Laubinger on Lie Algebras for Frölicher Groups

July 1, 2009

Laubinger on Lie Algebras for Frölicher Groups

Posted by Urs Schreiber

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[guest post by Martin Laubinger in the context of Smootheology: the study of generalized smooth spaces]

I have posted a preprint which contains the central new result I obtained in my dissertation.

The result may not directly relate to higher category theory. Still, I would appreciate feedback and problems for further investigation:

Martin Laubinger
A Lie algebra for Frölicher groups
(arXiv)

Here is a short description: The category of Frölicher spaces is a cartesian closed category which contains the category of smooth finite-dimensional manifolds as a full subcategory. The same is true for the closely related category of diffeological spaces. However, it is easier to define tangent spaces to Frölicher spaces than to diffeological spaces. Many groups have natural Frölicher structures, including all Lie groups, but also certain groups of mappings such as $C^\infty(M,G)$ or $\Diff(M)$ , which can not be given a manifold structure in general. A basic question is whether the tangent space (in the Frölicher sense) at the identity of these groups can be equipped with a Lie bracket. I have been able to construct such a Lie bracket, but there is an additional condition which has to be verified. This condition is very natural, but I have not found a general proof. In my thesis, I did not have an example for a group which satisfies the extra condition, but in the meantime I verified the condition for the additive group $\mathbb{R}^J$ (product of the reals) if J is not too big. This is explained in detail in the preprint.

Posted at July 1, 2009 10:19 AM UTC

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Re: Laubinger on Lie Algebras for Frölicher Groups

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Welcome Martin!

I’ve made it through the first few pages and I like it very much. Very clear writing. Anyone who can write something that penetrates my skull has some amazing talent :)

I had a very basic question about Frölicher spaces.

When I look at $(X,C,F)$ , I’m reminded of coordinate maps. If $X$ is a smooth $m$ -dimensional manifold we have coordinate maps

$\phi:U\subset X\to\mathbb{R}^m$

such that for any other coordinate map $\psi$

$\phi\circ \psi^{-1}:\mathbb{R}^m\to\mathbb{R}^m$

is smooth (where defined). Comparing this to Frölicher spaces, I’m tempted to consider a Frölicher $m$ -space $(X,C_m,F_m)$ , where

$C_m = \{c:\mathbb{R}^m\to X | (\forall f \in F_m) f\circ c\in C^\infty(\mathbb{R}^m,\mathbb{R}^m)\}$

$F_m = \{f:X\to \mathbb{R}^m | (\forall c \in C_m) f\circ c\in C^\infty(\mathbb{R}^m,\mathbb{R}^m)\}.$

Then a Frölicher space is a Frölicher $m$ -space for $m=1$ .

Does that make any sense?

Then you can imagine maybe crossing dimensions. For example, $(X,\mathcal{C},\mathcal{F})$ , where

$\mathcal{C} = \bigoplus_{0\lt m\le D} C_m,$

$\mathcal{F} = \bigoplus_{0\lt n\le D} F_n,$

$C_m = \{c:\mathbb{R}^m\to X | (0 \lt n \le D, \forall f \in F_n) f\circ c\in C^\infty(\mathbb{R}^m,\mathbb{R}^n)\},$

$F_n = \{f:X\to \mathbb{R}^n | (0 \lt m \le D, \forall c \in C_m) f\circ c\in C^\infty(\mathbb{R}^m,\mathbb{R}^n)\},$

and $D$ is to be thought of as the dimension of $X$ .

Does that make any sense?

Posted by: Eric Forgy on July 1, 2009 6:04 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

Martin and Eric, please excuse this brief off-topic comment, but I would very much like to get in contact with Eric Forgy.

Eric, I used to work as a quant and I think I now see a potentially new way to combine the work of two MIT researchers for obtaining a new approach to quantitative finance.

Roughly half my idea involves the work of Emily Fox from MIT who has just found a new way to discern order within complex systems. For example, consider her newest paper entitled “Nonparametric Bayesian Identification of Jump Systems with Sparse Dependencies” which is applied to the example of complex honey bee dances:

http://www.mit.edu/~ebfox/publications/sysid09_final.pdf

Eric, if you would please send me an email at this address below so that I can forward my suggestion to you for your consideration:

cstromeyer[at]post.harvard.edu

Thank you.

Posted by: Charlie Stromeyer on July 2, 2009 4:15 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

Sorry Urs et al. Please, feel free to remove these last two off-topic comments.

Posted by: Eric Forgy on July 2, 2009 5:09 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

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To answer Eric’s original question, all of your categories are actually the same (okay, isomorphic). The saturation condition ensures that all of the “plots” and “functions” in any bi-degree is completely determined by the restriction to lines. It’s easiest to see this for functions: a map $f: M \to \mathbb{R}^n$ is smooth if and only if all the compositions $p_i f : M \to \mathbb{R}$ are smooth, where the $p_i$ are the standard projections. The corresponding result for plots is the oft-quoted theorem due to J. Boman (Math Scand, sometime in the 1960s).

For Chen spaces or Diffeological spaces one can do something like this. I actually think that this is one of the weaknesses of these approaches - that dimension is something imposed from the outside rather than something intrinsic.

Posted by: Andrew Stacey on July 6, 2009 9:10 AM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

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Hi Andrew!

Good to see you back.

I actually think that this is one of the weaknesses of these approaches - that dimension is something imposed from the outside rather than something intrinsic.

We have talked about this before. Just for the record, I recall my reply to this.

The dimension of the test domains translates into the categorical dimension of higher categorical differential structure once we consider higher groupoids internal to smooth spaces.

That’s why it is important to have these higher dimensional plots around for applications where we are not just interested in smooth 0-categories, but in smooth $\infty$ -categories.

Take the category $CartesianSpaces$ , the full subcategory of manifolds on all cartesian spaces $\mathbb{R}^n$ .

An important 2-groupoid internal to $Sheaves(CartesianSpaces)$ is that of gerbes with connection. That cannot be recoverd from its image along the restriction map $Sheaves(CartesianSpaces) \to Sheaves(RealLine)$ , because the real line probe does not see the crucial 2-dimensional degree of freedom in that sheaf.

Anyone who wants to use generalized smooth spaces in order to take $\infty$ -groupoids internal to them as models for anything involving differential cohomology will need dimension $n$ -plots in order to probe the $n$ -categorical characteristics of degree $n$ differential cocycles.

As long as one does not care about any sheaves on smooth test spaces involving higher degree differential forms testing on the real line is fine. But if one does, it is no longer sufficient.

Posted by: Urs Schreiber on July 6, 2009 10:00 AM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

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Ah, yes. I was forgetting the “categorical dimension” aspect. I was thinking of dimension as a generalisation of dimension of manifolds (which was how I interpreted Eric’s question).

However, I’m still not completely convinced of the need for more than just lines. Rather, I’m happy with the idea of there being more than just lines but I don’t think that the collection of $\mathbb{R}^n$ s quite cuts the mustard (to use a, frankly, bizarre British expression).

The reason is that plots from $\mathbb{R}^2$ , say, into ones space, say $X$ , should be the same as plots from $\mathbb{R}$ into the corresponding path space, $PX$ - lines again. So saying that there are more plots from $\mathbb{R}^2$ into $X$ than one would expect from looking at the lines is the same thing as saying that $PX$ has more lines than one would anticipate from $X$ .

So there ought to be a way in which one can reformulate your insistence on taking all $\mathbb{R}^n$ s into a statement about the iterated path spaces of $X$ .

Saying that $X$ has categorical dimension $n$ then says that the higher path spaces are determined by the $n$ th path space.

I’m not saying that this is necessarily better than taking cartesian spaces, just alternative and equivalent. One thing it might help with is seeing what the correct “forcing” condition is on the plots (I’m sure everyone is well aware of my feeling about the sheaf condition here!).

Finally, I strongly suspect that it is actually possible to work with the iterated tangent bundles rather than the full path spaces, but this is just a hunch.

Posted by: Andrew Stacey on July 7, 2009 8:34 AM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

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So saying that there are more plots from $\mathbb{R}^2$ into $X$ than one would expect from looking at the lines is the same thing as saying that $P X$ has more lines than one would anticipate from $X$ .

Indeed. And that’s precisely the situation that occurs frequently when dealing with higher categorical smooth objects.

Let $X = \bar \mathbf{B}^2 U(1)$ be the classifying sheaf on $Diff$ (or similar) for trivial gerbes with connection (the bar is for “differential” not for “ $\infty$ -stackified”, therefore trivial gerbes, but with connection, just bear with me for a moment):

such that a map of generalized smooth spaces $U \to X$ is precisely a trivial gerbe with connection on $U$ .

Then $X$ has a single point, and a single line, the constant one on that point.

So we may not expect any lines in $P X$ . Yet, $P X$ has many interesting lines. That corresponds to the fact that $X$ has, while just a single point and a single line, many interesting surfaces.

This may make $X$ appear as a “generalized space” too weirdly general to bother with. It is not Frölicher. Its restriction to plots from the real line makes it disappear entirely and become indistinguishable from the ordinary point.

But it is these spaces $X$ that are important in higher categorical differential structures. Anyone intersted in these will not be able to follow your insistence to use only Frölicher spaces, nice as these may be.

Which doesn’t mean that people not interested in higher categorical smooth stuff ever need to bother with anything non-Frölicher.

But it might be good just to keep in mind that there are reasons, weird as they may seem, to go beyond Frölicher.

Posted by: Urs Schreiber on July 7, 2009 10:22 AM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

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Yes, that’s the sort of thing I had in mind (though I often feel that I’m still only seeing the black-and-white version while the rest of you have the 42” HD; or as Kryten would put it “Hmm, just the manual zoom then”).

I would like to argue that being Frölicher is not really anything to do with lines versus planes (and higher) but is really about the forcing condition. To give you a slogan:

Non-smoothness is detectable.

In an ordinary set-based theory, then if I have a map between the underlying sets which is not a morphism then there is a way to transfer that to a map between the underlying sets of two test objects so that it is not $C^\infty$ . The fact that this means that you can reduce the test spaces to lines is just a bonus.

(So my real fight is with the sheaf theorists who say that sheaves are so wonderful that if we can get sheaves on a site then we should.)

Now one could, using the exponential law, rephrase everything using higher Euclidean spaces in terms of lines but using higher “path spaces”. That would mean that everything becomes a statement about lines again. But that wouldn’t satisfy me. I’d want more structure to consider it as a true higher categorical version of a manifold. My hunch, from my last comment, tells me that this ought to mean that just as $M$ is completely determined by $C^\infty(\mathbb{R},M)$ so $^\infty M$ is completely determined by the sequence $C^\infty(\mathbb{R}, T^k M)$ , where the $T^k M$ are not necessarily the genuine tangent spaces but are some sort of generalised tangent space defining the higher categorical levels (and the categorical dimension is the point at which the sequence $T^k M$ stabilises in the sense that $T^{k+1} M = T(T^k M)$ ).

So an $\infty$ -Frölicher space ought to be something akin to a spectrum (in the sense of algebraic topology). A sequence of Frölicher spaces: $\mathcal{T}^k M$ together with smooth maps between $\mathcal{T}^k M$ and $T \mathcal{T}^{k-1} M$ (not sure what directions and what maps; probably ought to be so that $(\mathcal{T}^k M, T \mathcal{T}^{k-1} M)$ is the morphisms and objects of a category).

I’m in complete agreement with you that something beyond mere Frölicher spaces is wanted, I just disagree with how you’re going about it. If you just expand the test spaces to all Euclidean spaces then you have to weaken the forcing condition to get anything new. This, I strongly dislike. So I’d like a different way of expanding and that’s what I’m trying to feel my way towards in the above (only with the problem of being on a black-and-white 12” CRT).

Posted by: Andrew Stacey on July 7, 2009 3:00 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

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Hi Andrew,

I do get the intuitive point about higher tangent spaces that you make.

It reminds me of the fact that there is one further direction of generalization that we have yet to discuss in this context:

if we are being serious, then it is clear that we don’t just want sheaves (in a general sense) on a category of certain manifolds (say on the real line) but we want sheaves on cosimplicial such manifolds.

So maybe one way to realize what you are indicating here is to consider not just sheaves on the line, but on “DG-lines”, i. on (differential graded extensions of $C^\infty(\mathbb{R})$ $)^{op}$ .

Actually, of late I have come to wonder if [[ $\infty$ -Lie integration]] should be best understood as essentially the Yoneda embedding for [[derived $\infty$ -stacks]] on $SimplicialAlgebra \simeq DGCAs_+$

sending $A \in DGCA$ to $Y(A) : B \mapsto DGCAs( B \otimes \Omega^\bullet(\Delta^\bullet), A )$ with suitably completed $\otimes$ (i.e. Moerdijk-Reyes style).

Now that you mention the thing about insisting on curves, but with values in higher tangent spaces, that makes me think about whether restricting to DGCAs here which are “NQ-superlines” would actually be a good idea.

Certainly for instance that funny smooth space $\bar \mathbf{B}^2 U(1)$ that I mentioned above which is not entirely probeable by just lines, seems to be sufficiently probably by DG-lines (aka NQ-superlines).

(And I am saying this not despite the risk that David B.-Z. will see this and set me straight, but because of it :-)

Posted by: Urs Schreiber on July 7, 2009 4:06 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

cuts the mustard (to use a, frankly, bizarre British expression).

Bizarre, but certainly very familiar to those on the left side of the pond. At least it’s not rhyming slang.

Posted by: Todd Trimble on July 7, 2009 11:16 AM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

And, having just <insert name of favourite search engine>’ed it, also not British! Seems to have originated on your side of ‘t pond after all. Should have guessed from it’s complete inanity, I suppose …

Still, it’s odd that it’s American given that American mustard is so easy to cut, compared to, say, a really coarse-grain Dijon mustard.

Posted by: Andrew Stacey on July 7, 2009 3:08 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

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This gives me a strong sense of deja vu.

Does the set of curves $C$ and the set of functions $F$ contain enough information to determine the dimension of $X$ ? If I understood what you said, then it must. That would be interesting.

Posted by: Eric on July 6, 2009 7:36 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

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Your question is not quite precise because there are many ways to extend the definition of “dimension” beyond the realm of manifolds. So let me rephrase your question and answer that.

Suppose that $M$ is a manifold of some unknown dimension. Knowing $C^\infty(\mathbb{R}, M)$ and/or $C^\infty(M,\mathbb{R})$ , can I determine $dim M$ ?

Yes. The easiest method is if you know both the curves and functions. For then take a point $x \in M$ and define $C_x$ to be all smooth curves in $M$ which pass through $x$ at time $0$ . Now define $T_x M$ to be the set of equivalence classes of $C_x$ under the relation $\alpha \simeq \beta$ if $(f \circ \alpha)'(0) = (f \circ \beta)'(0)$ for all $f \in C^\infty(M,\mathbb{R})$ (note that I am only differentiating functions $\mathbb{R} \to \mathbb{R}$ here). This is the standard definition of the tangent space of $M$ at $x$ . I would like to read off its dimension. The only problem is that I don’t know the vector space structure yet.

To get this, I define:

$\lambda [\alpha] = [t \mapsto \alpha(\lambda t)]$

and

$[\alpha] + [\beta] = [\gamma]$

where $\gamma$ is a curve with the property that for all $f \in C^\infty(M,\mathbb{R})$ ,

$(f \circ \gamma)'(0) = (f \circ \alpha)'(0) + (f \circ \beta)'(0)$

(As $M$ is known to be a manifold, $\gamma$ always exists and $[\gamma]$ is uniquely determined.)

This defines the standard linear structure on $T_x M$ so I can read off its dimension.

I could have gone straight for the cotangent bundle by quotienting $C^\infty(M,\mathbb{R})$ . This would come with a linear structure already so that would enable me to skip a step in the above. However, for reasons to do with infinite dimensions I prefer to work with tangent rather than cotangent whenever I can.

If I only had access to $C^\infty(M,\mathbb{R})$ or $C^\infty(\mathbb{R},M)$ then I could still find out the dimension but it’s a little more intricate (unless I cheat and recreate the other family in my first step).

For more, see the Frolicher spaces entry on the n-lab.

Posted by: Andrew Stacey on July 7, 2009 8:48 AM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

That is neat. Thank you.

Posted by: Eric Forgy on July 7, 2009 5:02 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

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Just for convenience of reference, the Boman article mentioned is Boman - Differentiability of a function and of its compositions with functions of one variable.

Posted by: L Spice on November 19, 2019 4:10 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

Thanks for posting your preprint, Martin (and thanks to Eric Forgy for emailing me).

Since pro-Lie groups are Frohlicher groups does this mean that profinite groupoids might be some kind of Frohlicher groupoids?

Please see the 3 page paper by Andrew Baker entitled “Profinite Groupoids and their Cohomology” in which profinite groupoids are defined as the projective limit of automorphism finite groupoids:

http://www.maths.gla.ac.uk/~ajb/dvi-ps/cohgpds.pdf

Posted by: Charlie Stromeyer on July 4, 2009 12:04 AM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

Profinite groupoids are, more generally, internal groupoids in the category of profinite spaces. They and the corresponding profinite categories have been studied in various places (not just Andy’s work and the applications he has in mind) for instance:

J. Almeida and P. Weil, Profinite categories and semidirect products , J.
Pure Appl. Algebra, 123, (1998), 1–50,

Generally Almeida and his coworkers has been exploring profinite semigroups and their applications (surprising ones sometimes) and as old results of Ehresmann link inverse semigroups and `ordered groupoids’ this leads naturally to looking at groupoids.

Posted by: Tim Porter on July 4, 2009 8:27 AM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

Thanks, Tim. The paper you cite shows that a profinite category can be obtained as the projective limit of discrete topological spaces.

The string theorist Raphael Bousso once asked if string theory could be discretized in the following sense:

Is there a sequence of theories with finite dimensional Hilbert spaces such that string theory emerges in the infinite dimensional limit?

Of course, the first problem here would then apparently be how to define string theory on discrete topological spaces (perhaps via vertex algebras?)

Posted by: Charlie Stromeyer on July 4, 2009 12:15 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

There are other contributors to the Café who are better placed to comment on the String theory question. There are two points that I might have made earlier. One is that Andy Magid, years ago now, used profinite groupoids in his approach to Galois theory and this is also evident to some extent in the book by Borceux and Janeldize.

The other point is that there is a lot of discussion (in this café and in the nLab) on internal v. enriched categories and there are examples where a profinitely enriched category has been used. One tantalising area where these ideas seem to come to the surface is in the Drinfel’d approach to Grothendieck-Teichmuller theory. This uses a sort of profinite completion of the free braided monoidal category on one object, Braid. I am no expert but the links certainly look strong. (Please do not ask for details of the Grothendieck-Teichmuller theory, as I will just cry HELP! I used to know more on it than I do now, but that is not saying much.)

Posted by: Tim Porter on July 4, 2009 12:50 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

Speaking of vertex algebras, I found via Google Books an interesting statement in the book “Moonshine Beyond the Monster” by Terry Gannon.

WARNING: You will be sorely disappointed if you suspect that this book is about wickedly monstrous home-distilled alcohol !-)

Gannon notes that part of the discrete series of the Virasoro algebra has been related to the (co)homology theory of the universal cover of SL_2(R) given a discrete topology, but instead of attempting to explain this, Gannon instead cites reference number 164.

Next week, I will stop by the Harvard or MIT science libraries to find out what reference number 164 is because the Google Books entry does not give the references.

Posted by: Charlie Stromeyer on July 4, 2009 3:30 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

Charlie wrote: “Next week, I will stop by the Harvard or MIT science libraries to find out what reference number 164 is because the Google Books entry does not give the references.” —————–

I think the transition between discrete and continuous is an interesting boundary.

[164] J. L. Dupont and C.-H. Sah,
‘ Dilogarithm identities in conformal field theory and group homology’,
Commun. Math. Phys. 161 (1994) 265–282.

http://arxiv.org/abs/hep-th/9303111 (20 pages) –>[164]
Page 14/15: “Our discussion only pins down the fractional part of such
central charges while the integral parts apparently spread the central
charges out in a way that resembled the volume distribution of
hyperbolic 3-manifolds. In the present approach, these effective
central charges are volumes of certain 3-cycles in a totally different
space–the compactification of the universal covering group of P SL(2, R).
These 3-cycles can be viewed as “orbifolds” since they arise from the
finite cyclic subgroups of SL(2, R). It should also be noted that the
central charge of the Virasoro algebra is the value of a degree two
cohomology class while our description is on the level of degree three
group cohomology, but for the Lie groups viewed as discrete groups. The
precise relation between these cohomologies is not too well understood.
On the level of classifying spaces of topological groups, there is the
wellknown conjecture, see [M] and[FM]:
Conjecture of Friedlander-Milnor. Let G be any Lie group and let p be
a prime. Then Hi(BGd, Zp) ? Hi(BG, Zp) is an isomorphism
(it is known to be surjective). … to page 17 …

Another of the principal points in the present work is the fact that
Rogers’ dilogarithm has long been known to be connected with the second
Cheeger-Chern-Simons characteristic class which is represented by the
Chern-Simons form that appears in many current theoretical physics
investigations. This connection is related to the interplay between the
”continuous” picture and the ”discrete” picture. On the mathematics side,
we have a direct map on the level of classifying spaces for groups equipped
with two topologies: one discrete, the other continuous. The map is the one
that goes from the discrete to the continuous. On the physics side, the
passage from the discrete to the continuous is a subject of debate since
there does not appear to be a specific map (in the mathematical sense).
However, there are still a large number of unresolved issues on the
mathematical side. For example, the Virasoro algebra is typically viewed
as the algebraic substitute for the diffeomorphism group of the circle.
(More precisely, it may be viewed as the “pseudo-group” of holomorphic
maps on the sphere with two punctures). This contains P SL(2, R) which
acts as a group of diffeomorphisms on the circle through the identifiction
of the circle with P 1(R).) Our procedure replaces these infinite dimensional
(pseudo-) groups by the finite dimensional subgroups. However, it is also
accompanied by the use of the discrete topology. Although the process of
playing of one topology against another is familiar in foliation theory,
it is not explored in the present work.” …

Posted by: Stephen Harris on July 4, 2009 5:58 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

Thanks very much for this reference, Stephen. Also, what do you think about this paper entitled “Is Hilbert space discrete?” which is hep-th/0508039 by R. Buniy et al. Abstract:

We show that discretization of spacetime naturally suggests discretization of Hilbert space itself. Specifically, in a universe with a minimal length (for example, due to quantum gravity), no experiment can exclude the possibility that Hilbert space is discrete. We give some simple examples involving qubits and the Schrodinger wavefunction, and discuss implications for quantum information and quantum gravity.

Posted by: Charlie Stromeyer on July 4, 2009 8:51 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

Charlie wrote: “Thanks very much for this reference, Stephen. Also, what do you think about this paper entitled “Is Hilbert space discrete?” which is hep-th/0508039 by R. Buniy et al.” —–

I try to be helpful to those who make thoughtful contributions.

Actually, I tried to read this paper recently because of its reference to causality and I wondered if it would have any relevance to Fredkin’s and Wolfram’s claim that the universe is a cellular automata (though I don’t think it is). I got stuck at this paragraph on the 1st page.

http://arxiv.org/abs/hep-th/0508039 [page 1]
“However, in a spatially discrete universe there is no
experiment which can exclude discreteness of the coefficients
a_n, if that discreteness is sufficiently small. We argue as
follows. If the number of degrees of freedom is finite, so
is the set of possible distinct measurement devices one can
construct. (By “distinct” devices we do not mean different
in design or construction, but rather that they measure
distinct physical quantities – in other words, correspond to
different operators acting on the Hilbert space. See the
qubit example below.) Equivalently, the number of eigenstates
of all possible distinguishable operators is finite (recall
that with ultraviolet and infrared regulators present, the
spectrum, and hence the number of eigenstates, of any particular
operator is finite). Thus, the physics of this universe
can be described using a Hilbert space with only a finite number
of distinguishable states – that is, a discrete and finite Hilbert
space, in which the values of an are themselves quantized.” … ————–

SH: There was a question on the Linguist list, “Is [English] language infinite?”
This question has one answer if you just consider the physical aspects. No human can utter infinitely long sentences created from a finite amount of words in the language at any given moment.
However, most linguists agreed that at a more abstract level, how many sentences could possibly be uttered, that there were countably infinite sentences in language. So the properties of one level of abstraction do not necessarily carry over to properties of a different level of abstraction. Anyway, the reasoning seemed suspect to me;
“If the number of degrees of freedom is finite, so is the set of possible distinct measurement devices one can construct.”

However, the conclusion arrived at seems ok, that the universe can never be proven to be discrete for the principled reasons, Heisenberg and Pauli Exclusion, which came up in the DC Cassirer thread.

1. “Only a finite number of classical bits are required to
specify the state of a discrete qubit. Note that, because
we cannot directly measure its state, a single qubit can be
used to transmit or store only a single bit of classical
information. (This is a result of Holevo’s theorem [5].)
Nevertheless, a perfect classical simulation of qubits with
continuous Hilbert space requires an infinite number of
classical bits.”

SH: This seems to preclude the universe as some type of classical CA.

Posted by: Stephen Harris on July 4, 2009 11:19 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

Anyway, the reasoning seemed suspect to me;

If the number of degrees of freedom is finite, so is the set of possible distinct measurement devices one can construct.

Yes, I also found that bit of reasoning suspect.

Posted by: Toby Bartels on July 5, 2009 12:32 AM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

Toby wrote: “Yes, I also found that bit of reasoning suspect.” ————-

Finally, we agree on something! I tend to think of questions like this in terms of Turing Machines and computability, so these quotes seem relevant to the Hilbert space paper (to me), but I could very well be skating on this ice :-)

Simulating Physics with Computers by Richard P. Feynman
“By the way, on the right-hand side of the above formulas
the s a m e operators are written in terms of matrices that
most physicists find m o r e convenient, because they are Hermitian,
and that seems to make it easier for them. They have invented
another set of matrices, the Pauli o matrices:
1°) o:(0 01) o_(0 i -i)0
And these are called spin–spin one-half–so sometimes people
say y o u ’ r e talking about a spin-one-half lattice.

The question is, if we wrote a Hamiltonian which involved
only these operators, locally coupled to corresponding operators on
the other space-time points, could we imitate every quantum mechanical
system which is discrete and has a finite number of degrees of
freedom? I know, almost certainly, that we could do that for
any quantum mechanical system which involves Bose particles. I’m
not sure whether Fermi particles could be described by such a
system. So I leave that open. Well, that’s an example of what I
m e a n t by a general quantum mechanical simulator. I’m not sure
that it’s sufficient, because I’m not sure that it takes care
of Fermi particles.”

http://www.mext.go.jp/english/news/1999/04/990414.htm
“On the other hand, Fermi particles cannot allow the overlapping
of particles, and accordingly, no distinction between density and
dispersiveness of electrons arises in continuing electric current.
With respect to this phenomenon, a Nobel-prize physicist E.M.
Purcell predicted theoretically that if a similar experiment is
conducted for electrons, a correlation opposite to the case of
bose particles shall be obtained.

For more than 40 years after that, various experimental efforts
had been made, but Purcell’s prediction had not been verified.
This is because there had been no technique to produce and detect
high-dense electrons sufficient to observe a correlation under
the condition where about only one electron exists per 1 million
pieces, even by using a field-emission-type electron gun which
can produce the densest electron flow in vacuum.”

Posted by: Stephen Harris on July 5, 2009 3:49 AM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

Also, what do you think about this paper entitled Is Hilbert space discrete? which is hep-th/0508039
——————————–

I found a fairly expert analysis of a paper which strongly related to “Is Hilbert Space Discrete?” hep-th/0606062: “Discreteness and the origin of probability in quantum mechanics”.
——————————–

http://dabacon.org/pontiff/?p=1262
…”Finally there are the binary tetrahedral group, the binary octahedral
group, and the binary icosahedral group which are SU(2) versions of the
symmetries of the correspondingly named platonic solids. And thats it!
Those are all the finite subgroups of SU(2)! (There is a very cool
relationship between these subgroups and ADE Dynkin diagrams. For more
info see John Baez’s week 230) So if we require that we only have a
discrete set of states and that the group structure of unitary
operations be maintained, this limits us in ways that are very drastic
(i.e. that conflict with existing experiments.)”

Posted by: Stephen Harris on July 5, 2009 10:01 PM | Permalink | Reply to this

formal (co)limits in physics

$MathML-enabled post (click for more details).$

The string theorist Raphael Bousso once asked if string theory could be discretized in the following sense:

Is there a sequence of theories with finite dimensional Hilbert spaces such that string theory emerges in the infinite dimensional limit?

This is beginning to look very off-topic for this thread. On the other hand, if I understand correctly what you are trying to get at, there might be a good on-topic point here, which maybe we should make more explicit.

Something like this:

When we pass from [[manifolds]] to [[generalized smooth spaces]] in particular in the form of [[smooth spaces]] given by sheaves on ordinary test spaces, we are effectively passing to [[free cocompletions]] (subject to respect for gluing) of ordinary spaces, hence to generalized spaces that are “formal colimits” of ordinary spaces.

If the formal colimits in question are [[filtered]] then these would be [[ind-objects]] (but a dual logic for formal limits and pro-objects applied accordingly).

Now, maybe it is indeed noteworthy that

a) one of the main abstract motivation for considering generalized smooth spaces such as [[diffeological spaces]] or Frölicher spaces is that mapping spaces $[\Sigma,X]$ for $\Sigma$ and $X$ “ordinary spaces” are naturally of this kind,

b) that many of the infamous infinities in physics can be traced back to the appearance of such mapping spaces – notably since the [[path intergal]] purport to be about integrals over generalized spaces of the form $[\Sigma,X]$ .

So – and this is what I guess Charlie Stromeyer might be suggesting – maybe it is worth considering formal (co)limits in physics more seriously from this point of view.

For instance, the path integral over spaces roughly of the form $[X, \bar \mathbf{B}G]$ (with the bar denoting a differential refinement of the ordinary classifying space) are the ones of relevance in gauge theory, and the best-developed tool that physicists have to tackle these is to approximate $[X, \bar \mathbf{B} G]$ by finite dimensional spaces (in “lattice gauge theory”),

This gives a well defined theory for each single approaximation. However, it is still impossible or at least very subtle to take the limit.

So maybe one should take serious the idea that $[X , \bar \mathbf{B}G]$ is a formal colimit over ordinary spaces?

Whatever that means in detail, it would be in such a sense that discussion of Bousso’s idea might be on-topic for a discussin of Frölicher spaces.

Posted by: Urs Schreiber on July 5, 2009 2:46 PM | Permalink | Reply to this

Re: formal (co)limits in physics

Urs, thank you very much for making somewhat more concrete my vague intuition and for bringing us back on topic.

To help me better understand the meaning of Frohlicher spaces, might you please say a bit about how they might be related to a concept I already understand which is that of smooth, stable, etale groupoids as defined in this paper by D. McDuff which is Definition 2.1 on page 3:

http://www.math.sunysb.edu/~dusa/orbminedec2306.pdf

It will be very easy for you to also understand this definition. Thank you.

Posted by: Charlie Stromeyer on July 5, 2009 3:19 PM | Permalink | Reply to this

back to Frolicher

$MathML-enabled post (click for more details).$

To help me better understand the meaning of Frohlicher spaces, might you please say a bit about how they might be related to a concept I already understand which is that of smooth, stable, etale groupoids

Hm, why that example?

Let’s sketch the map from the bird perspective first, lest we get lost in small valleys.

You are familiar with how an orbifold is a stack of smooth manifolds I suppose.

So it’s someting that for each smooth test manifold $U$ gives you a groupoid, read as the groupoid of ways of mapping $U$ smoothly into your orbifold.

Now, for such a stack on $Diff$ to qualify as an orbifold, there are usually some constraints on.

Next, forget these constraints and consider all stacks on diff as “generalized smooth groupoids”.

Then, next consider the sub-collection of these that happen to take values just in sets (aka dsicrete groupoids).

Then what we have is sheaves on $Diff$ , regarded a generalized smooth spaces.

Now again impose a certain restriction on these sheaves called “Isbell self-duality”, never mind that. Finally let $Lines \subset Diff$ be the full subcategory of $Diff$ on the real line, and restrict your Isbell-self-dual sheaves along that inclusion. That’s Frolicher spaces.

Have a serious look at [[generalized smooth space]] and the links provided there.

(Yes, that entry is still suboptimal. We’ll try to improve on it.)

Posted by: Urs Schreiber on July 5, 2009 3:59 PM | Permalink | Reply to this

Re: back to Frolicher

“Hm, why that example?”

I thought this example might be interesting because it is used in the paper I referred to as the basis for a proof that sheds some light on the polyfold approach to constructing the virtual moduli cycle of symplectic field theory, wherein “polyfold” means an orbifold in the category of Hilbert spaces.

Another hm, I wonder what a symplectic Froelicher space would look like?

Posted by: Charlie Stromeyer on July 5, 2009 5:47 PM | Permalink | Reply to this

Re: back to Frolicher

$MathML-enabled post (click for more details).$

“polyfold” means an orbifold in the category of Hilbert spaces.

This doesn’t sound right to me.

Posted by: Eugene Lerman on July 6, 2009 3:58 AM | Permalink | Reply to this

Re: back to Frolicher

Perhaps you are correct for this reason? In the McDuff paper, it says on page 42:

“Hofer, Wysocki and Zehnder [10] define a notion of properness that yields a concept of ep polyfold
groupoid X that has all expected properties. In particular, these groupoids admit smooth
partitions of unity since they are modelled on M-polyfolds built using Hilbert rather than Banach spaces.”

However, from what I can discern, they actually restrict to M-polyfolds built upon sc-Hilbert spaces. But are both Hilbert spaces and sc-Hilbert spaces parts of the same category of Hilbert spaces?

Posted by: Charlie Stromeyer on July 6, 2009 12:12 PM | Permalink | Reply to this

Re: back to Frolicher

On second thought, it turns out that any orbifold groupoid can be faithfully represented on a continuous family of finite dimensional Hilbert spaces.

This is Theorem 4.1 on page 10 of the paper “Representations of Orbifold Groupoids” which is also arxiv paper 0709.0176 by J. Kalisnik.

The concept of a continuous family of Hilbert space is a generalization of hermitian vector bundles. (See Definition 2.2 on page 5 of the Kalisnik paper.)

Now, a vector bundle is a projective R-module which means that it is a projective object within its category, but which category? For example, the abelian categories _RMod or Mod_R where R is a ring?

Perhaps I need to look at the book “Rings and Categories of Modules” by F.W. Anderson and K.R. Fuller.

Posted by: Charlie Stromeyer on July 6, 2009 3:37 PM | Permalink | Reply to this

Re: back to Frolicher

$MathML-enabled post (click for more details).$

Charlie wrote:

Now, a vector bundle is a projective R-module which means that it is a projective object within its category, but which category? For example, the abelian categories _RMod or Mod_R where R is a ring?

This is Swan’s Theorem, which is explained here.

Like Urs, I donâ€™t see any coherence in this discussion â€” you seem to be throwing around quotes from various papers and books, without any evident goal.

Posted by: John Baez on July 6, 2009 5:50 PM | Permalink | Reply to this

Re: back to Frolicher

“This is Swan’s Theorem, which is explained here.”

Thanks, John.

“Like Urs, I don’t see any coherence in this discussion — you seem to be throwing around quotes from various papers and books, without any evident goal.”

Yes, I got sidetracked from trying to better understand various concepts. Now, I am trying to focus on what Urs said about formal (co)limits in such a way that it would involve Froelicher spaces.

Posted by: Charlie Stromeyer on July 6, 2009 6:06 PM | Permalink | Reply to this

Re: back to Frolicher

sc-Hilbert spaces are not just Hilbert spaces; they have extra structure.

More importantly “built using Hilbert spaces” is not the same as “modeled on open subsets of Hilbert spaces.” In particular the local dimension of an M-polyfold is not constant. I think it’s better to think of M-polyfolds as being modeled on possibly singular level sets of smooth functions, and these can be pretty wild.

Another comment: Hofer has plenty of lecture notes and papers on his web page. So rather than trying to figure out what polyfolds are from an aside comment in McDuff paper, go to the source.

Posted by: Eugene Lerman on July 6, 2009 4:49 PM | Permalink | Reply to this

Re: formal (co)limits in physics

Speaking of gauge theory, you already know that D-branes carry gauge fields, and Witten suggests in his paper hep-th/0007175 constructing a background independent open string field theory by starting with infinitely many D-branes.

But then how to condense out the finite case, and how might this relate to what you say above?

Posted by: Charlie Stromeyer on July 5, 2009 7:08 PM | Permalink | Reply to this

Re: formal (co)limits in physics

Urs wrote: “Is there a sequence of theories with finite dimensional Hilbert spaces such that string theory emerges in the infinite dimensional limit?
This is beginning to look very off-topic for this thread.”

I’m still learning Topos theory and maybe I’ve misapplied some ideas here. I think that it is widely held that nearly all if not all physical events are computable. So I think mathematics that bears this in mind is more likely to contribute to the “unreasonable effectiveness of mathematics”. That is my primary lens.
Maybe here I’m wrong about something but I think that since Laubinger’s paper utilized ‘cartesian closed categories’ then his Frolicher space structure is a topos and therefore computable.
…………..

“There is a natural categorical semantics for the simply typed
lambda calculus: namely, using cartesian closed categories.
Scott observed that any such model could be regarded as set-
theoretic by embedding it, via the Yoneda functor, in a topos:
a categorical model of constructive set theory.” …

“Category Theory for Computing Science is a textbook in basic
category theory, written specifically to be read by researchers
and students in computing science by Michael Barr and Charles Wells

Chapter 6 is an introduction to cartesian closed categories, which
have been a major source of interest to computer scientists because
they are equivalent in theoretical power to typed lambda calculus.
In this chapter, we outline briefly the process of translating
between typed lambda calculus and cartesian closed categories.”

A Lie algebra for Frolicher groups Authors: Martin Laubinger
“Frolicher spaces form a cartesian closed category which
contains the category of smooth manifolds as a full subcategory.
………………………….

Now, I don’t think the discrete Hilbert space paper constitutes a topos and I don’t think it describes a computable structure:
“In our discussion discreteness should
not be taken to imply regularity, either in spacetime or
the structure of Hilbert space. We are not suggesting
that continuous Hilbert space necessarily be replaced by
a lattice; instead, for example, the discreteness might be
due to an intrinsic fuzziness or uncertainty.”

In my admittedly beginning understanding of cartesian closed categories, I don’t think they are described properly as having “intrinsic uncertainty”.

Well, maybe still off topic… so I’ll desist.
Thanks to the pointer to Bousso’s idea. I remember in Bohm’s textbook he said that greatly increasing the frequency of discrete events would begin to approximate continuous phenomena.

Posted by: Stephen Harris on July 5, 2009 7:42 PM | Permalink | Reply to this

Re: formal (co)limits in physics

$MathML-enabled post (click for more details).$

Stephen Harris typed:

Urs wrote: “Is there a sequence of theories with finite […]

No, I didn’t write that.

I quoted somebody who quoted somebody who said that.

Generally in your above comment, I have, to be frank, trouble sorting out which part is (extensive) quote and which part is not.

You are encouraged to use the blockquote-tag to help make clear when you are quoting what.

[…] This is beginning to look very off-topic for this thread.”

This part is my voice. And I keep thinking that the discussion is lacking some focus.

I am open to discussing interesting speculative idea. But I need some sort of coherence, otherwise I don’t know what it is we are discussing.

Posted by: Urs Schreiber on July 5, 2009 9:25 PM | Permalink | Reply to this

Re: formal (co)limits in physics

Urs, I tried some to understand the path integral concept and the quantization of nonabelian gauge theories, but I am stuck on both concepts.

Now, I did read the Claymath description of the “Quantum Yang Mills Theory” by Jaffe and Witten, and at the beginning of Section 4.3 it says:

“One approach to the existence problem for four-dimensional gauge theory is to begin with a lattice regularization and to demonstrate the existence of a limit as the lattice spacing tends to zero. The gauge invariant action [39] can be defined on a toroidal Euclidean space-time, yielding a well-defined path integral. From this point of view, one must verify the existence of limits of appropriate expectations of gauge-invariant observables as the lattice spacing tends to zero and the volume tends to infinity.”

Is this the essence of the problem with lattice gauge theory, and what prevents category theory from framing the context for verifying the existence of these limits? (I don’t really know anything about lattice gauge theory which is why I am asking this question).

Posted by: Charlie Stromeyer on July 10, 2009 2:12 PM | Permalink | Reply to this

Re: formal (co)limits in physics

Ah, now I am starting to see what lattice gauge theory is some because it turns out to be dual to a concept that John Baez invented.

You see, roughly 11 years ago, John introduced the concept of spin foam and it turns out that a particular kind of spin foam model is dual to Euclidean lattice YM SU(N) theory in d=2 or greater.

This is shown in section 5 of F. Conrady’s gr-qc/0504059, and is improved upon in V. Bonzom’s arxiv paper 0905.1501, but I don’t see yet how this geometric approach would be expressed via category theory.

Posted by: Charlie Stromeyer on July 10, 2009 3:39 PM | Permalink | Reply to this

Re: formal (co)limits in physics

To make some progress with this question I need to better understand the category theoretic definition of a spin foam.

Someone (I don’t know who) once said that a spin foam is a functor between spin network categories, but what are spin network categories?

Also, a spin foam is a quantum fundamental groupoid which means a functor representation of CW complexes on rigged Hilbert spaces (aka Frechet nuclear spaces, aka Gelfand triple).

Posted by: Charlie Stromeyer on July 11, 2009 2:17 PM | Permalink | Reply to this

Re: formal (co)limits in physics

$MathML-enabled post (click for more details).$

Charlie wrote:

Ah, now I am starting to see what lattice gauge theory is some because it turns out to be dual to a concept that John Baez invented.

Yup.

To make some progress with this question I need to better understand the category theoretic definition of a spin foam.

The best currently available explanation of this is in my Winter 2005 QG Seminar notes. I’ll try to write something more terse and perhaps clearer in my discussion of Barrett and Westbury’s 1992 paper here. However, you’ll have to wait a couple of days for that.

Briefly, the idea is this:

Spin networks are a pictorial notation for morphisms in the category of vector spaces. They are equivalent to abstract index notation.

Spin foams are a pictorial notation for 2-morphisms in the 2-category of 2-vector spaces. They are equivalent to categorified abstract index notation.

A nice example of how the analogy works:

Given a semisimple algebra, the Fukuma–Hosono–Kawai construction lets you construct a 2d TQFT by triangulating spacetime and treating the graph dual to the triangulation as a spin network.

Given a semisimple 2-algebra, the Barrett-Westbury construction lets you construct a 3d TQFT by triangulating spacetime and treating the 2-dimensional complex dual to the triangulation as a spin foam.

I explained the Fukuma–Hosono–Kawai construction in exhaustive detail in the Fall 2004 QG Seminar, and more tersely and sketchily here. The Barrett–Westbury construction is a categorification of that.

I know you’re more interested in spin foams than in their application to TQFTs. In a sense the TQFTs are just a distraction — but so far, the most thorough discussion of the category-theoretic foundations of spin foams is mixed in with a discussion of this application.

Posted by: John Baez on July 12, 2009 12:05 PM | Permalink | Reply to this

Re: formal (co)limits in physics

Thanks, John. What I like about the definition of a spin foam as a functor representation of CW complexes on rigged Hilbert spaces is that I do not see any mathematical reason that would prevent viewing this functor representation as a (co)filtered limit.

Also, before I forget, I wanted to alert you to something off-topic: When either of you have some time for it, you or Mike Stay might want to check out quantum lambda calculi:

http://en.wikipedia.org/wiki/Quantum_programming#Quantum_lambda_calculi

Posted by: Charlie Stromeyer on July 12, 2009 1:34 PM | Permalink | Reply to this

Re: formal (co)limits in physics

This conversation is rambling too much for my taste. Nobody will ever look in this particular blog entry for discussion of the quantum lambda calculus. If you want to talk about quantum lambda calculi, it would better to do so in the blog entry devoted to that subject.

Posted by: John Baez on July 12, 2009 1:55 PM | Permalink | Reply to this

Re: formal (co)limits in physics

Here is the problem I have run into:

Is there some general way to prove that a sequence of weighted sums over CW complexes converges?

For example, perhaps by using some property of the fact that CW complexes are subsequential spaces?

Posted by: Charlie Stromeyer on July 14, 2009 2:17 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

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Profinite groupoids are, more generally, internal groupoids in the category of profinite spaces.

[…] for instance

J. Almeida and P. Weil, Profinite categories and semidirect products , J.

Pure Appl. Algebra, 123, (1998), 1–50,

From where I am right now I don’t seem to have access to the article.

but luckily there is the $n$ Lab.

I see from [[profinite group]] that these are just [[pro-objects]] in the category of finite groups.

Now C. Stromeyer asked

Since pro-Lie groups are Frölicher groups does this mean that profinite groupoids might be some kind of Frölicher groupoids?

How did we jump from pro-Lie to pro-finite here?

It would seem to me that for pro-finite objects the smooth structure is pretty much trivial, as these are “essentially finite”. No?

In which sense is a pro-finite group a pro-Lie group in more than the trivial sense that every finite group is trivially a non-connected Lie group?

Posted by: Urs Schreiber on July 5, 2009 2:13 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

The paper by Almeida and Weil that Tim mentioned is also available here:

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.5674

Urs, from what I can tell, you are correct that profinite groups are pro-Lie groups in the trivial sense you say.

The Lie theoretic analogue of profinite groups are pro-finite Lie rings. This is shown in this paper by L. McInnes and D.M. Riley:

Pro-finite p-adic Lie algebras. Journal of Algebra. v319(1), pp.205-234.

Posted by: Charlie Stromeyer on July 5, 2009 3:03 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

So pro-finite groups are pro-objects in the category of finite groups (or a cofiltered diagram of finite groups), and are also Froelicher groups.

We can also say that a pro-finite group is a compact Hausdorff topological group whose open subgroups form a neighborhood base of the identity.

By analogy, pro-finite Lie rings are compact Hausdorff topological Lie rings whose open ideals of finite index form a neighborhood base of 0. Each ideal of a pro-finite Lie ring L is a normal subgroup of (L,+). Thus, (L,+) is a pro-finite group.

The category of pro-finite groups is the category of topological groups that are filtered inverse limits of finite groups.

Does this suggest then that the category of pro-finite Lie rings would be something like the category of topological Lie rings that are inverse limits of Lie rings (which are a generalization of Lie algebras - an example being any Lie algebra over a general ring)?

Posted by: Charlie Stromeyer on July 5, 2009 11:32 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

$MathML-enabled post (click for more details).$

profinite groupoids are defined as the projective limit of automorphism finite groupoids

This seems a bit suspicious to me. This would include any set considered as a groupoid with no non-trivial arrows. As Tim said, groupoid objects in profinite spaces is better.

But how about the following suggestion: pro-tame groupoids, using the cardinality of the groupoid?

Or pro-essentially-finite groupoids - pro-objects in the essential image of the 2-functor $FinGpd \to Gpd$ ?

Posted by: David Roberts on July 8, 2009 4:05 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

Profinite things as topological things it is really a very odd link. The real objects are the pro objects in some nice category. Whether or not they give a topological thing in an inverse limit is largely irrelevant. You have results on prolocalic groupoids. You don’t take a space but may take a limit in a category of locales, but who cares, the beautiful pro-object is there all the time to remind you where everything came from!

Posted by: Tim Porter on July 8, 2009 5:40 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

Tim, I agree with you because the fundamental groups of algebraic topology are not usually profinite (and we already know that the constructs of algebraic topology are generally functorial).

Btw, Aviv Censor has extended the concept of cardinality of a groupoid to that of a measure a topological groupoid, but I don’t know how he did this.

Posted by: Charlie Stromeyer on July 8, 2009 7:11 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

I know there is a version of Haar measure for topological groupoids. One of the earliest that I know of is in

A.K. Seda, Haar Measures for Topological Groupoids. Proc. Roy. Irish Acad. Vol. 76A (5) (1976), 25-36.

But that may not be what you mean.

Posted by: Tim Porter on July 9, 2009 7:42 AM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

Thanks, Tim. Is it already known if such measures can be assigned to continuous groupoid homomorphisms?

Posted by: Charlie Stromeyer on July 9, 2009 12:52 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

I’m not an expert. I merely used to be a collegue of Tony Seda in Cork, who had done work on this. Have a look at his publication list.

Others have also done work on this. I believe there is a SLN by Jean Renault.

I googled this and found a paper

http://www.emis.de/journals/SMA/v01/a08.html

which has a reasonable list of references including the ones I had heard of and lots I had not.

Perhaps you can find more information there.

Posted by: Tim Porter on July 9, 2009 4:26 PM | Permalink | Reply to this

Re: Laubinger on Lie Algebras for Frölicher Groups

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Martin,

I hope that you’re still checking the comments on this post - discussions can tend to get a little wayward, I know! I’ve started reading through your paper (sometime I’ll take a more detailed look at your thesis but that’ll have to wait for now) and am recording my comments as I go through.

Differentiation in Topological Vector Spaces.

I’m curious as to why you use an iterative definition here rather than the “all in one” definition of Kriegl and Michor, especially given that you’ve already referred to their monograph. I was pretty much convinced by the “historical remarks” at the end of their first chapter (meaning that I haven’t looked into the matter much myself) that iterative definitions were doomed to failure outside the realm of Banach spaces. Thus the characterisation of $C^\infty$ -maps between metrisable TVS as being those that take smooth curves to smooth curves appears as a theorem and one is left wondering what happens for non-metrisable spaces. In particular, this implies (in the weak sense of “gives evidence for”) that the category of TVS with $C^\infty$ -maps is not a full subcategory of the category of Frölicher spaces.

This feeds into the slightly odd definition of “convenient” (definition 3). The treatment in Kriegl and Michor makes it clear (to me, at least) that the requirement of “convenience” is to ensure that “things that appear differentiable have derivatives”. It’s a bit like completeness of metric spaces: sequences that ought to have limits (Cauchy sequences) actually do. With Kriegl and Michor’s definition, Theorem 3 becomes a triviality (completeness implies local completeness, hence convenience).
Tangent “Spaces” of Frölicher Groups.

I had not realised that the tangent “space” of a Frölicher group is always a vector space. That’s a neat result.
Frölicher Structure on Tangent Bundle.

I’m curious as to the choice here. You note in Remark 3 that there are two ways to put a Frölicher structure on the tangent bundle of a Frölicher space, but you make a definite choice. Why do you make that particular choice? Indeed, is there actually a choice to be made here? Do you have an example where the two choices differ? (For the record, I guess that the answer to the first is “yes”, but an example doesn’t spring to mind.)

I would have gone for the following construction: the tangent bundle of $X$ is a quotient of the space of smooth curves in $X$ , $C$ . The curves $C$ has a natural Frölicher structure as it is $C^\infty(\mathbb{R},X)$ . Similarly, quotients gain Frölicher structures as the category is cocomplete. Thus $T X$ gets an obvious Frölicher structure. I think that this yields the same structure as you get, but I’ll not work out a detailed proof here (this is the sort of thing that belongs more in the n-lab than here).
The Bijection.

The bijection that the Lie bracket hinges on is extremely interesting. Do you have any sense of what it might mean for this to fail? For a convenient vector space, $E$ , then one certainly has that $T_0 E \cong E$ (for spectators, the notion of tangent space used here is the kinematic tangent space). If $E$ is a locally convex vector space, not necessarily convenient, with the Frölicher structure coming from the smooth curves then (I strongly suspect) that $T_0 E$ is the $c^\infty$ -completion of $E$ (Kriegl and Michor discuss $c^\infty$ -completions early in their book but by the time they get to tangent spaces they are working solely with convenient vector spaces). So a first step might be to try to prove that $\mathfrak{g}$ is a convenient vector space. The completion part shouldn’t be too hard; after all, $\mathfrak{g}$ is already the tangent space of something and that should have taken care of the completion part. By way of analogy, if I’m right that $T_0 E$ is the $c^\infty$ -completion of $E$ then $T_0T_0 E \cong T_0 E$ . Thus the crucial point seems to be the question of how the Frölicher structure on $\mathfrak{g}$ interacts with its vector space structure. Since both are defined using the functions on $G$ , I would suspect that this interaction is well-behaved.

Well, those are my initial thoughts. I hope that they are of some use to you! On first reading, it seemed a very nicely presented paper (I didn’t go through it looking for spelling/grammar mistakes as I occasionally do, though one howler stood out: “This paper contains the main result of our Ph.D. thesis”. I realise that it’s often considered bad form to use “I” in papers (why?), but the oft-substituted “we” reads badly here. Shortly afterwards you use “the author” which would be better here as well.)

Posted by: Andrew Stacey on July 8, 2009 10:32 AM | Permalink | Reply to this

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July 1, 2009