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April 6, 2008

This Week’s Finds in Mathematical Physics (Week 263)

Posted by John Baez

In week263 of This Week’s Finds, see the deep atmosphere of Titan:

Then read about the work of John Thompson and Jacques Tits, who won the 2008 Abel Prize. Learn how to build a group as a layer cake with simple groups as layers. And see an example of a Bruhat-Tits building!

Here are some questions I had when writing this Week’s Finds:

  • People think the atmosphere of Titan looks red because of tholins. But what are tholins, exactly? I want to see the actual chemical structure of some tholins! How are they related to polycyclic aromatic hydrocarbons (discussed in week258)? Do they both fit into some grand chemical cycle that takes place out in space? Why are places like Titan, Triton and Ixion so full of tholins?
  • What’s the importance of the result John Thompson proved for his PhD thesis — the nilpotence of Frobenius kernels? It seems cool, and related to groupoidification, but I don’t grok it.
  • What the heck is really going on with finite simple groups? Yes, that’s a very broad question… but I want to find some way to sink my teeth into this subject, and I’m not sure how to proceed. The classification theorem seems too technical for me to enjoy at this stage of my development. I really like thinking of finite groups as symmetry groups of ‘incidence geometries’, so maybe I should learn about Mathieu groups as symmetries of Steiner systems and codes, the Higman–Sims group as symmetries of the Higman–Sims graph, and so on. It’s a little intimidating to think about things like this, though:

    What are the overarching ideas? Maybe I should start by taking another crack at Conway and Sloane’s Sphere Packings, Lattices, and Groups, or the 2-volume Geometry of Sporadic Groups by Ivanov and Shpectorov.
  • What’s a definition of building that I can remember and explain? I’ve been through lots of examples, so I know the idea, but I haven’t thought about the definition lately, and I learned from writing this Week that it’s not crisp in my mind. I’d like to explain it clearly enough to provide an introduction for what Todd Trimble wrote about buildings! Hmm, maybe I should just reread what he wrote, and then explain it, so I remember it.

Given the depressingly puny response to my last list of questions I’m not sure it makes sense to continue asking them here — but maybe this is the sort of thing that you have to do for a while before people catch on.

(I would include these questions in the version on my website — that might work better — but I don’t really want more email. I’d like people to answer on this blog!)

Posted at April 6, 2008 5:24 AM UTC

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Re: This Week’s Finds in Mathematical Physics (Week 263)

These cocycles are studied in a subject called “group cohomology”.

[… ]

If [the coefficient group is not abelian] we need something more general called “nonabelian cohomology”.

Would it be helpful to say “nonabelian group cohomology” here, instead of “nonabelian cohomology”?

I have become accustomed to thinking of “nonabelian cohomology” as the classification of GG-torsors (GG-bundles) on spaces, for GG some nn-group, or equivalently as the descent with coefficients in an nn-group.

That seems to be the way the term has been used by J. Giraud, John Roberts, Ross Street, Larry Breen and others, and it is the way it seems to be used in the approach to \infty-stacks via simplicial presheaves by Jardine, Toën and others.

I am thinking of nonabelian group cohomology as the special case of equivariant nonabelian cohomology over a point.

It’s just a matter of terminology, of course.

Posted by: Urs Schreiber on April 6, 2008 9:24 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Urs wrote:

Would it be helpful to say “nonabelian group cohomology” here, instead of “nonabelian cohomology”? I have become accustomed to thinking of “nonabelian cohomology” as the classification of GG-torsors (GG-bundles) on spaces…

Eilenberg and Mac Lane’s big discovery was that the ‘group cohomology’ of a discrete group is the same as the cohomology of its classifying space (aka Eilenberg–Mac Lane space). The same is true for nonabelian cohomology: given discrete groups GG and HH, the nonabelian 2nd cohomology of HH with coefficients in GG classifies principal Aut(G)Aut(G)-2-bundles over the classifying space of HH.

So, I’m inclined to play down, rather than play up, the distinction between the cohomology of groups and the cohomology of spaces. As I pointed out later, groups are really just special case of nn-groupoids, which are the same as connected spaces, as far as homotopy theory goes.

If this is more confusing than helpful, maybe I should use a more standard terminology. Unfortunately, I don’t recall anyone ever talking about ‘nonabelian group cohomology’. The term I’ve seen most often is ‘Schreier theory’. I don’t like this very much, since it conveys absolutely no information to someone not already familiar with it.

I am thinking of nonabelian group cohomology as the special case of equivariant nonabelian cohomology over a point.

I find the ‘HH-equivariant cohomology of a point’ to be a slightly scary way of talking about the cohomology of the classifying space of HH. But, that’s because I’ve learned to love classifying spaces. I can easily imagine someone thinking the reverse. All these different ways of thinking are good.

What a tangled web of thoughts we weave! By the way, Bill Strossman in the physics department here is trying to get that nn-category wiki working. It seemed to work at first, but didn’t actually work, on Solaris. He’s rejiggering something now… Once we get this thing working I hope it’ll be easier to explain math in a systematic way online.

Posted by: John Baez on April 6, 2008 6:21 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Okay, I see.

If one uses a sufficiently general notion of cohomology, as people do, where we effectively have cohomology of “\infty-stacks” with values in “\infty-stacks” (following the Jardine, Toën approach of homotopy categories of simplicial presheaves) or of simplicial spaces with values in ω\omega-category-valued presheaves (following Ross Street), it all becomes one and the same anyway, since then the distinction between a category, its nerve and its realization is pretty much blurred away.

I just had this feeling that if I were seeing this for the first time, I’d be less confused if you went from “group cohomology” to “nonabelian group cohomology” and not magically removed the occurence of “group” in the process.

I don’t recall anyone ever talking about ‘nonabelian group cohomology’.

Somebody might have to start saying it! :-)

I have also not seen anyone talk about nonabelian differential cohomology, until I started doing so because it became clear to me in conversations that this is the kind of term that makes people think in the right direction when I tell them what I am thinking about.

Posted by: Urs Schreiber on April 6, 2008 6:45 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Urs wrote:

I just had this feeling that if I were seeing this for the first time, I’d be less confused if you went from “group cohomology” to “nonabelian group cohomology” and not magically removed the occurence of “group” in the process.

Okay, you just convinced me!

I’ll say “nonabelian group cohomology, usually called ‘Schreier theory’ (see ‘week223’).”

Posted by: John Baez on April 6, 2008 7:10 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

By the way, Bill Strossman in the physics department here is trying to get that n-category wiki working.

WHOO-HOO!!

Posted by: Todd Trimble on April 7, 2008 2:48 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Serre’s “Cohomologie Galoisienne” refers (if I remember that correctly) to articles by P. Dedecker (websearch, 2, 3) .

Posted by: Thomas Riepe on April 7, 2008 7:42 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

There is some stuff on non-Abelian group cohomology in the Menagerie notes that I made available last month via this blog.

The problem of the terminology is that (non-Abelian group) cohomology \neq non-Abelian (group cohomology). Long live the associative law!

The work of Dedecker is re-examined by Breen in various notes, and Ronnie Brown and I looked at some of the connections with work of Turing in a paper summarised in the menagerie.

Posted by: Tim Porter on April 7, 2008 11:25 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Non-commutativity in nomenclature can be equally dangertous:
Lie n-algebra versus n-Lie algebra
the latter has just one n-ary bracket satisfying one of two reasonable generalizations of Jacobi

Posted by: jim stasheff on April 8, 2008 1:36 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

There are now a variety of 3-ary algebras.
One of the most classic and recently revived is the Nambu bracket. It satisfies a generalized Jacobi identity which is not that of an L-infty algebra: [a,b,-] acts as a derivation on [ , , ]

Posted by: jim stasheff on April 13, 2008 12:30 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

There are now a variety of 3-ary algebras.

One of the most classic and recently revived is the Nambu bracket. It satisfies a generalized Jacobi identity which is not that of an L-infty algebra: [a,b,-] acts as a derivation on [ , , ]

Jacques Distler has a good review of that “recent resurrection” of the Nambu bracket (if that’s what it should be called) in the super membrane. (Though the Nambu bracket has always manifestly controlled the membrane action…)

I haven’t made up my mind yet if the situation found there lives up to the depth insinuated by the term “3-algebra”.

Does this have any homotopy-theoretic interpretation?

It feels like it should, but it’s not clear to me at this point. The main hint seems to be that we are looking at something that has two instead of one single “gauge parameter”.

If “Nambu bracket” is the answer, then I suspect we are looking at categorified symplectic geometry (“homotopy symplectic geometry, if you prefer”) controlled not by closed nondegenerate 2-forms but by closed non-degenerate 3-forms, as we once discussed in Gerbes, Strings and Nambu bracket and in Classical, canonical, stringy.

Posted by: Urs Schreiber on April 13, 2008 8:26 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Thanks for those links. The (?now defunct?)
String coffee table apparently had a lot of good stuff us more recent bloggers are unaware of.

“I haven’t made up my mind yet if the situation found there lives up to the depth insinuated by the term : 3-algebra.


Does this have any homotopy-theoretic interpretation?


It feels like it should, but it’s not clear to me at this point. The main hint seems to be that we are looking at something that has two instead of one single “gauge parameter”.


If “Nambu bracket” is the answer, then I suspect we are looking at categorified symplectic geometry (“homotopy symplectic geometry, if you prefer”) controlled not by closed nondegenerate 2-forms but by closed non-degenerate 3-forms, as we once discussed in Gerbes, Strings and Nambu bracket and in Classical, canonical, stringy.”

Yess, the fundamental identity says [a,b,-] acts as a triple bracket and this is NOT the fundamental identity for an L_\ifnty alg with only the 3-bracket non-trivial.

Notice that indeed prepending n- may have very differnt meanings depending on the context.

Caveat lector!

Posted by: jim stasheff on April 14, 2008 1:48 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Notice that indeed prepending nn- may have very differnt meanings depending on the context

Maybe somebody should have reserved the copyright on the proper usage of this terminology.

More seriously: did anyone before the recent membrane theorists refer to the “3-algebra” which they are discussing as such?

Is there a general theory or at least an accepted term for algebras with nn-ary brackets with the property that the unary operation obtained by feeding in (n1)(n-1)-arguments acts as a derivation?

Another question: since you seem to see a bigger picture behind what they have: what am I to think of the observation they make that the only known realization of this “3-algebra” is that where the structure constants of the triliniear bracket are nothing but the volume form on 4\mathbb{R}^4.

Recently there even seem to be arguments that no other solution should exist.

A final comment, concerning the existence of gauge transformations that depend on two gauge parameters:

if we start with an ordinary DGCA, then every contraction ι x\iota_x extended as a derivation yields a 1-parameter gauge transformation [d,ι x][d,\iota_x].

This is also known as the “first derived bracket”. With more contractions we can continue to form the higher derived brackets [[d,ι x],ι y] [[d,\iota_x],\iota_y] which then depened on more “gauge parameters”. Maybe their “3-algebra” is just a special case of such a situation.

I need to think about this.

Posted by: Urs Schreiber on April 14, 2008 10:41 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

For a thorough treatment of the two Fundamental Indentity alternatives for generalizing Jacobi to n-variables
see
MR1640457 (99k:17050) Vinogradov, Alexandre; Vinogradov, Michael On multiple generalizations of Lie algebras and Poisson manifolds. Secondary calculus and cohomological physics (Moscow, 1997), 273–287, Contemp. Math., 219,

Posted by: jim stasheff on April 14, 2008 1:35 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Urs wrote:
Recently there even seem to be arguments that no other solution should exist.

Meaning finite dimensional if true. references?

Posted by: jim stasheff on April 14, 2008 1:37 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Urs wrote:
Recently there even seem to be arguments that no other solution should exist.

Meaning finite dimensional if true. references?

Posted by: jim stasheff on April 14, 2008 1:38 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Are you talking about 3-algebra as in e.g. arXiv:0804.0913 or arXiv:0803.3218 ?

Also, is there some relation with M. Rausch de Traubenberg’s ternary algebra, see e.g. arXiv:0710.5368v1 ?

Posted by: Thomas Larsson on April 14, 2008 7:30 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

will check the 3 arXiv refs you sent
but no, none of those
rather B and L is 0711.0955
Bandres et al 0803.3242
van Raamsdonk 0803.3803
I find it easier to find articles by author than number

Posted by: jim stasheff on April 14, 2008 8:01 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Any discussion of the existence of examples
should be preceeded by Chatterjee and Takhtajan
hep-th/9507125

Posted by: jim stasheff on April 15, 2008 1:55 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Just arrived in Detroit, my connection to South bend goes any minute.

I think those membrane “3-algebras” are equivalent to DGCAs whose differential is of degree +2 instead of +1.

For {t a}\{t^a\} a basis of ( *) 4(\mathbb{R}^*)^4 set

dt a=ϵ a bcdt bt cwegdet d. d t^a = \epsilon^a{}_{b c d} t^b \wedge t^c \wegde t^d \,.

Then forming the derived bracket as I mentioned yields that “3-algebra”, it seems.

Posted by: Urs Schreiber on April 15, 2008 3:13 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

It is interesting to see what results by tampering a bit with the axioms of DGCAs.

We had discussed before that relaxing graded commutativity a bit gives rise to Clifford structures and spinors.

We now see that relaxing the condition that the degree of the differential be +1 takes us to Nambu brackets.

Another formally obvious but interpretationally highly mysterious step would be:

replace d 2=0d^2 = 0 with d 3=0d^3 = 0. And so on.

Posted by: Urs Schreiber on April 15, 2008 6:11 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Urs wrote:

Another formally obvious but interpretationally highly mysterious step would be:

replace d 2=0d^2=0 with d 3=0d^3=0. And so on.

Last summer we went to an interesting talk about this at ESI — delivered by a debonair Russian guy (I forget his name). Or at least: I went to the talk, and you were around at the time. I forget the punch line, but I remembered being pleasantly surprised at how interesting the results were: my expectations were very low, of course. I think a differential dd with d n=0d^n = 0 turns out to have something to do with quantum groups at an nnth root of unity. Or something like that.

Still, I think this is a rather strange direction of research — like walking into a dark jungle and hoping you’ll find a pot of gold.

Posted by: John Baez on April 15, 2008 7:41 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Last summer we went to an interesting talk about this at ESI — delivered by a debonair Russian guy (I forget his name). Or at least: I went to the talk, and you were around at the time.

Yes, and after the dinner that evening, we walked back to our hotel and I asked if anyone knew any good fundamental reason why the world is ruled by d 2=0d^2 = 0 and not by d n=0d^n = 0 for n>2n \gt 2.

I don’t remember the speaker having made a relation to quantum groups. I do remember that he talked about inserting roots of unities into formulas where we are used to inserting just signs. Did he come to any conclusion?

My impression was that he enjoyed giving his audience food for thought without giving away any grand answer.

Still, I think this is a rather strange direction of research — like walking into a dark jungle and hoping you’ll find a pot of gold.

Yes, but maybe somebody should be looking into it. Looking at differentials of degree higher then 1 is a similar enterprise – which apparently has now raised an enormous amount if interest.

For ordinary DGCAs we can see that the existence of dd itself encodes \infty-categorical structure morphisms, like compositors, associators, etc, while the condition d 2=0d^2 = 0 encodes their coherence laws.

Passing to d 3=0d^3 = 0 would hence seem to relax the coherence laws in a way.

And, for that matter, considering a differential of degree +2 corresponds to having a composition of 1-morphisms which is not defined on pairs of composable 1-morphisms, but on triples.

Possibly this would correspond to analogs of Kan simplicial complexes where simplices are replaced by higher nn-gons of sorts.

Posted by: Urs Schreiber on April 15, 2008 11:57 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

There is a long though sporadic history of interest in d^N=0, going back to the early days before Eilenberg-Steenrod brought coherence to the plethora of topological homology theories.

Posted by: jim stasheff on April 15, 2008 1:26 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Urs wrote:

I don’t remember the speaker having made a relation to quantum groups. I do remember that he talked about inserting roots of unity into formulas where we are used to inserting just signs.

He probably didn’t say the phrase “quantum group”. But, I vaguely seem to recall that he wrote down the qq-derivative operator familiar from the theory of quantum groups, and somehow used it to construct an operator dd with d n=0d^n = 0 when qq was an nnth root of unity. Or maybe he was just using roots of unity where we expect signs — a very ‘anyonic’ thing to do.

My memories are quite vague, but I remember thinking “hey, this is actually related to math I care about!”

There’s a whole other way of motivating d n=0d^n = 0 for higher nn, where you take John Wheeler’s motto “the boundary of a boundary is zero” and try to dream up weird geometrical ways of thinking where it doesn’t seem correct.

John Wheeler, alas, just died. It’s his writings that got me interested in differential geometry and quantum gravity!

Posted by: John Baez on April 15, 2008 1:43 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

take John Wheeler’s motto “the boundary of a boundary is zero” and try to dream up weird geometrical ways of thinking where it doesn’t seem correct.

Manifolds with boundaries and corners, maybe? Just a question, I don’t see the answer.

Posted by: Urs Schreiber on April 15, 2008 8:04 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

No, I’m pretty sure boundaries and corners don’t throw you off. In fact, in some formal ways the category of manifolds with (generalized) boundaries is nicer than the category of manifolds.

Posted by: John Armstrong on April 15, 2008 8:21 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

In extended QFT you want to distinguish between “physical boundaries” and “state boundaries” which usually meet at a corner. Only one of these you might want to count as a boundary proper.

For instance the open string zipping along without interaction would be the strip [0,1] 2[0,1]^2 with two opposite edges regarded as physical, the other two as “state boundaries”.

In that sense you might want to say that [0,1] 2\partial [0,1]^2 is the disjoint union of two copies of [0,1][0,1]. The boundary of that would be the disjoint union of four points. And the boundary of those would finally vanish.

Just an idea, I don’t claim that I see that this can be related to differential algebras with d 3=0d^3 = 0.

Posted by: Urs Schreiber on April 15, 2008 9:05 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Urs wrote:

Manifolds with boundaries and corners, maybe?

That’s a natural guess. Maybe there’s some sense in which the “boundary of the boundary” of a square consists of its 4 corners.

Of course in homology theory we define the boundary of a polytope to be a signed sum of its faces, ensuring that d 2=0d^2 = 0.

But, we could imagine not introducing these formal linear combinations and signs. One context where this might be a good idea is in geometric measure theory, where the Euler characteristic winds up being just part of a bigger story. Tom Leinster can tell you all about that…

Posted by: John Baez on April 16, 2008 4:45 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Of course in homology theory we define the boundary of a polytope to be a signed sum of its faces

I seem to remember that one proposal was that in formulas of this kind we replace signs by more general roots of unity. But I am not sure by which rule that should proceed. Nor if it can be related to manifolds with corners.

[…] geometric measure theory, where the Euler characteristic winds up being just part of a bigger story. Tom Leinster can tell you all about that…

Is that alluding to insights beyond those which have been discussed in various entries here on the Café?

If so, please somebody let us all know about the latest.

I am a big fan of the “Euler characteristic”of a category, though to me it seems that the really cool part is not just the cardinality itself, but the measure (the weighting) it arises from.

I think that BRST-BV formalism should be essentially nothing but the Lie version of the Leinster weighting prescription (pass to gauge orbits, divide out remaining auto gauge equivalences).

Posted by: Urs Schreiber on April 16, 2008 12:52 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

By the way, it’s a pity that the definition of simplicial set does not invoke 2=0\partial^2 = 0 more manifestly, though clearly that’s what the definition is all about.

What can be said about the relation between non-negatively graded chain complexes of free abelian groups and simplicial sets? Ought to be roughly the same thing.

Posted by: Urs Schreiber on April 16, 2008 10:23 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Urs:though clearly that’s what the definition is all about.

Jim: crucially NOT

Posted by: jim stasheff on April 18, 2008 3:24 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

crucially NOT

Okay, let me see:

let F:SetAb F : Set \to \mathbf{Ab} be the functor which sends any set to the free abelian group over that set.

By postcomposition this yields a functor from simplicial sets to simplicial abelian groups

F *:SSetSAb. F_* : S Set \to S\mathbf{Ab} \,.

Let

C:SAbCh +(Ab) C : S\mathbf{Ab} \to Ch_+(\mathbf{Ab})

be the functor which sends each simplicial abelian group AA to its chain complex C(A)C(A) with (C(A)) n=A n (C(A))_n = A_n and d= i=0 n(1) id i:(C(A)) n(C(A)) n1. d = \sum_{i=0}^{n} (-1)^i d_i : (C(A))_n \to (C(A))_{n-1} \,.

What information about a simplicial set SS is lost as we map it along SSetF *SAbCCh +(Ab)? S Set \stackrel{F_*}{\to} S \mathbf{Ab} \stackrel{C}{\to} Ch_+(\mathbf{Ab}) ?

Instead of the chain complex C(A)C(A) we could consider the normalized chain complex

N:SAbCh +(Ab) N : S\mathbf{Ab} \to Ch_+(\mathbf{Ab})

which appears in the Dold-Kan correspondence.

What information about the simplicial set CC is lost as we map it along

SSetF *SAbNCh +(Ab) SSet \stackrel{F_*}{\to} S\mathbf{Ab} \stackrel{N}{\to} Ch_+(\mathbf{Ab})

?

Posted by: Urs Schreiber on April 18, 2008 3:56 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

A simplicial space can be realized as a topological space.

What information about the space is lost as we pass to a chain complex?

How about \pi_1?

But I see what you really were after in saying d^2=0 cf Wheeler
Just a bit too brief

Posted by: jim stasheff on April 18, 2008 9:35 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Jim, thanks for the reply.

I realize that maybe my questions came across as being rethoric. But they weren’t meant that way. I really like to better understand this.

I’d really do want to better understand how the difference between

\;\;\;\; simplicial sets

on the one hand and

\;\;\;\; chain complexes in free abelian groups

on the other can be characterized.

The simplicial identities on a simplicial set say, heuristically, that the boundary of an abstract simplex behaves as we expect the boundary of a simplex to behave. In particular, it implies that the boundaty of the boundary vanishes, in that the face maps d id_i of the simplicial set may be assembled to a differential d= i(1) id i d = \sum_i (-1)^i d_i on the free abelian groups generated by it, which squares to 0.

So what if I’d only remember this property of a simplicial set, which says that the boundaries of the boundaries of its nn-simplices vanish. How much of the nature of the information in the simplicial set would be lost?

Posted by: Urs Schreiber on April 19, 2008 12:52 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

How about doing K-theory with only d^2 =0?

Posted by: jim stasheff on April 20, 2008 1:55 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

I agree that d 2=0d^2 = 0 is not sufficient to characterize simplicial sets.

But what I said was that it is a pity that the very definition of simplicial sets does not put d 2=0d^2 = 0 in more prominent position.

I think I am looking for a statement of the following kind:

a simplicial set is a complex in free abelian groups together with the extra structure XYZXYZ.

What is the minimum we have to take for XYZXYZ here? Does anyone know?

Posted by: Urs Schreiber on April 20, 2008 4:37 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

I have to say I’m very confused by this whole discussion. Isn’t the usual point that simplicial things are better than complexes and that the Dold-Kan correspondence tells us when the complexes are the same as the simplical stuff?

Posted by: Aaron Bergman on April 20, 2008 4:47 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Yes. How much better exactly?

I’d like to know: instead of starting with the usual definition of a simplicial set and deriving a complex in free abelian groups form it, can we formulate it the other way around: characterize a simplicial set as a complex of free abelian groups with extra information XYZ. What exactly is XYZ, then?

Sorry, I realize that I have apparently consistently failed to communicate the issue I am wondering about here. Let this be my last attempt. If that doesn’t work we’ll just decide that I am confused and ignorant and proceed. ;-)

Posted by: Urs Schreiber on April 20, 2008 6:23 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

I find the “H-equivariant cohomology of a point” to be a slightly scary way of talking about the cohomology of the classifying space of H. But, that’s because I’ve learned to love classifying spaces.

I vastly prefer the former, because the latter is intrinsically infinite-dimensional, whereas the former can be thought about finite-dimensionally, an easier and more traditional place to do algebraic geometry. (Not to mention easier to stick on a computer.) This was very important for us in this paper’s section 2.2.

The clearest time to prefer it, and abhor classifying spaces, is when thinking about K-theory rather than cohomology. One can talk about the K-theory of the Borel space, but it’s different (and not as good as) the equivariant K-theory.

Posted by: Allen Knutson on April 13, 2008 5:22 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

What is true is that “equivariant cohomology over a point” may sound much scarier than it should. It is a weird twisted term for a straightforward concept.

One way to put it is this:

cohomology of spaces XX is about ana-nn-functors on Disc(X)Disc(X), the discrete category over XX.

If XX has a group GG acting on it, then the equivarint cohomology of XX is about ana-nn-functors on the action groupoid X//GX//G.

If XX is a point, we find ourselves talking about nn-functors on one-object groupoids, which is precisely what John Baez and Mike Shulman consider in those lecture notes.

This way of thinking about it makes it clear that despite its name, “equivariant cohomology of a point” is in fact the simplest and not the most scary of all possible examples, especially when GG is finite (as it usually is when people talk about equivariance).

It’s just a scary name caused by an unjustified priority of spaces over categories.

Posted by: Urs Schreiber on April 13, 2008 9:35 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

So, given

For example, a group is a category with one object, all of whose morphisms are invertible. Similarly, an “n-group” is an n-category with one object, all of whose 1-morphisms, 2-morphisms and so on are invertible. We can build up n-groups as layer cakes where the layers are groups. It’s a more elaborate version of what I just described - and it uses not just “2-cocycles” but also “3-cocycles” and so on. I never really understand group cohomology until I learned to see it this way.

does this mean that a group cohomology ring in some sense classifies all oo-groups you can build using that group for the composition series parts?

Posted by: Mikael Vejdemo-Johansson on April 6, 2008 8:08 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Mikael wrote:

does this mean that a group cohomology ring in some sense classifies all \infty-groups you can build using that group for the composition series parts?

Alas, it’s not quite that simple.

Say we have a group GG acting on an abelian group AA as automorphisms. Then the cohomology

H n+1(G,A)H^{n+1}(G,A)

classifies spaces with π 1=G\pi_1 = G and π n=A\pi_n = A with the specified action of π 1\pi_1 on π n\pi_n.

So, group cohomology in its simplest form classifies connected spaces with only two nonvanishing homotopy groups: the first and just one other. We can think of these as \infty-groups — but only very special \infty-groups!

To classify more general connected spaces, or \infty-groups, we need the full machinery of Postnikov towers, as explained in those lectures I mentioned. Here the idea is to build a connected space XX using a tower of spaces X iX_i, each fibered over the previous one, with the iith one having nonvanishing homotopy groups only up to π i\pi_i. If we know π n(X)=A\pi_n(X) = A and we know X n1X_{n-1}, the possibilities for X nX_n are classified by

H n+1(X n1,A)H^{n+1}(X_{n-1},A)

In the special case mentioned before, where XX has only π 1=G\pi_1 = G and π n=A\pi_n = A nontrivial, then X n1=K(G,1)X_{n-1} = K(G,1), so this reduces to

H n+1(K(G,1),A)=H n+1(G,A)H^{n+1}(K(G,1), A) = H^{n+1}(G,A)

so we’re back to plain old group cohomology.

More general Postnikov towers should be thought of as involving ‘nn-group cohomology’. But, it’s all just about groups and the complicated ways they can interact.

Posted by: John Baez on April 6, 2008 10:25 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

A couple of typos:

“Thomposon” -> “Thompson”
“working on up to whole group” -> “working on up to the whole group”?

Posted by: logopetria on April 6, 2008 10:40 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Thanks! Fixed!

Posted by: John Baez on April 7, 2008 12:45 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

I’ve added a list of questions I’m stuck on. Please help me out!

Posted by: John Baez on April 7, 2008 2:10 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

What’s a definition of building that I can remember and explain? […] I’d like to explain it clearly enough to provide an introduction for what Todd Trimble wrote about buildings!

Mind, whatever I wrote was left in a dreadful state of incompleteness – that really would need an introduction, not to mention exegesis.

Should I try to explain it? Not long ago we were discussing, over at John Armstrong’s blog, the prospect of having Chris Hillman write an introductory piece on Tits’s work for which he was awarded, and then having me follow up with the enriched category view on buildings worked out between me and Jim Dolan, but nothing has come of that yet. Maybe now is a good time then?

Posted by: Todd Trimble on April 7, 2008 2:44 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Don’t wait for Chris Hillman to write something. I’d love for you to write an explanation of buildings, or polish your paper on them, or whatever… do it before the next Abel prize comes along!

Posted by: John Baez on April 7, 2008 6:06 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Presumably you know about Lieven le Bruyn’s blog, which includes plenty of material relevant to your questions, e.g., this post on Mathieu groups. Searching for Steiner gives you several Mathieuesque posts.

Posted by: David Corfield on April 7, 2008 9:21 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Excellent! One nice thing about Lieven le Bruyn’s post on Mathieu groups is that it presents them as quotient groups of PSL(2,)PSL(2,\mathbb{Z}) using the technique of ‘children’s drawings’, which is something I want to explain soon on This Week’s Finds. I’m not sure how much this illuminates the inner nature of Mathieu groups, but it’s cool. Also, le Bruyn has left some expository work for me, since by including a bunch of explicit formulas and leaving out some basics, he’s made children’s drawings seem a bit scarier, and a bit less childish, than they really are.

(Stuck here in the airport waiting for my flight — a great time to learn about Mathieu groups.)

Posted by: John Baez on April 8, 2008 8:23 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

The URL for

12) Kenneth S. Brown, What is a building?

should be http://www.ams.org/notices/200210/what-is.pdf

Posted by: David Corfield on April 7, 2008 9:39 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Thanks! Fixed!

Posted by: John Baez on April 7, 2008 5:26 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

“What the heck is really going on with finite simple groups?”

Well, depends what you want to know. Morally one can never truely understand a theorem, particularly one as deep and complex as CSFG, until one has at least two proofs of it and preferably more.

The closest thing to a complete proof we have revolves around the following simple idea: by the odd order theorem, any simple finite group has elements of order two (“involutions”). Consequently the proof (and thus in some sense our only known way to understand the whole situation) boils down to a close inspection of what these involutions look like and how they sit inside a simple group (“p-local group theory”). After a while of doing this you realise that larger primes (and in particular the prime 3) are also pretty interesting and after a very lengthy technical discussion you arive at the final conclusion.

In the spirit of “categorification” you may be interested to know that there are mathematicians out there (including toplogists and representation theorists) who are working on translating much of this theory into the language of categories using so called “fusion systems”. Lots of specific reference and details can be found towards the bottom of this.

Posted by: Ben Fairbairn on April 7, 2008 11:55 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Ben Fairbairn on April 7, 2008 11:55 AM:

In the spirit of “categorification” you may be interested to know that there are mathematicians out there (including topologists and representation theorists) who are working on translating much of this theory into the language of categories using so called “fusion systems”. Lots of specific reference and details can be found towards the bottom of this.

Has the fusion for finite groups anything to do with fusion on conformal field theories?

Posted by: Maarten Bergvelt on April 7, 2008 3:11 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Maarten wrote:

Has the fusion for finite groups anything to do with fusion on conformal field theories?

I wish it did, but I don’t see anything like that yet. This paper gives a definition:

For any finite group GG and prime pp, the fusion system F p(G)F_p(G) is the category whose objects are pp-subgroups of GG, and whose morphisms are homomorphisms between subgroups induced by conjugation in GG. People use techniques from homotopy theory to study these things, and use them to study finite groups.

Posted by: John Baez on April 7, 2008 5:45 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

All we have to do is understand what Mackey functors have to do with fusion. Then we can read what Ross Street and Elango Panchadcharam have to say about Mackey functors and we’re home and dry.

Hmm, is there a whiff of something similar to your groupoidification going on in the latter paper? Spans seem to be being sent to linear operators, and there’s transfer afoot.

Posted by: David Corfield on April 8, 2008 10:02 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Thanks for pointing out that thesis! Yes, the first part is very much related to groupoidification, especially the part related to permutation representations of groups.

Also, Proposition 0.0.1 of Chapter 0 plays a very important role in Jeffrey Morton’s thesis. It says that whenever we have a weak pullback square of groupoids, we can take a presheaf over the groupoid at one corner and either pull it back and push it forward, or push it forward and pull it back, to get a presheaf over the groupoid at the other corner — and the results are naturally isomorphic! So, just as we suspected, this result was already known.

Posted by: John Baez on April 9, 2008 10:27 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

Just so people aren’t confused, I linked to a paper which makes up a part of the thesis John is referring to.

Posted by: David Corfield on April 9, 2008 10:59 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 263)

I haven’t heard of these “fusion systems” before, it seems. But it reminds me of the following construction, which sounds similar and is related to CFT fusion:

for the WZW model with a string propagating on a group manifold, most everything of the theory is part of the representation theory of that group.

The conjugacy classes in the group, for instance, correspond to certain D-branes. These can be fused.

The operation on conjugacy classes is: for two conjugacy classes, form the set of products of elements in both classes. This set decomposes as a union of other conjugacy classes. This defines a fusion product structure.

I don’t know if that’s related to those “fusion systems”, though.

Posted by: Urs Schreiber on April 9, 2008 12:55 AM | Permalink | Reply to this

Funny Things Happen; Re: This Week’s Finds in Mathematical Physics (Week 263)

I’ve had interesting technical conversations with several of the key people who classified finite simple groups. But, when I teach very elementary classes (to high school or middle school students) i simply state:

“There are some very funny things that happen in math if you allow infinity. On the other hand, to those who work with infinity every day, there are some very funny things that happen if you only look at things that are finite.”

Different strokes. Your mileage may vary.

Posted by: Jonathan Vos Post on April 8, 2008 8:03 PM | Permalink | Reply to this

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