Klein 2-Geometry III
Update: I'm floating this post to the top again so that we don't lose it.
Time to begin the new month's posting on categorified geometry, continuing May and June. Fortunately John Baez, although now in Shanghai, is on broadband. I wouldn't fancy solo Kleincategorification (second and final comments). It's like when you're learning to ski, you can manage much trickier slopes with an expert to follow.
I guess the biggest worry in a venture of this kind is that all you achieve is a repackaging of what's already known. There's a discussion here, involving John, about whether Lie 2-algebras bring into the light anything new (cf. posts 9, 13 and 14). (The archives of sci.physics.research is full of delights. Here's another thread on 2-groups.)
In this discussion, John mentions his reasons for quitting his role as moderator of sci.physics.research. I'm not sure I've characterised all that well there what it is I'm looking for beyond individuals exposing their ideas in a free and informal way. John can see I want something a little more agonistic, but fears the tendency towards antagonism. I see agonism is a political position. One of its advocates has this to say:
Of course, you may be able to internalise the agon, by taking also the part of the opponent. Indeed, this is how I arrived at the idea behind this series on the categorification of Kleinian geometry. I imagined what someone highly dubious about the scope of worthwhile categorification might say. "Let us for the moment accept that the 'categorification' (such an ugly name) of arithmetic and combinatorial identities via groupoids and species has been worthwhile, what do you have to say about Euclidean geometry, the jewel of Greek mathematics. If you have nothing new to tell me about points, lines and circles, I shall remain unconvinced."
Time to begin the new month's posting on categorified geometry, continuing May and June. Fortunately John Baez, although now in Shanghai, is on broadband. I wouldn't fancy solo Kleincategorification (second and final comments). It's like when you're learning to ski, you can manage much trickier slopes with an expert to follow.
I guess the biggest worry in a venture of this kind is that all you achieve is a repackaging of what's already known. There's a discussion here, involving John, about whether Lie 2-algebras bring into the light anything new (cf. posts 9, 13 and 14). (The archives of sci.physics.research is full of delights. Here's another thread on 2-groups.)
In this discussion, John mentions his reasons for quitting his role as moderator of sci.physics.research. I'm not sure I've characterised all that well there what it is I'm looking for beyond individuals exposing their ideas in a free and informal way. John can see I want something a little more agonistic, but fears the tendency towards antagonism. I see agonism is a political position. One of its advocates has this to say:
Agonism implies a deep respect and concern for the other; indeed, the Greek agon refers most directly to an athletic contest oriented not merely toward victory or defeat, but emphasizing the importance of the struggle itself-a struggle that cannot exist without the opponent. Victory through forfeit or default, or over an unworthy opponent, comes up short compared to a defeat at the hands of a worthy opponent-a defeat that still brings honor. An agonistic discourse will therefore be one marked not merely by conflict but just as importantly, by mutual admiration. (Samuel Chambers)Bloggers of the world, forego antagonism, choose agonism.
Of course, you may be able to internalise the agon, by taking also the part of the opponent. Indeed, this is how I arrived at the idea behind this series on the categorification of Kleinian geometry. I imagined what someone highly dubious about the scope of worthwhile categorification might say. "Let us for the moment accept that the 'categorification' (such an ugly name) of arithmetic and combinatorial identities via groupoids and species has been worthwhile, what do you have to say about Euclidean geometry, the jewel of Greek mathematics. If you have nothing new to tell me about points, lines and circles, I shall remain unconvinced."
47 Comments:
After missing a lot of the previous action, here's another 2 cents. While the aim of this program is to follow Klein, I thought of the opposite problem, namely incidence geometry (influenced by a fellow student in my office).
Essentially an incidence geometry on a set S is a function t from S to the order {0,...,n} and a symmetric reflexive relation on S which satisfy a whole lot of nasty conditions making things like dimension and intersection make sense. The order gives us the dimensions of figures and we require t^{-1}(0) nonempty and t^{-1}(n) nonempty and unique. A symmetric reflexive relation isn't nice until one realises that that is precisely what spans look like. The spans would be in some lattice of figures, with inclusion as arrows. The `maximal' object spanning a pair is the figure they intersect in.
All this talk of double cosets puts us squarely with cospans involving epis so I take hope in that, and I would hope there is some contravariant functor taking an incidence geometry to a Klein geometry, thinking of both of them as a category
A naive first attempt at categorifying incidence geometry, without any details, would be a category C equipped with a functor to a (complete?) 2-order A such that the fibres over the 0 and 1 are not empty. C should be some sort of `category of spans', but I can't see what this would be. A is the category of types of figures here, but when we go to Klein geometry, I don't know what this turns out to be.
Actually I just realised I've gone two steps there - should we try for some straight generalisation like:
An incidence geometry is an object of a topos plus a map to a (complete) Heyting algebra object in that topos satisfying the requisite conditions. Then we can consider the dual, Klein situation. The topos object should probably be some internal object of spans.
Just for completeness, the conditions I avoided mentioning above are mostly to ensure nondegeneracy, like `there exists a point in our space not on a given hyperplane', and `given a figure and a point off it, there is a figure meeting both' (plus 2 others in the same vein). There is one, however that cries out to be dropped: `incident objects of the same dimension are equal'.
All this talk of double cosets puts us squarely with cospans involving epis
Is there anything more specific to say about double cosets in this context?
John said last month:
One annoying thing is that these 2-groups seem to be built using both G and H. To get a 2-group whose "Klein 2-geometry" is a "turn-the-crank categorification" of the Klein geometry associated to G, I'd ideally want to build a 2-group using just G. Or maybe using G and all its subgroups? Anyway, something functorial in G. This could be too much to ask.
So why not say that the 2-group associated to a geometry normally thought of as corresponding to the group K, is the one which in the notation of the June post has G = K, H = K, t = id_K, a = conjugation?
Then the sub-2-group of symmetries preserving a figure has G = K, and H the stabilizer of that figure.
That last paragraph is of course only right for Abelian K.
What we need to look at is a very simple nonabelian K. I think 1d Euclidean geometry should be simple enough. Transformations can be thought of as lines at 45 degrees to the x-axis. Left multiplication by -I corresponds to reflection in the x-axis. {+-I}//E(1) is isomorphic to two copies of R with a pair of arrows between corresponding points on the two lines. Now for the symmetry 2-group...
John wrote:
To get a 2-group whose "Klein 2-geometry" is a "turn-the-crank categorification" of the Klein geometry associated to G, I'd ideally want to build a 2-group using just G.
David wrote:
So why not say that the 2-group associated to a geometry normally thought of as corresponding to the group K, is the one which in the notation of the June post has G = K, H = K, t = id_K, a = conjugation?
This is one of several 2-groups we can build starting from a group K. Let's call it C. The funny thing about this C is that it's equivalent to the trivial 2-group!
The reason is that, as a category, C is equivalent to the category with one object and one morphism.
To see this, it suffices to show that C is a codiscrete category: a nonempty category where any pair of objects is isomorphic in a unique way!
And this, in turn, follows from the fact that t is one-to-one and onto.
Huh?
Well, remember that G is the group of objects of C, while H is the group of morphisms starting at the identity object, and t: H -> G sends any object to its target. So, if t is 1-1 and onto, there is a unique morphism from the identity object to any other object. So, any pair of objects is isomorphic in a unique way!
I've shown that C is equivalent as a category to the category with one object and one morphism. There's only one way to make the latter category into a 2-group - the trivial 2-group. But, you may be suspicious: just because C and the trivial 2-group are equivalent as categories, why should they be equivalent as 2-groups?
The point is, given any equivalence of categories
F: C -> D
and a way of making C into a 2-group, we can transport this 2-group structure along F and make D into a 2-group, thereby promoting F to a (weak) 2-group homomorphism - indeed, an equivalence of 2-groups!
(In general, the 2-group structure on D may not be strict, but in our example it clearly is, since D is the terminal category, which has only one 2-group structure: the trivial one, which is strict.)
Since this post is getting a bit technical I'll stop now, and talk in another post about whether it's inevitably boring to do Klein 2-geometry with a 2-group equivalent to the trivial one.
Don't we leave this triviality behind when we start to look at sub-2-groups of the 2-group you've called C? We wouldn't expect the full 2-group to preserve any figure. Just as the full Euclidean 1-group doesn't preserve any figures, except perhaps for the whole plane.
What got me thinking of this was the additive group R^3. If you take the subgroup of translations by integer amounts in the x-direction and do what we did earlier, you get a groupoid of figures R^3//Z with invariance sub-2-group G = R^3, H = Z. On the other hand, if you take the subgroup of tranlations by integer amounts in y- and z- directions, you get a groupoid of figures R^3//Z^2 with invariance sub-2-group G = R^3, H = Z^2.
In this abelian case, it looks like we're interested in sub-2-groups of the form G = R^3, H = a subgroup of R^3.
Taking Gelfand's advice to study the simplest nontrivial example, why don't we look at the group S_3. Now S_3//A_3 is a groupoid with 6 objects arranged as two triangles, a single arrow going between vertices of the same triangle. A first guess of the symmetry 2-group of this groupoid would have G = S_3, H = A_3.
Now try it for the subgroup generated by (12), let's call it C. S_3//C is a groupoid with 6 objects, arranged as 3 pairs with a single arrow between members of each pair. Looking for the 2-group of symmetries, again you'd think there were S_3 many autoequivalences, but now the non-normality of C kicks in, and you can't get any nontrivial natural transformations between them.
Was I being too restrictive about the autoequivalences of S_3//C? On the face of it, there are 6 x 4 x 2 = 48 possible choices for a groupoid of this shape.
Continuing the above, at least we find that the incidence groupoid C\\S_3//C has two components, reflecting the fact that two vertices of a triangle are either the same or different. It looks like the components are not equivalent though. One has two objects and two arrows between any pair, the other has four objects and a single arrow between any pair. Groupoid cardinality 3/2 as it should be.
I had a bit more time to think about Klein geometry while sitting in a Starbucks on Huaihai Zhong Road last night... and I think I made some progress. Unfortunately most of it is not quick to describe. But, I feel more confident that we can get somewhere if we keep at it. And, I think this strategy of yours is a good one:
Taking Gelfand's advice to study the simplest nontrivial example, why don't we look at the group S_3. Now S_3//A_3 is a groupoid with 6 objects arranged as two triangles, a single arrow going between vertices of the same triangle. A first guess of the symmetry 2-group of this groupoid would have G = S_3, H = A_3.
Now, when I first read this I was a bit dismissive, because you seemed to be categorifying a bad idea. Here's the bad idea: you're doing Klein geometry, so you take a group G and a subgroup H, you form the quotient G/H, and then you look at the group of automorphisms of this as a set. This loses lots of information!
But, I now think we just need to firmly grab ahold of some 2-group and work out its Klein 2-geometry. And, the 2-group above is as good as any other - nice and simple, maybe a bit trivial, but so what?
In other words, now we take a group G and a subgroup H, form the weak quotient G//H, and look at the 2-group automorphisms of this as a groupoid. That loses a lot of information - but it gives us a nice 2-group, acting on a groupoid, and we can just work with that.
Indeed, there's a huge amount of very interesting Klein geometry to be obtained simply from the automorphism groups of finite sets - I could talk your ear off about this, but I already have: it's basically the theory of "species".
So, there should be an even huger amount of very interesting Klein 2-geometry to be obtained simply from the automorphism 2-groups of finite groupoids.
So, let's dig in!
I'll start by checking that you worked out the automorphism 2-group of S_3//A_3 correctly. Then let's work out various "sub-2-groups" of this and see what kinds of "figures" they "stabilize".
Hmm, but now I should have dinner.
David wrote:
Now S_3//A_3 is a groupoid with 6 objects arranged as two triangles, a single arrow going between vertices of the same triangle. A first guess of the symmetry 2-group of this groupoid would have G = S_3, H = A_3.
Well, now Lisa is the one taking too long to finish her work so I have a bit more time before dinner....
Let's see. The groupoid S_3//A_3 has 6 objects, namely the 6 permutations of
{1,2,3}
and there's a morphism from one object to another precisely when we can get from one permutation to another by multiplying it on the right by a cyclic permutation. So, there's at most one morphism between any pair of objects. Any pair of even permutations has exactly one morphism going between them, and ditto for any pair of odd permutations. There are no morphisms going between the even permutations and the odd ones.
So yes, the picture of our 2-group S_3//A_3 is just as you said! Two triangles of dots, with the edges of the triangles being double-headed arrows: morphisms and their inverses.
(In category-speak: the coproduct of two copies of the codiscrete category on 3 objects!)
Now we get a choice: to look at the "strict" automorphism 2-group of S_3//A_3, or the "weak" one.
The strict automorphism 2-group has strictly invertible functors
f: S_3//A_3 -> S_3//A_3
as objects, while the weak automorphism 2-group has weakly invertible functors
f: S_3//A_3 -> S_3//A_3
usually known as equivalences, as objects. They both have natural isomorphisms as morphisms.
The weak one is the morally correct one: strength in weakness is the moral of higher category theory.
But first let's think about the strict one. The objects here are strictly invertible functors
f: S_3//A_3 -> S_3//A_3
and these really look just like symmetries of our picture.
What's the symmetry group of two identical equilateral triangles.
We can rotate our triangles, or reflect them, but we can also switch the two!
Rotating and/or reflecting a single triangle gives us the dihedral group D_3, which happens to be the same as the permutation group S_3.
That matches your guess.
But, we also have the ability to switch our two triangles! Switching gives the group S_2 = Z/2.
So, by definition, we get the wreathe product of Z/2 and S_3, which is a group with 6 x 6 x 2 elements, if I'm not mistaken. Either triangle can be rotated/reflected however much you like (6 x 6 choices), but you also can switch them or not (2 choices).
So, we're getting a pretty big symmetry group!
Do you agree with me? Were you indeed trying to work out the strict automorphism 2-group of S_3//A_3?
The next step is to work out the morphisms in our strict automorphism 2-group: natural isomorphisms between the functors I just described. I believe two such functors are naturally isomorphic iff they either both do, or both don't switch the triangles... and that they're naturally isomorphic in at most one way.
Does this sound right?
(Hmm. Could you have gotten a different answer because you were thinking about the symmetries of S_3//A_3 as a 2-group instead of as a mere groupoid? I would prefer to think of it as a mere groupoid, following my desire to simply take a nice little groupoid, work out its symmetry 2-group, and then do Klein 2-geometry with that.)
Could you have gotten a different answer because you were thinking about the symmetries of S_3//A_3 as a 2-group instead of as a mere groupoid?
Yes, I was thinking of S_3//A_3 as a groupoid of weak figures (in this case orientations), and then imagining S_3 acting as the objects of a 2-group of symmetries.
You will note, however, that I raised the possibility that we might need to look at all the automorphisms of the weak quotient groupoid, in my last but one post when I said "Was I being too restrictive about the autoequivalences of S_3//C? On the face of it, there are 6 x 4 x 2 = 48 possible choices for a groupoid of this shape." This was for a 2 element subgroup C.
Doing it this way, though, surely there's no reason to call the quotient S_3//A_3. After all the groupoid's just an equivalence relation on the set {1,..,6}, with a ~ b if a - b = 0 mod 2.
John wrote:
Could you have gotten a different answer because you were thinking about the symmetries of S_3//A_3 as a 2-group instead of as a mere groupoid?
David replied:
Yes, I was thinking of S_3//A_3 as a groupoid of weak figures (in this case orientations), and then imagining S_3 acting as the objects of a 2-group of symmetries.
Sorry, I asked if you were thinking of S_3//A_3 as a 2-group instead of a groupoid, and you said "yes, I was thinking of it as a groupoid...." That's a bit confusing.
I guess you're saying this: you were thinking of it as a groupoid but only looking at some of its symmetries. I'm drastically changing the subject by looking at all of its symmetries, because now I just want to take a few groupoids, work out their automorphism 2-groups, and work out the Klein 2-geometries of those.
Doing it this way, though, surely there's no reason to call the quotient S_3//A_3. After all the groupoid's just an equivalence relation on the set {1,..,6}, with a ~ b if a - b = 0 mod 2.
Right, it's just a groupoid and you can describe it however you like. You could
call it
(Z/6)//(Z/3)
i.e. integers mod 6, weakly mod the "even" ones.
Or, you could say it like I did: the coproduct of two copies of the codiscrete category on 3 elements.
Or, intuitively: 3 uniquely isomorphic things, and 3 other uniquely isomorphic things.
Anyway, whatever you call it, I just like it as an example of a small groupoid with lots of symmetries. Let's call it X.
The strict automorphism 2-group of X has as objects the wreathe product
Z/2 $ S_3
and its morphisms are described as in my previous post.
The trick is to think of all this geometrically.
Our groupoid X is a close relative of the two-point set. Indeed, they're equivalent as categories.
The two-point set is fairly pathetic as Klein geometries go, since its symmetry group Z/2 has only two subgroups: the whole group and the trivial group! The corresponding types of figures are an empty figure and a frame - a complete labelling of every point. You see, in this particular case labelling one point is equivalent to labelling the other!
I hope you can get comfy with the ideas of the "empty figure" and a "frame", which play important roles in Klein geometry. In Klein geometry based on a group G, a type of figure is a subgroup of G, but we should think of it as some extra structure that we can put on a situation - the more structure, the smaller the subgroup preserving it.
The empty figure corresponds to no extra structure, and thus the biggest possible subgroup - G itself.
A frame corresponds to maximal extra structure, and thus the smallest possible subgroup, the trivial group.
Note that the set of empty figures is the one-point set G/G, while the set of frames forms a G-torsor, G/1.
Anyway, that's what one can learn from G = Z/2. The current case should be similar but at least superficially a lot more complicated. In particular, sub-2-groups of G = Aut(X) will correspond to ways of putting extra stuff on X. Sometimes this extra stuff will just be extra structure, but not always!
Q: can you tell when the extra stuff will simply amount to extra structure?
For example, take the full sub-2-group H of G = Aut(X) whose objects fix the first 3 objects of X, but not necessarily the second 3.
Q: What extra structure, or stuff, does this preserve?
Or, take the sub-group H' of G = Aut(X) with all the same objects as G, but only identity morphisms.
Q: What extra structure, or stuff, does this preserve?
Sorry, I asked if you were thinking of S_3//A_3 as a 2-group instead of a groupoid, and you said "yes, I was thinking of it as a groupoid...." That's a bit confusing.
Oh, when you asked whether I was
thinking about the symmetries of S_3//A_3 as a 2-group instead of as a mere groupoid
I took you to mean the symmetries were what was forming the 2-group, not the weak quotient itself. But as you point out I was only thinking about this 2-group in terms of the structure of S_3.
A frame corresponds to maximal extra structure, and thus the smallest possible subgroup, the trivial group.
This is why they call four non-collinear points in the projective plane a frame.
Regarding subgroups of our 2-group, we'll find one which just corresponds to labels on our triangles, another which picks out relative orientation, and so on.
So this is all very Galois. We picture a 2-group G = Aut(X) for some groupoid X. Corresponding to G' a sub 2-group of G, there is X' a groupoid with an arrow to X. Then this X'-->X gets analysed in terms of faithfulness, fullness, and essential surjectivity. We need to translate these qualities over to find corresponding qualities in terms of sub 2-groups.
Your first example labels the triangles, then labels the points of the first of these. I'll dwell on your second over breakfast.
Perhaps two paragraphs above I should be phrasing things more in terms of the 2-groupoid of groupoids isomorphic (equivalent?) to X - the 2-groupoid of empty figures. That corresponds to the 2-group G. Then corresponding to 1 is the 2-groupoid of frames, where there is only one arrow between frames, and one 2-morphism from such an arrow to itself.
Now, your second example is beginning to bug me. We can strip things down to absolute basics. Start with the groupoid, Y, of two objects and a single arrow between any pair of objects. G = Aut(Y) is the 2-group which is isomorphic as a groupoid to Y, with obvious multiplicative structure. G has two sub 2-groups, the trivial one and the discrete 2-group on two objects, H. Now, to the former corresponds the labelling of the objects, one will do. But what form of labelling could correspond to the latter? How can you introduce extra structure or stuff so that there are two automorphisms, but no natural equivalences? Is there something strange here in that as we pass from G to H to 1, the groupoid cardinality goes from 1 to 2 to 1?
If you have nothing new to tell me about points, lines and circles, I shall remain unconvinced.
These were the words of my imaginary agonistic partner at the end of the main post. Now in Starting with SL(2,R) group we find Vladimir Kisil returning to Klein to provide
some surprising conclusions even for such over-studied objects as circles.
Kisil is showing us how much there is still to do in the ordinary Erlangen Program with as basic a group as SL(2,R). He remarks:
The Erlangen program has probably the highest rate of (praised/actually used) among mathematical theories not only due to the big nominator but also due to undeserving small denominator.
What rate can we expect for the Erlangen 2-program?
Note also some interesting thoughts on the EPH (elliptic, parabolic, hyperbolic)- classification.
John wrote:
A frame corresponds to maximal extra structure, and thus the smallest possible subgroup, the trivial group.
David wrote:
This is why they call four non-collinear points in the projective plane a frame.
Good! I didn't know that. I was thinking of some other examples. In general relativity, a frame is an ordered orthonormal basis of an inner product space. If we let our group G be the orthogonal group (e.g Lorentz group) acting on our vector space, such a frame is stabilized by just the trivial group.
In classical mechanics or special relativity we often use the term inertial frame of reference for the same sort of concept, but now G is the Galilee or Poincare group, so we have to specify an origin before specifying a basis.
From Wikipedia:
In mathematics, a projective frame in projective geometry is an (n + 2)-tuple of points in general position in the space K^(n + 1) (K is a random field) from which a projective space KP^n has been projected, one can take the first n + 1 points to form a basis, and the last to be the sum of the others.
Okay, now let's look at sub-2-groups of Aut(X), where X is our favorite little groupoid, consisting of 3 uniquely isomorphic dots and 3 more uniquely isomorphic dots. X looks like two triangles, roughly speaking.
David wrote:
Regarding subgroups of our 2-group, we'll find one which just corresponds to labels on our triangles, another which picks out relative orientation, and so on.
Ah, you may be falling into a little trap here - precisely what I wanted.
All the sub-2-groups you mention come from equipping X with extra structure. This is familiar in ordinary Klein geometry, where X was a set. But there will also be sub-2-groups that come from equipping X with extra stuff! This is new to Klein 2-geometry.
So this is all very Galois.
Yes - as James Dolan has explained, the Erlanger program is secretly the same as Galois theory.
We picture a 2-group G = Aut(X) for some groupoid X. Corresponding to G' a sub-2-group of G, there is X' a groupoid with an arrow to X. Then this X'-->X gets analysed in terms of faithfulness, fullness, and essential surjectivity. We need to translate these qualities over to find corresponding qualities in terms of sub 2-groups.
You've got the right idea, but I think you're making a little level slip here. It's the inclusion of G' in G that we need to analyze, not the inclusion of (some generally nonexistent) X' in X.
Let's decategorify it all one notch - then you'll see what I mean.
When we equip a set X with some extra structure S, this structure is preserved by some subgroup G' of its permutation group
G = Aut(X)
The inclusion map
i: G' -> G
is 1-1. G and G' are one-object groupoids, so i is really a functor. G is equivalent to the groupoid of "unstructured X-sized sets". G' is equivalent to the groupoid of "X-sized sets equipped with extra structure S". The functor i is faithful and essentially surjective, but not full. Using our dictionary, this means i forgets purely structure - not stuff or properties.
Ordinary Klein geometry deals with nothing but this: a single object with extra structure, and the process of "forgetting purely stucture". We could easily enhance it to study the process of forgetting properties and stuff, and more objects, by generalizing it from symmetry groups to symmetry groupoids.
But now we're being a bit bolder, and generalizing Klein geometry to symmetry 2-groups. So, we'll see a somewhat different range of phenomena.
Your first example labels the triangles, then labels the points of the first of these. I'll dwell on your second over breakfast.
Okay, let's see if I agree. I had written:
For example, take the full sub-2-group H of G = Aut(X) whose objects fix the first 3 objects of X, but not necessarily the second 3.
Yes, I agree! This H preserves the structure you describe. We can think of this structure as giving distinguishable labels to the first 3 objects of X. It's not unlike a structure on a finite set where we label or "color" some of its points.
My second example should be more interesting, since I think it illustrates something that only shows up in Klein 2-geometry.
You also didn't answer my very first question, which was supposed to be a clue.
Vladimir Kisil writes:
The Erlangen program has probably the highest rate of (praised/actually used) among mathematical theories not only due to the big nominator but also due to undeserving small denominator.
The Erlangen program is in action throughout 20th century mathematics - most people just don't know it well enough to see it staring them in the face! For example, most people don't know the connection between Galois theory and the Erlangen program - and they don't see how logic, symmetry and topology fit together in this neat circle of ideas.
Lots of people seem to implicitly understand bits of the story - for example, you can find it in Grothendieck's work on covering spaces and the fundamental group, and Toen's recent followup. But the only person I know who has consistently and explicitly explained the power of the Erlangen program is James Dolan. And alas, he does so mainly in conversations - he's only written a little on this issue, and it merely scratches the surface.
It's one of my duties in life to fix this situation. I've started writing up some lecture notes on n-categories and cohomology theory which sketch how Galois theory and Erlangen program fit together, and then sketch their generalization from groups to n-groups and n-groupoids.
In writing this, I found to my horror that the only English translation of Klein's famous announcement of the Erlangen program seems to be lurking here:
Felix Klein, A comparative review of recent researches in geometry, trans. M. W. Haskell, Bull. New York Math. Soc. 2 (1892-1893), 215-249.
Doesn't anyone know a source that's easier to obtain???
I think I've just acquired another duty, which is to put a translated version of Klein's talk on my website. The original German version is available here. Scroll down to see the actual text.
David writes:
What rate can we expect for the Erlangen 2-program?
I guess I have to write my book HDA for our ideas to avoid the obscurity that befell Klein's.
By the way, to poor DM Roberts, who wrote the first post in the July thread - I think your "categorified incidence geometry" approach is interesting, and if I had the energy I'd pursue it in parallel to what I'm doing with David. I'm just too busy!
If you found a nice specific example of a categorified incidence geometry, maybe I could work out its symmetry 2-group and then feed it into the Klein 2-geometry machine that David and I are gradually tuning up. Right now we're using this machine on a sort of silly 2-group, just to get started. Eventually we'll want some nicer examples.
You also didn't answer my very first question, which was supposed to be a clue.
The question being:
Q: can you tell when the extra stuff will simply amount to extra structure?
So we have an inclusion of 2-groups, which are 1 object 2-groupoids. There are three levels to think about, but at the object level we just have one object. As this is an inclusion we're not worrying about faithfulness, but failure to be full can apply at the level of the 1-morphisms or 2-morphisms.
Presumably you're guiding me to think that the answer to your question is whenever G' has a pair of 1-morphisms, all 2-morphisms between them in G are there in G'. This is an ungainly way of saying that the functor between the looped versions of G and G' (i.e., thought of as 1-groupoids) is full. I can't remember whether you have a name for this in your Chicago lectures.
In our running example, this fullness fails, so we have to find a groupoid of figures equipped with extra stuff which allows all 72 morphisms but only identity 2-morphisms. This is what I was trying to do with my stripped down Y with little success. But emboldened...
I begin to see what's confusing me. Back at the level of 1-groupoids, I think of adding extra structure having the effect of cutting down on possible morphisms, e.g., not all set morphisms are group morphisms. And I think of adding extra stuff having the possible effect of allowing extra morphisms. E.g., for the projection on the first component of Set x Set --> Set, there are many maps from [A,C] to [B,C] which map down to the same map from A to B, if C has more than 1 element.
Now here, while we are up a level, still I can't see how adding stuff is going to cut down on our 2-arrows.
I suppose you could just force the conditions 'by hand'. Label our 6 objects, allow any of the 72 morphisms permuting these labels in allowable ways, but then only allow 2-morphisms between identical morphisms. Is there not a more natural picture?
So, we've got a 2-group G, another 2-group G', and we're thinking of them 2-groupoids with one object. And we've got an inclusion
i: G' -> G
and I asked:
Q: can you tell when the extra stuff will simply amount to extra structure?
David replied:
...whenever G' has a pair of 1-morphisms, all 2-morphisms between them in G are there in G'.
Right!
I can't remember whether you have a name for this in your Chicago lectures.
Usually people say a 2-functor is 2-full when it satisfies the condition you just wrote down. Since we know that
i: G' -> G
is faithful and essentially surjective, just by virtue of being an inclusion of 2-groups, if it's 2-full it can only lack one property in this list:
0. essentially surjective (very roughly, "surjective on objects")
1. full (very roughly, "surjective on morphisms")
2. 2-full (very roughly, "surjective on 2-morphisms)
3. faithful (very roughly, "surjective on 3-morphisms", which are equations between 2-morphisms here)
... namely, it can only fail to be full. Now, there's this other list of definition:
0. A 2-functor between 2-groupoids that only fails to be essentially surjective is said to forget purely properties.
1. A 2-functor between 2-groupoids that only fails to be full is said to forget purely structure.
2. A 2-functor between 2-groupoids that only fails to be 2-full is said to forget purely stuff.
3. A 2-functor between 2-groupoids that only fails to be faithful is said to forget purely 2-stuff.
This is supposed to explain why your answer is right, given that you already understand the yoga of properties, structure and stuff.
But I see you're still a bit puzzled:
I begin to see what's confusing me. Back at the level of 1-groupoids, I think of adding extra structure having the effect of cutting down on possible morphisms, e.g., not all set morphisms are group morphisms.
Actually, adding extra structure has two effects! It can increase the number of possible objects - there are lots of ways to make a set into a group. And, it can decrease the number of possible morphisms - a function between sets may not be a homomorphism between groups.
(I hope you see how this is related to the concept of Euler characteristic. Euler realized that there are two ways to increase the Euler characteristic of an archipelago: either build a new island, or remove a bridge between them.)
And I think of adding extra stuff having the possible effect of allowing extra morphisms.
Actually, adding extra stuff has two effects! It can increase the number of possible morphisms. And, it can decrease the number of possible 2-morphisms.
Now here, while we are up a level, still I can't see how adding stuff is going to cut down on our 2-arrows.
Well, you must at least agree that it makes for a nice consistent pattern. It's also easy to see why you may never have noticed this possibility. If you're dealing with mere groupoids, decreasing the number of possible 2-morphisms is indistinguishable from increasing the number of possible morphisms - since the only 2-morphisms are equations between morphisms! But in the world of 2-groupoids, they really are different (though intimately connected) phenomena - and they both happen.
You may feel happier if you think of "adding extra stuff" to objects as the same as "adding extra structure" to their morphisms - and note that when we do this, it's possible that a 2-morphism between them no longer preserves this extra structure.
So, I think that what I just should help us solve the puzzle that's bugging you:
Or, take the sub-group H' of G = Aut(X) with all the same objects as G, but only identity morphisms.
Q: What extra structure, or stuff, does this preserve?
Especially the bit about "adding stuff on objects is the same as adding structure on morphisms".
Let's see if it does...
Structure on arrows, eh? Well, I had already thought of putting labels on them, but had rejected this due to the thought that to preserve a labelled arrow is to preserve its source and target, in effect to label them. But I think I see why that's wrong now.
Our running example must already be a case of adding extra stuff. The 2-group on two triangles we are considering with its 72 morphisms and its 2-morphisms is a sub 2-group of the 2-group with G = S_6 = H. This is the 2-group of symmetries of the coarse groupoid on 6 objects, i.e., one morphism between every ordered pair of objects. As this 2-group inclusion is not 2-full, structure must have been added to arrows during the specification that the objects form 2 triangles. I guess this structure is the colouring of the 18 arrows in Z_6//Z_3 which then had to be preserved.
Is this pointing us to the remark I made earlier to the effect that the 2-geometry corresponding to a group K is the story of the 2-group with G = K = H and its sub 2-groups? So, projective plane 2-geometry is the study of G = PSL(2,R) = H and its sub 2-groups, from the amorphous empty figure preserved by the whole 2-group, right up to the trivial 2-group which preserves the specified frame of 4 labelled points in general position.
there's a huge amount of very interesting Klein geometry to be obtained simply from the automorphism groups of finite sets - I could talk your ear off about this, but I already have: it's basically the theory of "species".
So, there should be an even huger amount of very interesting Klein 2-geometry to be obtained simply from the automorphism 2-groups of finite groupoids.
So if species are categorified series N[X], what are we finding here? Do we need a sequence of groupoids, just as we have a sequence of sets to put structures on? The coarse finite groupoids might do, but then why must we restrict ourselves to a maximum of one arrow between two objects?
More generally won't our 2-geometry be more interesting if we allow more than one 2-morphism between pairs of morphisms? Hmm, the fundamental 2-group of the sphere has its interesting structure up at the 2-morphism level.
At some point you're going to have to initiate us into the mysteries of quotienting 2-groups. If with ordinary groups, we can lose information by looking at a set of cosets G/H rather than the groupoid G//H, with 2-groups will we have 3 levels of information, a set G/H, a groupoid G//H, and a 2-groupoid G///H?
I missed what John said about this, but let me see.
Given a group G and a subgroup H, we can form the action groupoid of H acting on G. Objects are elements of G, unique morphisms go from g to hg.
This groupoid is a 2-group precisely if H is a normal subgroup of G.
(Because then we can consistently specify the action of G on H to be by conjugation.)
This 2-group corresponds to a crossed module coming from the embedding
H --t--> G.
Like every strict 2-group, this 2-group is equivalent to the skeletal one with coker(t) of objects and ker(t) of automorphisms of every object.
In the present situation, ker(t) is the trivial group and coker(t) is the quotient group that we are after.
So we find that iff H is normal in G then the action groupoid of H on G is a 2-group equivalent to the discrete 2-group with group of objects G/H.
Right?
That's what we need to categorify.
Ok, so all we need to do is to formulate the situation cleanly, then the formalism does the rest.
So what's an action groupoid?
Given a space X and a group G acting on it, the action groupoid is the category with objects being X and with space of morphisms being G x X, with the obvious source, target and identity morphisms.
So now we have a 2-space, namely some 2-group X, and a 2-group acting on it, namely some sub-2-group G. We get the same diagrams as before. But now these diagrams live in Cat. Both our object of objects, as well as our object of morphisms, are categories.
So I guess we here need to be dealing with double categories.
So we find that iff H is normal in G then the action groupoid of H on G is a 2-group equivalent to the discrete 2-group with group of objects G/H.
Right?
That's what we need to categorify.
Yes, but we're not just interested in normal H. One of our starting points way back in May was H = stabilizer of a point, sitting in G = Euclidean group. Then G/H is the homogeneous space of points.
We're interested in any 2-group and its sub 2-groups.
Sorry, I should have followed your discussion more closely.
I think I was making two points: I made a simple remark on how to define an action 2-groupoid, saying that it seems like it will in fact have to be something like a double groupoid.
And I remarked that the ordinary action groupoid of a subgroup is a 2-group precisely if the subgroup is normal, in which case the action 2-group is equivalent to the quotient group.
So just ignore this part. :-)
I made a simple remark on how to define an action 2-groupoid, saying that it seems like it will in fact have to be something like a double groupoid.
Sounds plausible. If we have h--k-->kh, acting on g--m-->mg, then do we find this in an action double groupoid:
hg ---k--->khg
.|.....................|
hlh^-1.........khl(kh)^-1
. |.....................|
V....................V
hlg---k---->khlg
Off to dinner in the old town of Tuebingen now, so one last thought. Using our old notation E for Euclidean group, P for stabilizer of a point, and letting K-K be the 2-group G = K = H, might it be that the 2-(double?) groupoid of weak points we sought is E-E//P-P, and incidence relations are P-P\\E-E//P-P?
A long time ago David wrote:
I guess the biggest worry in a venture of this kind is that all you achieve is a repackaging of what's already known.
Actually I'm not at all worried about that. True, homotopy theory is secretly the study of infinity-groupoids, so everything we are doing can be translated into the language of homotopy theory, but while this is very interesting in itself, when we finally get our hands on some interesting Klein 2-geometries after having built up the strength to understand them, we're bound to think of all sort of questions that homotopy theorists would never imagine.
To my mind, the biggest worry in a venture of this kind is that we'll get worn out from doing the necessary spadework before we hit gold.
Or, to be really precise, I'm worried that you'll get worn out. I'm used to how these things go: I spent about 5 years messing with 2-groups and higher gauge theory before everything fit together thanks to a lot of work by various people including Alissa Crans, Danny Stevenson and Urs Schreiber. The basic plan was simple and robust: use 2-categories to describe parallel transport along 2-dimensional surfaces. The final results are simple and robust, too. But somehow getting from one to the other took a frightening number of mathematician-years.
As Piet Hein put it:
Problems worthy of attack
prove their worth by fighting back!
But, I'm impressed by how persistently you're going after this Klein 2-geometry stuff. In fact, tonight I'm too tired to keep up with you... you have a big general conjecture about categorifying ordinary Klein geometries to get Klein 2-geometries:
...the 2-geometry corresponding to a group K is the story of the 2-group with G = K = H and its sub-2-groups.
... and it sounds worth studying, but right now I don't have the energy to test it!
I'll say this, though, for what it's worth:
There's another 2-group canonically associated to a group K, namely the automorphism 2-group AUT(K), with G = Aut(K) and H = K, where t: H -> G sends each element k in K to the operation of conjugating by K, and the action of G on H is the obvious one.
When people tried to categorify the theory of K-bundles, without quite knowing at first what they were doing, they basically invented the theory of AUT(K)-2-bundles - but under a different guise: they called it the theory of "nonabelian K-gerbes".
It turned out to be a useful idea. So, at least in one context, the natural way of "boosting" a group K to a 2-group was to use AUT(K).
Note that when K has trivial center and all its automorphisms are inner, this 2-group is the same as the one you're talking about!
This happens, for example, when K is SL(2,C) (but not SL(n,C) for n > 2) or the permutation group S_n (unless n equals 6).
when we finally get our hands on some interesting Klein 2-geometries after having built up the strength to understand them, we're bound to think of all sort of questions that homotopy theorists would never imagine.
I guess one could define a spectrum of 'repackagings', from pointless rewording right up to powerful reformulation. The fear I was expressing concerned the lower end.
As for my interest waning, if some kind industrialist could fund this research I think I would be happy to stick around for quite a while. As it is, it's a welcome relief from the spinning head that information geometry and statistical learning theory, topics of my day job, are giving me.
Back to 2-groups, how much has been worked out? There's a fairly obvious definition of normal sub 2-group. Is there anything like a Jordan-Holder theorem? Are there structure theorems, as with groups? More relevantly for us, what could the action ?-groupoid of a sub 2-group acting on a 2-group be?
John Baez wrote:
"Note that when K has trivial center and all its automorphisms are inner, this 2-group is the same as the one you're talking about!
This happens, for example, when K is SL(2,C)"
And, for example, for E8 -- unless I am dreaming.
Is it obvious how to think of the AUT(K) 2-group as the symmetry 2-group of a geometry? Most of the way through the discussion I've been thinking of the morphisms as the symmetries and the 2-morphisms as the symmetries between symmetries. I can see that when K has trivial center and all automorphisms are inner that AUT(K) is the same as what I called K-K. But, when there are outer automorphisms, can we think of AUT(K) in terms of symmetries and symmetries between symmetries, other than as a sub 2-group of M-M where M = permutations of the underlying set of K?
David asked
"can we think of AUT(K) in terms of symmetries and symmetries between symmetries"
I would reply to this with "Yes, sure", but possibly my way of thinking about this is not precisely what you are looking for.
Take any category C and look at its automorphism 2-group
Aut_Cat(C).
So the single object is C, and a 1-morphism is an automorphism of C, hence an invertible functor C->C, hence a symmetry of C.
Now, these 1-morphisms can themselves be regarded as objects, and can have isomorphisms between them. This are invertible natural transformations, and they form the 2-morphisms of Aut_Cat(C).
In the special case where C = Sigma(G) is a group regarded as a category with a single object, Aut_Cat(C) is the Automorphism 2-group of G.
So 1-morphisms are automorphisms of G, which can justly be regarded as the symmetries of G.
But some of these symmetries differ only by an inner automorphism. Composing with an inner automorphism is like a symmetry operation on the collection of 1-morphisms, even though this now yields a groupoid instead of a group.
Hm, that was possibly too tautological to explain anything.
Yes, a tad tautological. Perhaps it doesn't amount to much, but when you think of the 2-group S_6-S_6 (in my notation) acting on the coarse groupoid on 6 objects, then extend S_6-S_6 to the 2-group AUT(S_6), which as you say collects symmetries between symmetries of Sigma(S_6), the interpretation seems very different. Perhaps I should look through John's page for an answer.
Pardon the high level of redundancy in my posts; I just imagine someday someone will read this stuff we're writing and have trouble understanding it unless it includes lots of repetition.
A Tale of Two 2-Groups
1. One 2-group built from a group K has K as objects and K as the morphisms of form
f: 1 -> x
(where f = k is in K and in fact x = k as well).
This group is weakly trivial: it's equivalent to the trivial 2-group. BORING! (Or maybe not?)
2. Another 2-group built from a 2-group K has Aut(K) as objects and K as the morphisms of form
f: 1 -> x
(where f = k is in K and x is the automorphism given by conjugating with k).
This 2-group is called AUT(K) and it plays a basic role in the theory of nonabelian gerbes. INTERESTING!
But sometimes the two 2-groups are isomorphic! How can something weakly trivial be interesting? That's the big puzzle. But first, let's remember when this funny situation happens.
Not surprisingly, the two 2-groups are isomorphic when the map sending k in K to the operation of conjugating by k is 1-1 and onto.
This map is 1-1 when K has trivial center. It's onto when every automorphism is inner.
Getting both of these at once is an interesting challenge. I made some mistakes in my last post about this, so let me try again.
It works for the permutation group S_n unless n = 2, when there's a nontrivial center, or n equals 6, when there's an outer automorphism.
Urs wrote:
And, for example, for E8 -- unless I am dreaming.
No, you're not dreaming!
Let's think about how it works for simple Lie groups in general. We'll start with complex simple Lie groups to keep things manageable, but at some point we'll be forced to switch to real ones.
To make the center trivial, we need to take the simply-connected form of our simple Lie group and mod out by its center, which is a discrete group, Z. The result is the adjoint form of our simple Lie group; its fundamental group is Z.
For example, if we do this with SL(n,C), we get
SL(n,C)/Z = PSL(n,C)
I forgot to do this modding out in my last post on this subject. I also forgot something else, as you'll see.
How to make sure all the automorphisms of this group are inner? Since the adjoint form of a simple Lie group can be canonically defined in terms of its Lie algebra, automorphisms of the adjoint form are the same as Lie algebra automorphisms. So, we need to make sure all the Lie algebra automorphisms are inner.
You'll read various places that outer automorphisms of a simple Lie algebra come from symmetries of its Dynkin diagram, and this is basically true. So, we need to make sure our Lie algebra has a Dynkin diagram with no symmetries!
This rules out all the Dynkin diagrams except A_1, B_n, C_n, G_2, F_4, E_7, and E_8.
So, these are the simple Lie groups where our two Lie 2-groups match.
Not quite! This theorem about outer automorphisms of the adjoint form coming from Dynkin diagram symmetries applies only to complex-analytic automorphisms! Complex conjugation also gives an outer automorphism of our Lie groups! If we take that into account, there are no complex simple Lie groups with trivial center and all automorphisms inner!
There are different ways to get around this, though:
1) Work with the compact real adjoint form of our Lie group. In the A_1 example this means using
PSU(2) = SO(3)
instead of
PSL(2,C) = Lorentz group.
Or, we can use the compact real adjoint form of E_8. In this example the adjoint form equals the simply-connected form, so we don't have to fuss about that particular issue.
2) Declare that we're working with groups (and 2-groups) not in the category of smooth manifolds, but in the category of complex-analytic manifolds. Then PSL(2,C) is okay, as is the (unique) complex form of E_8.
SL(2,C) is still no good, of course: it has center +-1.
Okay, so much for that. The really interesting puzzle is how a boring 2-group can seem interesting: for example, how a weakly trivial 2-group can have nontrivial sub-2-groups.
More on that later.
The really interesting puzzle is how a boring 2-group can seem interesting: for example, how a weakly trivial 2-group can have nontrivial sub-2-groups.
Doesn't the boring old plane contain the circle, 3-space contain all the knots? And the boring old set of all strings of letters contain the collected works of Shakespeare?
But I guess there's containment and containment, and being a sub-2-group's quite a restricted form of containment.
On the other hand, we can wonder about the internal structure of points, as Cartier does. On p.403 of A Mad Day's Work he says:
We begin to suspect that not all points are alike - there are several species of monads. The question thus arises of finding out whether a point can have symmetries.
John Baez wrote:
"The really interesting puzzle is how a boring 2-group can seem interesting:"
For instance, a flat 2-bundle with structure 2-group G->G is nothing but an ordinary and in general non-trivial G-bundle with non-trivial connection.
Even though 1->1 and G->G are equivalent 2-groups, the latter suggests other deformations than the former.
For instance if G has a central extension H, we may form the 2-group H->G, which, at least as a 2-group in Set, is equivalent to U(1)->1.
Every H->G 2-bundle is the lifting gerbe of the attemted lifting of a G-bundle through 1->U(1)->H->G->1 to an H-bundle.
So here we gain something by regarding a G-bundle as a trivial G->G 2-bundle: it allows us to perform the lift to an H-bundle - albeit possibly a "twisted" H-bundle.
From this point of view, the fact that G->G is equivalent to the trivial 2-group says that every G-bundle produces a trivial lifting gerbe when not lifted. ;-)
A similar way of thinking is helpful for thinking about Chern-Simons gerbes (->).
So I guess what I am saying is that it may happen that remembering irrelvant details is useful.
It's like remembering the action groupoid of the action of a subgroup on a group may be more useful than just remembering the quotient.
In a sense, this might be an example of what David refers to as "internal structure of a point", I guess.
We might want to replace the point (say the 2-group 1->1) by something which is less trivial, but still equivalent to a point (the 2-group G->G, say).
Some people in the blogosphere have recently discussed this in a different context: it is usually more convenient to study gauge theory _without_ dividing out by gauge transformations.
In the end, gauge symmetries are just internal structure of a point, in a sense. But it pays to keep track of it and not collaps everything to the point that it is equivalent to.
Actually, there is more to say about the use of weakly trivial higher groups in the context of higher parallel transport.
One nice way to think about higher parallel transport along p-dimensional volumes is in terms of higher order pseudofunctors from the fundamental 1-groupoid of base space to some partly weakly trivial (p+1)-group.
In a way this is just a reformulation of what I said in my previous comment, but maybe a more suggestive one.
Consider ordinary connections. Recall from a previous discussion that Lawvere teaches us that we should probably think of these as pseudofunctors on the fundamental path groupoid.
Namely, to any two points in the same connected component we associate a group element, which is to be interpreted as the parallel transport (in a trivial bundle, say) along a fixed "straight" path between these two points.
While composing two sides of a triangle is, in the fundamental groupoid, the same as the third side of the triangle, this is in general not true fro the composition of the parallel transport along these three sides, unless our connection is flat.
Instead, the transport along two of the sides differs from that along the third edge by the integral of the connection's curvature over any surface cobounding these paths.
Technically, this means that the connection, together with its curvature, is a pseudofunctor from the fundamental 1-groupoid to the 2-group G-->G.
Next consider a trivial 2-bundle with 2-connection and structure group H--t-->G. I believe (but you should check this before believing it) that we always get a 3-group
ker(t)--> H --t--> G .
We can describe the 2-connection as a pseudofunctor from the fundamental 1-groupoid to this 3-group.
The fact that the 3-curvature of this 2-connection is bound to take values in ker(t) is actually precisely what is found when looking at 2-connections as 2-functors from 2-paths to the 2-group H-->G.
This pattern should continue for all higher n-connections.
So in all of these cases it is useful to take some n-group and blow it up to an (n+1)-group which is partly redundant.
Maybe we should return to our running example. Recall we were looking at the symmetries of a pair of triangles, and found a 2-group with 72 objects, for each of which there are 36 arrows.
This is a non-2-full sub-2-group of what we have been calling S_6-S_6, a group with 720 objects and 720 x 720 arrows. Factoring at the level of objects is straightforward - we see the 10 components of the action groupoid, corresponding to the 10 ways of choosing a pair of triangles out of 6 points. But what happens at the level of arrows? You'd kinda think that you'd end up with (720 x 720)/(72 x 36) = 200 'components' of arrows, represented by 2 arrows between any ordered pair of components of objects.
There's something '1/2'-like about this quotient.
Continued here.
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