Galois theory has already been n-categorified to powerful and still growing effect.
But you're right - nobody seems to have thought hard about Klein geometry with Lie 2-groups (or higher) replacing Lie groups. Somehow people have skipped straight to categorifying principal bundles, even though principal bundles are a stripped-down way of thinking about Cartan geometries, which generalize Klein geometries! Sometimes ontogeny fails to recapitulate phylogeny. So, maybe the "punters" should be handed a nice specific Lie 2-group, some 2-spaces on which it acts, and be asked to study the "incidence relations" between these figures. Incidence geometry could be given a whole new lease on life!
Best, jb
DC:
In a pleasanter world I'd be funded to think longer about such things. For those with the leisure time, you can read about incidence geometry at TWF 178, which treats incidence relations in projective geometry in terms of the Dynkin diagrams An. Perhaps we should start with projective rather than Euclidean geometry. Is there an obvious candiate for a Lie 2-group one step up from SL(n, C)? What then is projective 2-geometry? What are Dynkin 2-diagrams?
Or doing things axiomatically, perhaps we can categorify the axioms of projective geometry, such as those for the projective plane, taken from week 145:
A) Given two distinct points, there exists a unique line that both points lie on.
B) Given two distinct lines, there exists a unique point that lies on both lines.
C) There exist four points, no three of which lie on the same line.
D) There exist four lines, no three of which have the same point lying on them.
TL (Tom Leinster): David asks (or rather, has a hypothetical character ask) what categorified lines and circles are. John points out that this could be taken in various directions. Here's a possible beginning of an answer.
We have to decide which aspects of lines and circles we're interested in. Let's treat them as metric spaces. Then the "categorified circle" should be a categorified metric space. OK, so what's a categorified metric space?
Here we can follow Lawvere, who's done a lot to develop the thesis that everything is a category. (I'm exaggerating, but see the first page proper of his metric spaces paper.) If a thing can be regarded some kind of category, that increases our chances of being able to perform some useful sort of categorification. This is ironic, as Lawvere seems to disapprove of categorification...
Anyway, Lawvere interprets metric spaces as being categories enriched in R, the poset of non-negative reals ordered by >=, made monoidal by its additive structure. So it looks as if our task is to categorify R - for then a categorified metric space could be defined as a category (weakly) enriched in the categorified R.
As far as I know, there's no really compelling answer yet to "what is the categorification of the reals?"
DC: Could you also recapitulate Descartes' coordinate-based approach to geometry? With categorified reals you could carve out subcategories of R^2 in terms of isomorphisms such as X^2 + Y^2 is isomorphic to 1. Seems like you wouldn't be too far from Joyal's species.
JB:
> Is there an obvious candidate
>for a nice specific Lie 2-group?
I list a bunch of nice 2-groups in HDA5, and if I were going to categorify Klein geometry I would I would look at a bunch in parallel. The fun of course is seeing the effect and significance of the 2-morphisms (sorry, now I'm thinking of a 2-group as a 1-object 2-groupoid). We've got the ordinary groups, with no 2-morphisms to speak of; then we've got the 2-groups with only one morphism and an abelian group of 2-morphisms.In between these extremes, and more interesting, are groups built from a group G acting on an abelian group A; a great example is the "Euclidean 2-group" where G = SO(n), A = R^n. Or, it might be nice to let G be the whole Euclidean group and A some abelian group on which it acts: this would decategorify to the Euclidean group.
DC:
> I list a bunch of nice 2-groups in HDA5, and if I were
> going to categorify Klein geometry I would I would look
> at a bunch in parallel. The fun of course is seeing the effect and
> significance of the 2-morphisms (sorry, now I'm thinking
> of a 2-group as a 1-object 2-groupoid).
I misread this when I first read it, but even the misreading raised a question. I thought you were talking about 2-morphisms BETWEEN 2-groups (should be 3-morphisms I suppose). Back on the level of Kleinian 1-geometry, what role could group homomorphisms play in the Erlangen program? I guess what's already treated is group inclusions, e.g., projective transformations within affine transformations within Euclidean transformations. But is there scope for more interesting homomorphisms?
JB: Good point. Sure! Each group determines, or we could say "is", a Klein geometry, but groups form a category - in fact a 2-category, since groups are categories - so we get a 2-category of Klein geometries.
Ignoring the 2-morphisms for a moment, though they're interesting and important, let's think about the morphisms: group homomorphisms, viewed as morphisms between Klein geometries.
The examples you mention are already very interesting, because they show how geometry is a unified subject, not just a bunch of isolated "geometries". They show that including a little group in a big one can be seen as making a geometry "more flexible", by adding new transformations.
But, there's another way inclusions of groups show up: in Klein geometry, a "figure" is more or less given by the subgroup that stabilizes it. But, a subgroup is an inclusion of groups! So, the examples you listed of group inclusions can also be seen as "figures"! The most famous example is getting affine geometry by taking projective geometry and restricting to the transformations that stabilize a "point at infinity", or "line at infinity", or...
I hadn't noticed this dual viewpoint, for some reason.
Anyway, some other examples come from outer automorphisms of groups: the outer automorphism of SL(n) is the "duality" that switches points and lines, and for Spin(8) one has 3! acting as outer automorphisms, called "triality". For simple Lie groups one can read these off from the Dynkin diagramn
symmetries.
There are also inner automorphisms, which are just "changes of reference frame". The difference between inner and outer automorphisms is nicely handled by the 2-categorical structure of Grp. (Ever figured out what a natural transformation between a functor betwen groups is?)
This leaves the epimorphisms of groups: quotient maps. When you mod out the Euclidean group by the translation subgroup, you get the rotation group. What does this short exact sequence mean for the corresponding 3 Klein geometries?
This is a lot of fun, and it will be even more fun to categorify.
JB:
If you want to keep talking about this, maybe you should pick a 2-group, and I'll tell you about it, and then we can start trying to develop the corresponding "Klein 2-geometry". Or, we could work in abstract generality. Or both: lots of generalities, but with a key example or two to light the way. That's usually what I do.
Of course the fun will start as soon as we try to generalize the concept of "figures" familiar from ordinary Klein geometry. Ordinarily, a "type of figure" is a subgroup H of our symmetry group G, and the space of figures of that type is G/H. So, the concepts of "sub-2-group" and "categorified quotient space" will soon need some clarification. And, we'll soon start wondering if in addition to "figures" there are some new "2-figures", or "morphisms between figures", or something....
For this it'll be good to describe the theory of ordinary Klein geometry quite clearly using category theory, so we can see how it categorifies.
DC: The "Euclidean 2-group" where G = SO(n), A = R^n looks like a good candidate. So, we need to find sub-2-groups and work out what their quotient 2-spaces are. A couple of obvious cases are G = SO(n), A = {0}, and G = {id}, A = R^n. And then there's G=SO(n-1), A = R^n. Hmm, have you worked out the theory of 2-cosets yet? Or is that co-2-sets, which I suppose ought to be cocategories, but is already taken.
19 Comments:
David writes:
The "Euclidean 2-group" where G = SO(n), A = R^n looks like a good candidate.
Okay. Of course, we should be careful: despite its name, this group may wind up acting quite different from the Euclidean group, since when we decategorify it we get not the Euclidean group but SO(n).
So, we need to find sub-2-groups and work out what their quotient 2-spaces are.
Right. But again, it's good to be prepared for surprises. We'll need to figure out the appropriate generalization of subgroups and quotient spaces, and see what they amount to in this case, and maybe some other cases too.
A couple of obvious cases are G = SO(n), A = {0}, and G = {id}, A = R^n. And then there's G=SO(n-1), A = R^n.
Right, these are all sub-2-groups in the sense of being monoidal subcategories closed under inverses for both objects and morphisms. They have very different flavors, and none of them except the last is remotely like anything we're familiar with in the decategorified case... since in the first two cases we're picking "all the objects but only identity morphisms" and "only the identity object and all its endomorphisms".
Hmm, have you worked out the theory of 2-cosets yet?
Not quite, though it's probably lurking in the literature - that's part of why this is fun! There's certainly a theory of 2-groups acting on categories, and that's probably what we should focus on.
Indeed, let's not try to be too sophisticated too soon. The relation of geometry and group theory began with some realization sort of like this: the group G of Eulidean transformations acts on everything in the plane, in a way that preserves all geometrical structure. So, it acts on the set of points, the set of lines, etc.. In fact it acts transitively on these two particular sets: you can carry any point to any point, or any line to any line, by a Euclidean transformation. This is why these sets are of the form G/H for certain subgroups H, where H is the stabilizer of a figure: the subgroup of transformations preserving that figure.
Only later did people get so sophisticated as to realize that what matters most is the subgroup H.
It might be fun to recapitulate the historical development Klein geometry, keeping an eye on later phases but not trying to rush them unduly.
In other words: instead of looking for sub-2-groups H of aome 2-group G and then trying to figure out the quotient 2-spaces G/H, we could look for "categories of figures" on which G acts - "transitively", in some sense - and then try to figure out their stabilizers H.
Actually it's good to both what you suggested (working out the theory of quotient 2-spaces G/H) and what I'm suggesting now (looking around for plausible categories of "figures" and seeing what their stabilizers are) - the two approaches should meet at some point.
So, here goes. Consider first the group G = SO(n). This acts on the sphere S^{n-1}, so a nice type of figure is "point on the sphere". The stabilizer of a point is H = SO(n-1), and we get
S^{n-1} = G/H
tying everything up nicely.
Now let's try the Euclidean 2-group E. This also acts on the sphere S^{n-1}, which we think of a category with a sphere's worth of objects and only identity morphisms. Objects of E, namely elements of G = SO(n), act as functors from this category to itself (namely, rotations). Morphisms acts as natural transformations between these (necessarily the identity).
Which sub-2-group H of E is the stabilizer of a point on the sphere? Do we get
S^{n-1} = E/H
in some sense?
I think the 2-group E also acts on the category X with one object an R^n as morphisms. I could be wrong. If I'm right which sub-2-group of E is the "stabilizer" in this case? Here it's less obvious what "stabilizer" should mean.
Which sub-2-group H of E is the stabilizer of a point on the sphere? Do we get
S^{n-1} = E/H
in some sense?
You'd think the objects of H would be SO(n-1) and the arrows between any two objects R^n.
It's hard to see that there's any scope to get the A = R^n in on the action. Is transitivity more subtle at the 2-level? I guess for some other choice of G and H, it would be possible to have a 2-coset of the form -
objects: lines in R^n, arrows: transformations mapping between lines.
I think the 2-group E also acts on the category X with one object and R^n as morphisms. I could be wrong. If I'm right which sub-2-group of E is the "stabilizer" in this case? Here it's less obvious what "stabilizer" should mean.
Something like *--a-->* gets sent to *---phi.a.phi^(-1)-->*, for a rotation phi? And only trivial natural transformations?
Recapitulating the history of group theory from further back, it might be worth deriving the theory of finite 2-groups, perhaps as permutation 2-groups. I wonder what the simplest nontrivial example might be. There's surely an equivalent of Lagrange's theorem about the order of a element dividing the order of the group. Another early situation concerns automorphisms of field permuting roots of equations.
There's a lot to say, but I just arrived at the Perimeter Institute so I'll say just a little.
David wrote:
Recapitulating the history of group theory from further back, it might be worth deriving the theory of finite 2-groups, perhaps as permutation 2-groups. I wonder what the simplest nontrivial example might be.
Just as every set has an automorphism group, usually called a permutation group, every category has an automorphism 2-group. Given the category C, the 2-group AUT(C) has invertible functors
F: C -> C
as its objects, and natural isomorphisms between these as its morphisms. The "multiplication" in this 2-group comes from composing functors, so if we multiply
F: C -> C
and
G: C -> C
we get
FG: C -> C
There is also a slightly less obvious way to multiply morphisms in AUT(C), riding on the back of this multiplication of objects. For details, try section 8.1 of this paper.
If C is finite, AUT(C) will be finite, so one gets tons of finite 2-groups this way.
Let's consider an example: let C be the "walking isomorphism". This has two objects x and y, an isomorphism f: x -> y, and nothing else. Well, of course it has identity morphisms and the inverse f^{-1}: y -> x, but these are required to exist by virtue of C being a category and f being an isomorphism. That's why we call C the "walking isomorphism" - it's a category with nothing but what's required whenever you have an isomorphism!
So, what's AUT(C)? Well, its objects are the "symmetries" of C - the invertible functors
F: C -> C
There are two: the identity, and the one that switches x and y and turns f around. Call these 1 and F, respectively.
We clearly have
F^2 = 1
So, so far we've just got the group Z/2, the obvious symmetry group of this category.
But in fact AUT(C) is a 2-group, with some interesting morphisms, since there's a natural isomorphism between 1 and F!
Indeed, AUT(C) has four morphisms, namely
1_1 : 1 => 1
1_F : F => F
A: 1 => F
and
A^{-1}: F => 1
It turns out that while this 2-group AUT(C) isn't isomorphic to the trivial 2-group, it's equivalent to the trivial 2-group, unless I'm making a mistake. And, ultimately, this is because C is equivalent to the trivial category: it has two uniquely isomorphic objects, but that's no more interesting than having one object!
(There are some subtleties I'm glossing over here, which I'll be glad to discuss with anyone interested.)
Anyway, instead of tackling these:
There's surely an equivalent of Lagrange's theorem about the order of a element dividing the order of the group. Another early situation concerns automorphisms of field permuting roots of equations.
which could easily keep us happily entertained for months, I think we should stick to Klein 2-geometry. And, I think the best way to do that is to focus on an example. But, as I said before, I think the best way to start is get some nice examples of "geometries" and figure out their symmetry 2-groups.
Normally a geometry, in the naive sense I'm using it here, is a set of "points" together with various relations (e.g. "collinearity") or functions (e.g. "distance") or subsets (e.g. "lines") that we want to be preserved by any transformation that deserves to be called a "symmetry". And, from this we work out the group of symmetries. Klein geometry is the elegant way to work backwards from the group....
But let's not be elegant; let's be simple-minded!
So, now we want a CATEGORY of points with some extra relations or operations or subcategories....
And here's a very simple example, just to get going. Normally we can form the plane R^2 as the quotient
R^2 = E(2)/O(2)
where E(2) is the group of Euclidean transformations and O(2) is the stabilizer of a point, namely the group of rotations and reflections.
This quotient is just a set.
But now let's look at the weak quotient
E(2)//O(2)
as defined on page 9 of this PDF file (called page 55 in the handwritten notes in the file). This is a groupoid instead of a set... so now it's a plane which has points and "morphisms of points". I think this will be good! Then we can put some geometrical structure on it, maybe, and see what it's symmetry 2-group is.
Let's see - can you guess what
E(2)//O(2)
is actually like?
Hmm, this is more than just a little, but honestly, it's just a fraction of what there is to say!
For example, I didn't explain what I'm doing here: showing that we can automatically categorify any homogeneous space G/H by using the weak quotient G//H instead. This should set up some interesting link between Klein geometry and Klein 2-geometry! And we need some link or other to guide us....
I think we should stick to Klein 2-geometry
Yes. Speculation is so much easier than calculation.
Let's see - can you guess what
E(2)//O(2) is actually like?
The reference you give defines the weak quotient of the action of a group on a set. So presumably we're to think of O(2) acting on the *set* E(2), and let us say by left multiplication. Then E(2)//O(2) is the groupoid whose points are elements of E(2), and whose arrows are labelled (e,g) running from e to g.e. From the identity, there's a single arrow going to each element of O(2). Each component of this groupoid will have precisely one object corresponding to a translation. So the decategorification of this groupoid is R^2. So then do two distinct components determine a line?
Are we to think of this happening for different copies of O(2) inside E(2)?
I wrote:
I think we should stick to Klein 2-geometry.
David wrote:
Yes. Speculation is so much easier than calculation.
Well, I don't mind the fact that it's easy - I just don't like how it leaves me without a sense of accomplishment. I much prefer figuring something out without any real work and still getting the feeling that I accomplished something.
So, let's get cracking!
The reference you give defines the weak quotient of the action of a group on a set. So presumably we're to think of O(2) acting on the *set* E(2), and let us say by left multiplication.
That's right - when we're playing the Klein game, we've got a subgroup H of a group G, and we consider the action of H on G by left multiplication - or right multiplication, if you prefer. These are actions of H on the mere *set* G. These actions don't preserve the *group* structure in G, since h(gg') is not equal to (hg)(hg'). So, the quotient is a mere *set* in general.
(There's also the action of H on G by conjugation, and this is really an action on the *group* G, but it's not what want in the Klein game.)
Indeed, E(2)/O(2) is the plane when we let O(2) act on E(2) by right or left multiplication, since guys in E(2) give points in the plane by letting them act on the origin, and two such guys give the same point if they differ by a rotation (since a rotation fixes the origin).
Okay? So, now we consider the weak quotient E(2)//O(2). This amounts to saying two guys in E(2) give isomorphic points if they differ by a rotation!
So, in this new categorified view of geometry, rotations do not "fix" the origin! They only weakly fix it - they carry it to another isomorphic point.
So, maybe we should imagine the origin (and thus every other point in the plane) having a little dial on it, so when we rotate it, we can see that something happened. We say "hmm, that point's not quite the same anymore... it's been rotated! But, I guess it's isomorphic".
Then E(2)//O(2) is the groupoid whose points are elements of E(2), and whose arrows are labelled (e,g) running from e to g.e. From the identity, there's a single arrow going to each element of O(2).
Right, we have one point for each Euclidean transformation e of the plane - think of this as the point obtained by applying the transformation e to the origin. Two points e and e' are decreed isomorphic if there's a rotation g with e' = ge... first rotating, and then doing e, we get e'.
This agrees with what I was saying.
So the decategorification of this groupoid is R^2.
Right! To decategorify, we forget the little "dials" on our points - points that were isomorphic are now be decreed "equal".
This is always how these weak quotients work: the decategorification of X//G is X/G. This is why weak quotients are better than ordinary quotients - you can always get the ordinary quotient back by forgetting some information!
What's cool is that we can automatically categorify any Klein geometry by this trick!
Just use G//H instead of G/H.
I have a feeling in my gut that this will be a great source of Klein 2-geometries. We'll need to understand G//H and then its 2-group of symmetries.
So then do two distinct components determine a line?
I guess we need to make up our minds what a line is. But there's an obvious thing to try. The set of points was E(2)/O(2), and we got our spiffy new groupoid of points by taking E(2)//O(2) instead. So, the same trick should work to define the groupoid of lines.
Puzzle: do you know what subgroup to use now?
Puzzle: can you say anything about what the groupoid of lines is like?
I bet we can then categorify the incidence relation "the point P lies on the line L" in some fairly automatic way... which when understood, will generalize to pretty much any Klein 2-geometry obtained from a Klein geometry by this "automatic categorification" trick.
It may turn out at some point to be a little better to use projective geometry instead of Euclidean, since the symmetries of the projective planes are precisely those transformations that preserve the point-line incidence relation...
... but anyway, I'll stop here for now.
So, maybe we should imagine the origin (and thus every other point in the plane) having a little dial on it, so when we rotate it, we can see that something happened.
Isn't this all very reminiscent of gauge theory, and principal bundles?
We'll need to understand G//H and then its 2-group of symmetries.
Presumably we have to take into account that G//H isn't just a groupoid. I.e., there's extra structure (smoothness, etc.) which needs preserving under symmetry. You would think that for the 2-group of symmetries of E(2)//O(2), the objects would be the whole of E(2), shifting between points and twirling the dials. Then what is the Abelian group's worth of natural transformations?
the same trick should work to define the groupoid of lines.
Puzzle: do you know what subgroup to use now?
How about the 1-dimensional subgroup of translations in one direction (perhaps also put in the reflection about that direction).
Puzzle: can you say anything about what the groupoid of lines is like?
Each component of this weak quotient is a line. A line is a recipe to move somewhere, orient oneself, then move forward and backward in a certain direction.
David wrote:
Isn't this all very reminiscent of gauge theory, and principal bundles?
You're just saying that because I started talking about "little dials". But it's true - there's an O(2) bundle over the plane lurking in my description of the "groupoid of points" in 2d Euclidean geometry. The total space of this principal bundle is the space of objects in E(2)//O(2), and O(2) acts on this, and the quotient is the plane. I'm not sure how general this is, and I'm feeling too lazy to work it out now...
... oh, but it's just obvious, I can't help working it out: given any Lie groups G and H, the space of objects of G//H is just G, and this is indeed a principal H-bundle over G/H.
So, yeah - you're right!
I should have noticed this myself, because when I was trying to make the groupoid E(2)//O(2) very easy to visualize, I not only grasped at the image of "points with little dials attached", I also thought of saying that it's like the usual plane, but where each point now has "internal symmetries".
And, that's just what gauge theory is about.
I had written:
We'll need to understand G//H and then its 2-group of symmetries.
and you replied:
Presumably we have to take into account that G//H isn't just a groupoid. I.e., there's extra structure (smoothness, etc.) which needs preserving under symmetry.
Right! Very important!
Understanding G//H as a groupoid is as basic and crude as understanding G/H as a set. Knowing the plane as a mere set isn't nearly enough to figure out the Euclidean group - we need to understand it as something like a metric space. Similarly for our categorified plane.
And this is why I said we might prefer to switch to projective geometry at some stage. In projective geometry we have a set of points, a set of lines, and then an incidence relation "the point P lies on the line L". To get our symmetry group, we just need to find maps sending points to points and lines to lines preserving this incidence relation. Euclidean geometry is not so simple.
But enough chat - now let's get to work and figure out the groupoid of lines in categorified Euclidean geometry. It will be of the form E(2)//H for some subgroup H, and I asked:
Puzzle: do you know what subgroup to use now?
You replied:
How about the 1-dimensional subgroup of translations in one direction (perhaps also put in the reflection about that direction).
Right - translations and reflections are the elements of E(2) that stabilize a line, so these form our subgroup H. Technically, H is the semidirect product of R and Z/2.
(I may have been a bit sloppy: in the case of a point, the stabilizer is not just the rotation group, but the rotation/reflection group O(2). This is because my Euclidean group E(2) includes reflections. You can get rid of them if you prefer - doesn't much matter yet.)
And next:
Puzzle: can you say anything about what the groupoid of lines is like?
Each component of this weak quotient is a line. A line is a recipe to move somewhere, orient oneself, then move forward and backward in a certain direction.
... and possibly flip over. Yes!
Or in other words:
In categorified Euclidean geometry, for every ordinary point in the plane we now have lots of isomorphic copies, since now we decree that rotating or reflecting about a point does not send it to the same point, but merely an isomorphic one. Similarly, for every ordinary line we have lots of isomorphic lines, obtained from the original by translation or reflection.
So, our geometrical figures now have "inner life", or if that's too poetic for you, "internal symmetries".
Next we should figure out how to categorify the point-line incidence relation.
As a little warmup, let me test your telepathy:
What might you get when you categorify a relation?
I had an instant gut reaction when I pondered this question last night, and I'm wondering what yours will be. Then I decided my gut reaction may not have been appropriate for this particular adventure we're on today. But, I'm not sure.
Since you like the philosophy of real mathematics, you may also like to think about the trick we're using here: the "method of the vivid example". We're really studying an arbitrary group G, an arbitrary collection of subgroups H, relations between the quotient spaces G/H, and now categorified relations between the weak quotients G//H.
But, instead of approaching this problem in a completely abstract way, we seize upon a vivid example, where G is the Euclidean group, and our G/H's are things like the space of all point or the space of all lines in the plane - things we've known since childhood!
The vividness of the example is supposed to whet our appetite and spur our thinking forwards, but we're not supposed to get too caught up in its particular features - it's just a stand-in for the general case. It's sort of like when a magician or talk show host takes a member out of the audience and gets them to do things - they may feel special, and in a way they are, but in a way they could be anybody.
For example, I will be completely ruthless and cold if it turns out that projective geometry serves my purposes better - out with Euclidean geometry, in with projective! One can always come back to Euclidean geometry later if it's actually important.
As a little warmup, let me test your telepathy:
What might you get when you categorify a relation?
Well a relation between two sets S and T is just a subset of S x T. So a categorified relation between two categories ought to be a subcategory of their product. So we need an incidence sub-category of the product of the groupoid of points and the groupoid of lines. Probably, an incidence sub-groupoid. Now I wonder how that little internal arrow on the point is going to relate to how far a line has been moved along itself.
But yes, I agree projective geometry might well be the thing just now. So playing the same game with SL(2,C) and the subgroup of upper triangular matrices, i.e., the stabilizer of a point, what do we find? Well, we were happy once upon a time imagining the quotient as the Riemann sphere. But now we know that if we look more closely at each point on the sphere, there's C^2's worth of stuff there. Now what's the 2-group that's going to act on this weak quotient?
With a little more room for a manoeuvre, looking at SL(3,C), is it the case, back at the level of 1-geometry, that two distinct subgroups isomorphic to the matrices
* * *
0 * *
0 * *
determine a subgroup isomorphic to those of the form
***
***
00*
and two of the latter determine one of the former?
Why isn't the whole world (or at least a large part of it) trying to do stuff with 2-groups? It might have been the case that although groups are everywhere in math, 2-groups added little. But the evidence is there. E.g., the representation theory of the Poincare 2-group goes beyond the 1-level.
For some notes on projective geometry by a master geometer try here.
SL(3,R) seems rather roomy for the group of transformations of a plane geometry. It's 8d, and so there are 6-dimensions of weak points equivalent to a given weak point, and likewise for weak lines.
Risking the snares of Euclidean gemetry for the moment, we might pose ourselves the question of what happens as we vary two weak lines meeting at a weak point. It could be that the dial on the weak point is measuring the angle at which the lines meet, in which case sliding the lines along themselves won't change the pointer. On the other hand, treating the lines like a pair of scissors will. Presumably then, two weak points would determine a weak line, by the direction of the interval between them, and also the length of this interval. Moving the dials on the points would make no difference to the weak line, but moving the points themselves further apart would.
That would neatly give us an answer to one of the first questions I posed. "When we say the intersection of any pair of altitudes of a triangle is the *same* point as that of any other, is there room for weakening?" Now we know that three equivalent weak points occur as these intesections. Although there is a relationship between their pointers, they need not be identical.
Back in the projective case, there are 12 dimensions worth of pairs of weak points equivalent to any given pair. How many dimensions worth allow a weak line to pass through them?
I was distracted this weekend by a bad cold and a strong urge to understand the relation between three-strand braids and SL(2,Z). Something very beautiful is going on there.
But anyway, David asked:
With a little more room for a manoeuvre, looking at SL(3), is it the case, back at the level of 1-geometry, that two distinct subgroups isomorphic to the matrices
* * *
0 * *
0 * *
determine a subgroup isomorphic to those of the form
* * *
* * *
0 0 *
and two of the latter determine one of the former?
Yes, that's right. Groups of the first sort stabilize a point in the projective plane; groups of the second sort stabilize a line. Since we already know two lines intersect in a unique point, a transformation that stabilizes two lines must stabilize a unique point - their intersection. So, the intersection of the stabilizers of two lines must lie in the stabilizer of a unique point! So, the intersection of two groups of the first sort must lie in a unique group of the second sort!
Ain't logic wonderful?
But be careful - in Klein geometry, the relevant notion of "isomorphic" subgroups is not that they're isomorphic as abstract groups, but that they're conjugate within the big group!
Indeed, the group of determinant-1 matrices of this form:
* * *
0 * *
0 * *
is isomorphic as an abstract group to those of this form:
* * *
* * *
0 0 *
But, we don't say points are lines!
We say that points are dual to lines, since there's an outer automorphism of SL(3) that maps matrices of the first form to those of the second form.
I always find this a bit confusing - I keep thinking there must also be an inner automorphism (= conjugation) that does the job. But, I know that SL(3) has a nontrivial outer automorphism, thanks to the symmetry in its Dynkin diagram:
o-----------------------o
and this symmetry interchanges the two figures "points" and "lines", which correspond to the two dots in the diagram. And, I know this automorphism consists of taking the transpose of a matrix.
I just keep thinking there must be some inner automorphism that would do the job, like conjugating by the matrix that switches two coordinates. Okay, that doesn't have determinant 1, but maybe throw in a minus sign or something.
I think I once laid this paradox to rest with some explicit calculations, but it keeps bouncing back like Dracula.
I will post something separate about our real problem: how we categorify this business.
Okay, now let's think some more about categorifying the point-line incidence relation. This is the real crux of the matter - if we succeed in doing this nicely, we'll instantly know that there's a vast subject of incidence 2-geometries waiting to be explored. The key word is nicely - we want something that sheds new light on geometry, not some ad hoc baloney.
You've made some good guesses about how two weak points might determine a weak line, and so on... but I feel a bit uneasy about it all, so I'd like to try a more formal tactic.
In ordinary incidence geometry, say we have a symmetry group G and two figures corresponding to subgroups H and H'. Then I believe we should say H and H' are incident if they have nontrivial intersection. In other words, some symmetries preserve both figures.
The idea of "intersecting" sub-2-groups is a bit tricky, since it requires some weakening. In general, given subcategories H and H' of some category G, it's silly to start defining their "intersection" to have objects that lie in both H and H' - this is imposing an equation on objects, since we're asking for an object in H to equal one in H'. We should
instead have an isomorphism.
Allowing myself a bit more jargon: an intersection of sets is defined using a pullback, but the concept of pullback needs to be weakened when we go up to categories, to what the Australians call a "pseudo-pullback".
But instead of grinding away and working out the pseudo-pullbacks of sub-2-groups of the 2-group we're considering, it's better to think about what the hell this would actually mean, and see what it's trying to tell us.
First, suppose we understood the symmetry 2-group of the categorified plane. Is there some obvious choice? It depends - are we categorifying projective or Euclidean plane?
Oh well, let's just fake it for now: it's probably sort of like some well-known symmetry group, but with new "transformations between transformations".
Next, we need to grok the stabilizer 2-group of a "weak point", as you call it. In fact we even need to think about what "stabilizer" means here - I think this concept too must be weakened: it's okay if a transformation sends our weak point to an isomorphic one. (Did I say this already somewhere? I forget).
But I can make a wild guess: I bet "rotating" our weak point will give an element of its weak stabilizer - since rotating a weak point gives an isomorphic point.
Third, we need to grok the weak stabilizer for a line. I guess this should include translations where we slide the line along itself.
Fourth, we need to grok what it means for an object in the weak stabilizer of a point to be isomorphic to one in the weak stabilizer of a line. Here we'd need to understand the "transformations between transformations". I have no feel for these right now, so I'm getting stuck, and it's starting to look really attractive to quit and go get some lunch.
David wrote:
Why isn't the whole world (or at least a large part of it) trying to do stuff with 2-groups?
Perhaps the above ruminations explain why? I don't know lots of people who are willing and able to think about these issues. It takes some training in 2-categories, and a willingness to take familiar concepts and say "now I will think about them in a way where equations are systematically replaced by isomorphisms". Most people run away howling mad after a bit of this.
I was coming to the conclusion that the 2-group will have E(2) as 1-morphisms, with only identities for 2-arrows. But then I remembered the Poincare 2-group having 1-morphisms SO(3,1) and 2-morphisms between identical 1-morphisms R^4? Why not then have Euclidean 2-group with O(2) as 1-morphisms and R^2 as 2-morphisms between identical 1-morphisms. (The action being the defining representation of O(2) on R^2.) Then a sub 2-group with the same 1-morphisms, but trivial 2-morphisms would fix a point, while a sub 2-group with 1-morphisms the subgroup of order 4 of O(2) generated by reflection in a line and rotation by 180 degrees, with 2-morphisms the translations in the direction of the line.
Hmm. In this set up it's the 2-morphisms which are like the elements of the Euclidean group. Why aren't we just looking for sub-2-groups which share a non-trivial 2-morphism. E.g., rotation about the origin by pi is in both the stabilizer of the x-axis and of the origin. 2-morphisms don't have weakened identity relations.
Notes to self:
Spherical geometry might be an easy case to think about.
From what we said earlier about gauges/principal bundles, is there a 2-group of gauge transformations here?
Has the categorification of projective geometry anything to do with invertible linear maps of a 2-vector space, factored by something to make a PGL type thing? If so, whose version of 2-vector space?
How do you quotient 2-groups? In the case of 1-groups, G and subgroup H, the old-fashioned way was to form the set of cosets G/H. We didn't want to throw away information so looked instead at G//H the weak quotient which is a groupoid which records H's action on G as 1-sided multiplication.
Now we're wondering about 2-groups, G and sub-2-group H. Is there a category G/H, and then a 2-category G//H? So a 1-morphism of H acts on a 2-morphism of G, to give us an arrow in the 2-category G//H, which has 2-morphisms of G as objects. And then a 2-morphism in H acts so as to mediate between two 1-morphisms in G//H.
Hopefully the idea is clear enough.
Hi,
I was kindly invited to participate in this discussion.
In fact, I did follow it and take interest in it. But last time I submitted a comment to this blog it did not seem to pass the moderation process (maybe something went wrong?), and that somewhat demotivated me from submitting further comments.
Anyway, I wanted to make a comment on what Tom Leinster said, somewhere in this thread, namely this:
"Anyway, Lawvere interprets metric spaces as being categories enriched in R, the poset of non-negative reals ordered by >=, made monoidal by its additive structure. So it looks as if our task is to categorify R - for then a categorified metric space could be defined as a category (weakly) enriched in the categorified R."
Tom Leinster then went on to say:
"As far as I know, there's no really compelling answer yet to "what is the categorification of the reals?" "
My comment on this would be the following:
There is indeed a problem with categorifiying fields, like the real or complex numbers, as anyone notices who tries to do so.
On the other hand, there is a straightforward categorification of semi-rings: abelian monoidal categories.
Better yet, abelian monoidal categories in their very role as semi-2-rings already show up in quite a few interesting examples, which reassures one that this is the way to go.
So if we are, as in Tom Leinster's comment, concerned with an ordinary semi-ring in the first place (the non-negative real numbers), I would assume that any abelian monoidal category would be a useful categorification of that.
I have mostly pondered this question from the point of view of categorified linear algebra. Of course most I ever had to say about this was copied from or at least inspired by what John Baez has already said.
In that context we might be tricked into believing that we want a categorification of the category of K-modules, where K is some field.
But this is really a prejudice. We can do most of ordinary linear algebra with rings and even with semi-rings. So let's not restrict generality too much. Instead we should assume that what we are looking for is a categorification of the category of semi-ring modules.
Well, _that_ is easy. The right notion now is clearly the 2-category of semi-2-ring modules.
So the objects (the categorified vector spaces) are now module categories for some abelian monoidal category C.
Morphisms are 2-linear maps, functors that respect the left C action.
And there are 2-morphisms now, of course.
Thinking of 2-categories of 2-modules this way turns out to be the right thing in many situations.
I have talked about that several times over at the String Coffee Table. Mainly, my point is that this gives the right notion of categorification of quantum mechanics useful for the description of 2-dimensional field theory (string mechanics).
More recently, I noticed that, somewhat in disguise, this notion of categorified linear algebra already appears in major projects, such as Langlands program.
See my remarks at
http://golem.ph.utexas.edu/string/archives/000805.html
for more on that.
In this context one always uses C=Vect as the semi-2-ring.
But, as I don't get tired of mentioning, J. Fuchs, I. Runkel and Ch. Schweigert (here) have figured out that by replacing Vect by any modular tensor category, all the known constructions from topological 2D field theory carry over to the richer world of 2D conformal field theory.
This allows to interpret lots of phenomena encountered in CFT in terms of relatively simple ideas of categorified linear algebra.
For instance, a phenomenon known as Kramer-Wannier duality in 2D field theories amounts essentially to some sort of categorified version of orthogonality condition on categorified linear morphisms.
You can see this being hinted at in cond-mat/0404051. The more detailed version of this paper will appear in the near future.
Anyway, my point is that it seems that we do have a robust guess for the right categorification not of the real numbers, but of the non-negative real numbers.
Hi again,
I have more comments
here.
Please note that I have no intention of moving the discussion away from your blog. I just thought this somewhat lengthy elaboration deserves to sit in an entry on the Coffee Table.
If your blog supported trackback pings (it doesn't, does it?) it'd be slightly more natural to have a discussion spread over several blogs.
In ordinary incidence geometry, say we have a symmetry group G and two figures corresponding to subgroups H and H'. Then I believe we should say H and H' are incident if they have nontrivial intersection. In other words, some symmetries preserve both figures.
Is this a case of low dimensional Euclidean geometry's grip being too strong? In less rigid geometries, even in low dimensions, non-incident figures can be fixed. E.g., in projective plane geometry, 3 points can be fixed and there is still room to move another point. So certainly there are transformations which fix a line and a non-incident point, i.e., their stabilizers intersect.
If you think that the stabilizer of a line in the real projective plane is 6 dimensional, as is the stabilizer of a point, there's going to have to be non-trivial intersection in an 8 dim group. There must be something subtler happening in the case of incidence.
For instance, in the real projective plane, if point and line are incident, then the intersection of the stabilizers is isomorphic to the upper triangular 3x3 matrices of determinant 1. If they are not incident, the intersection is isomophic to GL(2,R).
The discussion continues.
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