Baratin Freidel Beef Baez and the Higher Yum-Yums

[marcus]

something new is happening, indicated by John Baez starting that "Baratin Freidel" thread

I am starting this thread so that I can be excited without risking inadvertently spamming the other thread (with preliminary MISunderstanding and misguided attempts to be helpful)

it looks to me like they are finding the bones of a new dinosaur that they didnt know was in the ground

Baez has been to the dig and he knows what is emerging and so he started a thread to encourage us to understand twogroups ahead of time.

because if something new is coming out it helps to have a little preparation so that it does not sound totally strange

Beef (BF theory) seems to be important and in the usual BF theory the differential forms can be Lie group/algebra valued and I dimly suspect that you could have a BF theory that was twogroup-valued.

We had company to the house and all my arranged piles of papers all over the livingroom had to be picked up and put in a large pile in the study (which I never use) and for two days I can't find any paper. That is my excuse. We all have our excuses.

the main thing is that our whole civilization is built on the symmetry onegroup and we can't imagine anything more fundamental beautiful and basic than that----and when we were young impressionable college kids they told us about onegroups----and they NEVER MENTIONED TWOGROUPS. when we first heard twogroups mentioned it seemed that this was just another case of mathematicians doing what they always do and that it could not possibly be important[joke]. BUT NOW IT SEEMS POSSIBLE THAT TWOGROUPS MIGHT ACTUALLY BE a basic thing that is important to how the universe is being run. so I began to experience twogroup shock, and was actually wondering why, when I was young and happy and learning some of the most beautiful things in the world why didn't my teacher, Ralph Abraham, a very cool person, why did he not tell us about twogroups. so then we would have been prepared for this shock, if it is really happening. (he would have had to travel into the future to know that he should be introducing those as well as ordinary kind, so of course the omission is forgivable)

 

[marcus]

when you look in JB "higher Y-M" paper, or in the Perimeter talk notes, you see this petal-diagram-----this is an icon of a onegroup

the single object is the dot in the middle, and the petal-outlines are the arrows looping out from that central dot and coming back-----the petals are {curved} single arrows ->

that is the ordinary idea of a onegroup (which in the old day we used to call a GROUP and was for us the holy transubstantiated Madeleine cookie of mathematics, but now we call onegroup instead, to acknowledge that there is not just one kind of holy cookie)

and on the same page you can see the icon of a twogroup
it is the same petal-diagram but with a new layer of arrows
which are drawn as double arrows =>

each double arrow, instead of taking the central dot to itself,
is taking one petal to another
so it is the same flower icon but with cycled petals, or pied petals. (pied as in Gerard Manley Hopkins, or as in magpie)

it turns out these icons for onegroup and twogroup are NOT in the Higher Y-M paper but in another printout I mislaid at the same time---JB notes for the Perimeter talk: "Higher-dimensional Algebra, a Langauge for Spacetime". I just found that one a few minutes ago. It seems to have been a hectic couple of days but maybe things will settle down.

 

[marcus]

now I have to forget about these two flower icons because I don't get the message of them well enough

and think about crossmodules.

the reason is that whenever you meet something new you should always, at least once, try to imagine it in the dumbest, least abstract, most clunky possible way.

then later you can imagine it in more elegant ways

a crossmodule is the clunkiest way to look at a twogroup

as we know from the flower icon, a twogroup has TWO LAYERS OF ARROWS, and morally each layer of arrows is associated with an ordinary classical onegroup-----a group standing for the onemorphisms and a group standing for the twomorphisms

so the twogroup is a pool in which a pair of Koi are swimming, called G and H. If we can join these two Koi, we can reconstruct the pool.

you can actually make a twogroup by puttng together two ordinary onegroups in a certain way

to do this you need a homomorphism of H into G
and an action of G upon H.
this seems like a fairly simple reciprocal coupling between two groups. now maybe I should let the dust settle, maybe someone else wants to discuss this.

I think I want to talk some either to myself or other people about this crossmodule coupling of two ordinary onegroups

and besides that, there are several other things we should try to cover, hopefully before the Baratin Freidel paper comes out

 

[Mike2]

now I have to forget about these two flower icons because I don't get the message of them well enough

It always helps me to understand the contexts and relevance of things in order to get it to stick in my memory.

So is a two-group something we would use to transform a U(1) group into a SU(2) group which would then show us how two diffent types of particles/force can ultimately be unified by a two-group? Or some such thing as that.

 

[marcus]

It always helps me to understand the contexts and relevance of things in order to get it to stick in my memory.
...

me too

So is a two-group something we would use to transform...

I suspect not but can't say for sure. can't serve as teacher or authority about twogroups. with luck someone more certain will help adjust your understanding (and mine)

 

[Hurkyl]

Have you heard of the fundamental groupoid of a nice topological space? This is a category built as follows:

The objects of this category are the points of the space. (Though you only really need to use one point from each connected component)

The arrows of this category are the "rubber paths" that connect two points. By this, I mean that if we had two different paths, such as:

  -->--
 /     \
*       *
 \     /
  -->--

where we can take the first and bend and stretch it (but keeping its endpoints fixed!) until it becomes the other path, then we want to consider them to be the same path.

We multiply two paths by concatenation -- we simply attach the endpoint of the first path to the startpoint of the second path, and then detatch that point from the space (so that the path remains fixed only at its endpoints)

Note that every path as an inverse: simply reverse its orientation! If you multiply them, then it can "snap back" to the constant path that never leaves its origin!


The fundamental group is formed by doing the above, but using a single point, so all of the arrows are actually loops.

It turns out that every group is equivalent to a fundamental group, so we can actually take those flower diagrams literally! (The same is true for groupoids)


The fundamental 2-groupoid is built in the same sort of way. We use points, paths between points, and rubber sheets between paths. (Note that we're now using rigid paths that can't move around)

If we use only a single point of our space, then we're looking at its fundamental 2-group.

Again, any 2-groupoid is equivalent to a fundamental 2-groupoid -- I think (but am not sure) the same is true about 2-groups.


This is a nice, geometric picture. I feel like there should be a nice, algebraic picture, but the classification theorem doesn't seem particularly intuitive yet.

 

[marcus]

it's nice, Hurkyl. You put something real into it.

 

[Hurkyl]

Oh, and part of the connection to the algebraic depiction you're looking at can be cleanly seen.

Suppose I have a fundamental 2-group. This can be described as you've noted by a group G, an abelian group H, and some other data.

G is simply the fundamental 1-group -- it's what we get by "rubberizing" all of our paths: we change isomorphic paths (meaning that there's a rubber sheet between them) into equal paths.

H is simply the set of rubber spheres through our point. These can be thought of as the rubber sheets that connect a constant path (which look like a single point) to itself.

I have trouble picturing the other stuff geometrically.

 

[marcus]

cheers
this is better than watching television
good pictures!

You just reminded me that, at least in one example I've looked at, when a twogroup is constructed as a crossed-module from a pair G and H, one of these is ABELIAN (meaning that multiplication is commutative)

Hurkyl, you could try simply stating the definition of a crossmodule to me. I am still finding it too slippery to grasp from the available print-out.
================

I think I have identified a couple of confusing typos on page 10 of "Higher Y-M"

At the bottom of page 10, in example 6, it says "G acting on H via RHO" when it means "via alpha".
One can go on a goosechase looking for what rho could be.
and also in example 6 it says "with t:G -> H the trivial homomorphism" when it should probably say
"with t: H -> G the trivial homomorphism"

 

[marcus]

the way it looks to me now is this

there are two or three ways to define a twogroup and in particular the Poicare Lie twogroup.

One way is if you understand TWOCATEGORIES. Then a twogroup is a twocategory with a single object and two layers of morphisms

the bottomlayer onemorphisms are morally the usual lorentzgroup and the upperstory twomorphisms are the usual poincaregroup.

in the other thread there is a post where Baez says something like "a twogroup is a twocategory with a single object and both layers of morphisms are invertible."

=========
that is one way, but it is rather abstract. there is another way
that he uses in "Higher Y-M" paper.
it uses ORDINARY category theory.

In that case you have the objects C₀ and the morphisms C₁ and an identity-finding map from former to latter, and source and target maps s, t from latter to former------the kicker being that these cees are groups and these maps are homomorphisms.

this is looking better. In the Poincare twogroup case it seems to me that the objects C₀ might turn out to be the ordinary Lorentz and the morphisms C₁ turn out to be the usual Poincare

the puzzle is saying to yourself what the identity finding map should be and the source and target maps.
===========
the third way is cross modules and the example I would most like to understand is example #9
on page 11 of "Higher Y-M"
there we have a group G = usual Lorentz
and an abelian group H = R⁴
with usual action of Lorentz on R⁴
and he says form the semidirect product of G with H (using that action) which is the usual Poincare'
and then let THAT be C₁, while C₀ is just G alias usual Lorentz group.
And in this way he gets back to the earlier definition of a twogroup where you have cees and the three special maps.
==============

time for supper!

 

[marcus]

when he builds the Poincare twogroup using G and H, where G is the usual Lorentz and H is R⁴
he has a semidirect product of H and G with generic element (h,g)
then recalling that C₀ = G = usual Lorentz
and C₁ = semidirect product = usual Poincare
and the identity-finding map C₀ -> C₁
just sends g to (0,g) it is the inclusion as subgroup

and the source and target maps are very simpleminded
maps C₁ -> C₀
which simply FORGET THE TRANSLATION PART
(h,g) -> g
I think it turns out like that because in example 6 he says to use the TRIVIAL homomorphism from H -> G (that is where the typo occurred)
so that would always give the G identity
and then further up the page he specifies the target map in a way in which, if that homomorphism really is trivial, would just say
(h,g) -> g.
================
and then when he comes to example 9 which is really where he does Poincare twogroup, he says to do it AS IN example 6
================
so the way I picture it the TWOCATEGORY picture of the Poinc twogroup would be like an icecream cone with a single object * which is the point of the cone---the low-end tip of it.
and on top of that scoop one of morphisms which is the usual Lorentz
and on top of that scoop two of morphisms which is the usual Poincare.

and somehow the usual Poincare will allow itself to be treated as morphisms OF the scoop underneath which is usual Lorentz

by, I guess, a very simpleminded device that every upperlevel morphism is of the form (h,g) and when you want to know its source and target you just throw away the h information and both the source and target are g.

all commonsense says it can't be that simple. maybe something clever happens when you start checking the horizontal and vertical composition rules

well we are supposed to watch a Gilbert and Sullivan tonight.
========
I still have misgivings about this. Usual-type groups make such good sense. there is the group of the square where you flip your textbook or the mattress on the bed. Where is there a twogroup that is so inescapably obvious as that? Baez did not promise. He indicated it MIGHT work out that higher algebra would be the key to finally doing QG right. My superstition is that there must be a very good chance of that. It's dark and there is no road and somebody has a compass with a glow-in-the dark needle, so i listen to that person.
I guess we will know better soon when (if all goes as expected) Baratin and Freidel show us their new paper.

 

[Hurkyl]

I looked at the definition of a crossed module, and it didn't sink in... and I didn't bother looking at it again because I was happy with the category theoretic POV. (I've become fairly interested in "higher algebra")

I'm really tired, so I won't look at it again now, but maybe I will later. (I don't think I have the paper you mention, BTW)



Incidentally, there is a natural algebraic origin of some 2-groups, which might be related to what you're looking at. (I haven't really looked at the papers on Lie 2-groups yet)

If you have something like a group or a category, its automorphisms actually form a 2-group. For example, in the case of a group G, we have:

The objects of AUT(G) are the isomorphisms of G to itself.
The arrows of AUT(G) are conjugations.

In particular, an arrow between two isomorphisms p and q is simply an element g of G such that:

g p(x) = q(x) g

for every x. (I might have those arrows pointing in the wrong direction!)


In the general case, the objects of AUT(C) are invertible functors (or maybe weakly invertible functors!) from C to itself, and the arrows are natural transformations.


I notice that R^4 is a vector space -- that's just a group with some extra structure. The Lorentz group is a subgroup of the automorphisms of R^4. I imagine the Poincaré 2-group, then, will be a subgroup of the (2-group of) automorphisms of R^4! Maybe if we consider enough extra structure on R^4, like a bilinear form, then its automorphism 2-group really will be the Poincaré group!


Bah, I'm going to have to look again at how to go from $(G, H, \rho, \alpha)$ to an actual 2-group... maybe that will make all clear. Have you read Baez's papers on "Higher dimensional algebra"? Issue V is specifically on 2-groups, and VI is on Lie 2-algebras.

 

[selfAdjoint]

If you have something like a group or a category, its automorphisms actually form a 2-group. For example, in the case of a group G, we have:

The objects of AUT(G) are the isomorphisms of G to itself.
The arrows of AUT(G) are conjugations.

In particular, an arrow between two isomorphisms p and q is simply an element g of G such that:

g p(x) = q(x) g

for every x. (I might have those arrows pointing in the wrong direction!)


In the general case, the objects of AUT(C) are invertible functors (or maybe weakly invertible functors!) from C to itself, and the arrows are natural transformations.

I was thinking of AUT(G) last night as an example of how you could think of one object (G) with many morphisms (the automrphisms if G), and yes then they form a group so that direction checks out for the general category definition of a group, but now we need the other direction.

So here's a question for you algebraists: is it known that ANY group can be expressed as the automorphisms of some other group? Surely if that's known it's somebody's famous theorem?

 

[marcus]

...
So here's a question for you algebraists: is it known that ANY group can be expressed as the automorphisms of some other group? Surely if that's known it's somebody's famous theorem?

what a nice question! This is hunch and not knowledge: I suspect it can't be (but as you know am sometimes incautious about making a fool of myself). Shhh, don't tell Hurkyl, or it will get him distracted from higher algebra.

 

[marcus]

... Have you read Baez's papers on "Higher dimensional algebra"? Issue V is specifically on 2-groups, and VI is on Lie 2-algebras.

No I have not yet.
What i was looking at this morning is the Perimeter lecturenotes
"Higher-dimensional algebra: a language for spacetime"

If you feel like it, post some links. I will hunt for something just called "Higher-dimensional algebra" with some issues numbered like that---probably it will turn up quickly.

for people just looking into the thread, the main link is
http://math.ucr.edu/home/baez/quantum_spacetime/

and what I meant by the Perimeter lecturenotes "A Langauge for Spacetime" is what you get if
you click on "transparencies"----the lecture slides serve kind of as notes because they are fairly complete

[edit: mistaken guess]A/H/H! W/h/a/t/ H/u/r/k/y/l/ was talking about is [edit: not]another link from that same page:
http://math.ucr.edu/home/baez/planck/
"Higher-Dimensional Algebra and Planck-Scale Physics "
it is from January 1999 and it has half a dozen numbered sections

So you don't have to give me the link, Hurkyl. I see what it is, and i will have a look.

 

[Hurkyl]

Actually, I was thinking about this:

http://math.ucr.edu/home/baez/hda5.pdf

(You can get the others by playing with the number! A google search on the title works too)

 

[marcus]

Actually, I was thinking about this:

http://math.ucr.edu/home/baez/hda5.pdf

(You can get the others by playing with the number! A google search on the title works too)

thanks! today risks becoming a Printout-feast. must use restraint.
was also looking just now at the Baez/Schreiber paper "Higher Gauge Theory". briefly decided to throw restraint to the winds.

 

[Hurkyl]

So here's a question for you algebraists: is it known that ANY group can be expressed as the automorphisms of some other group? Surely if that's known it's somebody's famous theorem?

This certainly seems wrong. In fact, I think Z_3 is a counterexample, but I can't prove it.

I still have misgivings about this. Usual-type groups make such good sense. there is the group of the square where you flip your textbook or the mattress on the bed. Where is there a twogroup that is so inescapably obvious as that? Baez did not promise.

This has been bothering me. So much, that since I thought about replying, that I've spent a bunch of time thinking about it when I should be sleeping! I think I've identified a couple points about why it's so hard to picture these things.


The first is that we can visualize groups through a group action. In fact, that's exactly what you're describing in your paragraph! An action of a group G on a set X is simply a reasonable way to "apply" elements of G to elements of X to make new elements of X.

If we stir in some abstract nonsense, we realize that we're simply writing a homomorphism G ---> Aut(X), where X is something we can "picture".

So, to visualize a 2-group, we probably want to look at (2-)homomorphisms G ---> AUT(X). But here we have a problem -- sets don't have automorphism 2-groups! What sorts of things can we picture that do have automorphism 2-groups??



The other problem is "level shifting". e.g. we can think of a group as a set with an operation, or a category with one object. We can think of a 2-group as a category with an operation, or a 2-category with one object. We can think of a topological space as a bunch of points, or as a bunch of loops, or as a bunch of paths. There's some potential for level shifting on the geometric side too, I think. (I forgot my example as I was rewriting this)



Now... there are some 2-groups that are very easy to picture like this. Let's return to the field of topology -- there is a very natural way to extend the category of nice spaces into a 2-category: add in homotopies. We have a 2-category built from:

objects -- nice topological spaces
1-morphisms -- continuous maps
2-morphisms -- homotopies of maps

A homotopy between two maps f and g is essentially a choice of, for any x in the source, a path from f(x) to g(x) in the target.

Maybe this picture is adaptable to the current situation?



But wait, there's more -- I previously had a thought on 2-groups that I now realize might really be useful for picturing stuff. When trying to understand the classification theorem, I had the whimsical thought that a special 2-group looks superficially like a Lie group:

If we just look at the "lower" elements of our special 2-group, they are just an ordinary group. The automorphisms of each point in our special 2-group are sort of like a "tangent group" at that point. Multiplication by a "lower" element at a specific point can be lifted to a homomorphism of the tangent groups.


So maybe that's what we need to do -- we need to imagine a differential-manifold like thing, but where we don't have tangent spaces, but instead have "tangent group"s? Then, maybe it's more clear what the upper morphisms are doing?

 

[Hurkyl]

I had the whimsical thought that a special 2-group looks superficially like a Lie group:

Maybe that's what the crossed module is capturing? Just like we talk about Lie groups by looking at them as a group, looking at the tangent algebra at the identity, and how group elements act on the tangent algebra... maybe a crossed module is telling us the group, the "tangent" group at the identity, how group elements act on the tangent group, and some other technical detail?

In fact, any Lie group is a 2-group!

The lower morphisms are simply the points on the Lie group.
The upper morphisms at a point is simply the tangent space at that point.

To put it in a more concrete way, if you don't mind me treating the tangent space as infinitessimals...

The upper morphisms are simply things that look like:

g (1 + d)

Where g is an element of the Lie group, and d is an element of the Lie algebra. (This product is simply an element of the tangent space at g)

Then, vertical composition is (I like using # for operations):

g (1+d) # g(1+e) = g (1+d+e)

(note that (1+d)(1+e) = 1+d+e+de = 1+d+e)

And horizontal composition is

g(1+d) . h(1+e) = g[(1+d)h](1+e) = g[h + h (h^(-1) d h)](1+e)
= gh(1 + h^(-1) d h) (1+e)
= gh(1 + h^(-1) d h + e)

Well... if there's any justice in the world at all, this is a 2-group. :tongue:

I suppose the simpler way to write these things is like this: the upper morphisms are pairs (g, d) where g is a Lie group element and d is a Lie algebra element. Then:

(g, d) # (g, e) = (g, d + e)
(g, d) (h, e) = (gh, h^(-1) d h + e)