## May 28, 2007

### This Week’s Finds in Mathematical Physics (Week 252)

#### Posted by John Baez

In week252 hear about the possibility of oceans on Neptune billions of years from now:

Learn the latest about hot Neptunes in other solar systems. See the electromagnetic snake at the center of the Galaxy. And, continue reading the Tale of Groupoidification! In this episode, with a nod to the work of Georg Frobenius and William Burnside, we begin to tackle the theme of "Hecke operators".

Posted at May 28, 2007 2:37 AM UTC

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### Re: This Week’s Finds in Mathematical Physics (Week 252)

In 3 billion years the Andromeda Galaxy will collide with our galaxy. Many solar systems will be destroyed.

How many?

The distances between stars are quite large compared to the size of solar systems. On the other hand, there are many stars, and the number of collisions probably goes as the number of stars squared.

Can we estimate the number of collisions? What is the probability the solar system will be involved in such an event? Is our position within the galaxy lucky or unlucky in this respect?

Posted by: Squark on May 28, 2007 6:59 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Squark wants to know:

How many?

I don’t know! Christine Dantas has studied galactic gravitational dynamics; she might have a better guess, or know someone who does. A key question is how much gravitational perturbation it takes to break a solar system.

I imagine research on this topic will become more fashionable when Andromeda gets closer.

For now, you can look at some pictures of colliding galaxies. In the Antennae Galaxies, huge numbers of bright blue-white young star systems are being formed by the collision of gas and dust clouds:

However, I’ve read that the end result of a collision is an elliptical galaxy where few new star systems form. The end of new life? I wish I knew more.

You can also watch a movie of the collision between the Milky Way and Andromeda! Simulated, of course.

You’ll note that the galaxies go through each other a couple times before merging… it’s not as if they shoot through each other only once. Surely this increases the chance of solar system disruptions.

Posted by: John Baez on May 28, 2007 7:38 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

As far as I can tell, the amount of ejected material, tidal debris, etc, will depend on the collision scenario (orbital energy of the galaxies) as well as on their mass distribution (baryonic and non-baryonic) and possibly on other finer details such as the amount of diffuse mass of the Local Group (these latter are important for taking into account dynamical friction effects).

I have never simulated such a detailed collision between the Milky Way and Andromeda, but Cox and Loeb did.

(My previous work with N-body simulations focused on merging elliptical-like objects in the context of the origin of the “fundamental plane” relation, and were intended as to study the overall features of the mergers, not their detailed interactions. If memory serves, however, I could have about ~ 15% of mass ejections in some cases, or even more, depending on the “violence” of the merger, that is, how strongly the gravitational potential would fluctuate during the collision).

Christine

Posted by: Christine Dantas on May 30, 2007 12:36 PM | Permalink | Reply to this

### planetary and galactic; Re: This Week’s Finds in Mathematical Physics (Week 252)

Excellent “Week’s Find” – about which I am not now commenting on the lovely Math exposition, but rather on the planetary atmospheres [I was a house-mate of Andy Ingersoll] and futures point.

(1) “… UK researchers from University College London, along with colleagues from Boston University, have found that the hotter than expected temperature of Saturn’s upper atmosphere – and that of the other giant planets – is not due to the same mechanism that heats the atmosphere around the Earth’s Northern Lights. Reporting in Nature (25th January [2007]) the researchers’ findings thus rule out a long held theory.

A simple calculation to give the expected temperature of a planet’s upper atmosphere balances the amount of sunlight absorbed by the energy lost to the lower atmosphere. But the calculated values don’t tally with the actual observations of the Gas Giants: they are consistently much hotter…. We need to re-examine our basic assumptions about planetary atmospheres…”

(2) The first published suggestion thast a Jovian planet could have its mass mostly ablated away by supernova, leaving a metal-rich pseudo-terrestrial planet was by Science Fiction author Poul Anderson (who’d originally wanted to be an astrophysicist). When such a planet was found orbiting a pulsar, and he was footnoted, it was one of the happiest days of his life. I spoke with him several times about that.

(3) Colliding galaxies: trickier subject than it seems, due to non-classical issues of dark matter, magnetic fields, and cold hydrogen. Colliding galaxies, as seen from civilizations embedded therein, was also first discussed in Science Fiction. Surely a reader here can identify the source [exercise for the reader]…

Posted by: Jonathan Vos Post on May 28, 2007 8:56 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Regarding the issue of linear vs. permutation representations in this week’s TWF, I’ve always felt a bit of wonder about the power of linear representations in the study of finite groups. It doesn’t surprise me that permutation representations are useful. After all, finite groups are group objects in the category of finite sets. But of the millions of other categories in the night sky, why is the category of finite-dimensional vector spaces the one whose objects it is so useful to make finite groups act on?

There are probably lots of other examples of theorem X about groups of type Y being proved by looking at actions on objects in a category Z, but I just don’t know them. Do other people know? It would be fun to make a little table of it all.

Posted by: James on May 29, 2007 9:02 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

But of the millions of other categories in the night sky, why is the category of finite-dimensional vector spaces the one whose objects it is so useful to make finite groups act on?

I expect that if there is a good answer to this question, then it is related to the fact that representations of groups on vector spaces are a degenerate case of equivariant vector bundles.

Posted by: urs on May 29, 2007 11:47 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

But of the millions of other categories in the night sky, why is the category of finite-dimensional vector spaces the one whose objects it is so useful to make finite groups act on?

Thanks! And this is close to one of my naive questions: “why everything is abelian?”. I mean that the large part of mathematics is built on the notion of abelian (additive) group: rings, fields, bodies, modules, vector spaces, etc. I think this is unnatural without some mathematical explanation… But I don’t know how to formulate my question correctly. So, your question is better then my.

Posted by: osman on May 29, 2007 12:38 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

“Thanks! And this is close to one of my naive questions: “why everything is abelian?”. I mean that the large part of mathematics is built on the notion of abelian (additive) group: rings, fields, bodies, modules, vector spaces, etc.”

I think the answer is: for historical reasons only.

There are mathematical fields (like analysis in metric spaces, see the work of Gromov) where the natural notion replacing a vector space is a simply connected Lie group whose Lie algebra admits a positive graduation (aka Carnot group). A particular example is the Heisenberg group.
See http://xxx.arxiv.org/abs/0705.1440
for a definition of linearity in this context.

200 years ago there was a single geometry:
euclidean. Is it risky to say that in 100
years from now we shall have non-euclidean analysis, which is not based on abelian groups?

Posted by: marius on May 30, 2007 8:16 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

There are mathematical fields (like analysis in metric spaces, see the work of Gromov) where the natural notion replacing a vector space is a simply connected Lie group whose Lie algebra admits a positive graduation (aka Carnot group). A particular example is the Heisenberg group.

Since I’m not mathematician and, consequently, I don’t know some explanational theorems, I see the “ghost of abelianity” in all above words. For example, “Lie group” is associated for me with “smooth manifolds” wich require abelianity (and, moreover, differentiation!)… Same for metric spaces…

Posted by: osman on May 30, 2007 10:08 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Not sure if this is relevant for these questions, but the concept “differentiation” and everything derived from it – like Lie algebras of Lie groups – is by definition about approximating things by the abelian groups $\mathbb{R}^n$ (“linear approximation”).

But there is no lack of attempts to generalize geometry which is locally modeled on $\mathbb{R}^n$ to geometry locally modeled on more non-abelian things.

It’s just that geometry modeled on $\mathbb{R}^n$ is already immensely rich. It will take a while (millenia? :-) until all the non-commutative or non-whatnot generalizations of everything here have been put to paper.

Posted by: urs on May 30, 2007 10:44 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

It’s just that geometry modeled on $\mathbb{R}^n$ is already immensely rich. It will take a while (millenia? :-) until all the non-commutative or non-whatnot generalizations of everything here have been put to paper.

I don’t need such generalizations. I just want (categorified, if you like) answer on my question ‘why’? Why additive-group-geometry is so rich? Is it just a chance? Why you (I mean cafe) try to categorify everything using additive categories? How unconscious you attempts are? For example, groups are exactly groups of authomorphisms of something general (say, universal algebras). Abelian groups are exactly - … what? Please don’t say ‘abelian automorphisms’ :)

Posted by: osman on May 30, 2007 11:03 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Osman wrote this:

And this is close to one of my naive questions: “why everything is abelian?”. I mean that the large part of mathematics is built on the notion of abelian (additive) group: rings, fields, bodies, modules, vector spaces, etc. I think this is unnatural without some mathematical explanation…

and also this:

I just want (categorified, if you like) answer on my question ‘why’? Why additive-group-geometry is so rich? Is it just a chance?

I think these are interesting questions, well worth pondering. In fact, these are the types of questions that drew me to category theory in the first place.

I don’t think there is one simple answer. But I think category theory can shed some light on some partial explanations.

Let’s agree with virtually all mathematicians that the notion of ‘collection’, in particular of finite collections, is basic to mathematics. Now already there is some ‘commutativity’ lurking within that simple notion: for example, in drawing a two-element subset, the order in which you draw the elements out is irrelevant. To put it in more sophisticated terms: given two elements of a collection, there is a symmetry of the collection which interchanges the two elements [despite the fact that the symmetry group is non-abelian].

This simple observation is at the root of some other observed commutativities. Let $A$ and $B$ denote sets. The operation of taking disjoint union $(A, B) \mapsto A + B$ is again ‘commutative’ in the sense that there is a canonical isomorphism

$A + B \stackrel{\sim}{\to} B + A.$

It wasn’t until the 20th century, when Eilenberg and Mac Lane introduced categories, that we had a fully satisfying explanation of this and various other ‘canonical commutativity isomorphisms’ in terms of ‘universal properties’. Nowadays we may say that the direct sum $A + B$ is characterized by the fact that a function $A + B \to C$ is uniquely determined by two functions: an function $f: A \to C$ and a function $g: B \to C$. The ‘commutativity’ of disjoint sum then derives from the fact that we can interchange $f$ and $g$: the above-noted ‘commutativity’ lurking within the very notion of collection itself.

A similar observation shows that the cartesian product $A \times B$ is also ‘commutative’ (up to canonical isomorphism): a function $C \to A \times B$ is uniquely determined by a set consisting of two functions $f: C \to A$ and $g: C \to B$, and again the commutativity arises from an act of interchanging $f$ and $g$.

The commutativity of addition and multiplication of natural numbers is then a decategorification of the ‘commutativity’ of the operations of disjoint sum and cartesian product as applied to finite collections. From there we derive commutativity of addition and multiplication on integers, rational numbers, and so on. The same type of thing shows up in much more sophisticated mathematics too, for example when we consider natural ‘commutative’ operations on vector bundles (e.g. Whitney sum), and their decategorified expressions in terms of K-theory.

Another source of commutativity is in various ‘interchange laws’. There is for example a famous lemma of Eckmann-Hilton which shows that the higher homotopy groups of spaces are abelian. [In some sense, this lemma is rooted in ancient or childhood experience: if you have some marbles spread out on the ground, you can interchange the positions of two of them by rolling them around without having them bump into each other.] The relationship between interchange and commutativity allows for the fact that two homomorphisms of abelian groups $f, g: A \to B$ can be added to form another homomorphism $f + g: A \to B$, and this has some very nice structural consequences for the world of abelian groups: in addition to direct products, it has internal homs and tensors and lots of interactions between these operations. The world of abelian groups is thus incredibly rich, but at the same time much more tractable, and therefore much more completely analyzed and developed, than the world of groups generally [as is clear when you consider the theory of finite abelian groups as versus the theory of finite groups].

These are just some very partial explanations. I think I can understand what you mean when you say that abelian-ness is unnatural from certain points of view (as we have learned from quantum mechanics for instance, but also just from contemplating the very simplest symmetry groups and categories generally). But from other points of view, abelian-ness is of course highly natural and expected.

Posted by: Todd Trimble on May 30, 2007 3:55 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Thank you. I pointed below in this thread that decategorification of coproduct (and disjoint union as particular case) is one of possible directions. I also know some theorems from algebraic topology (like fundamental group of topological group is abelian). In fact, I found many partial explanations. :) But it is not enough for me, and I don’t understand why. I’m sorry to spend your time and attention.

Posted by: osman on May 30, 2007 4:20 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

That’s all right. Excuse me if I over-explained some things already well known to you.

Buried within my post was a remark whose hidden point is that the category of abelian groups is a closed category (unlike the category of groups). This means that the category of abelian groups (or the category of vector spaces) serves as the base or foundation for a theory of categories whose homs carry such structure, so-called “categories enriched in abelian groups”. Similarly, a linear category is a category whose homs are enriched in vector spaces.

So, what I was hinting is that the category of abelian groups or of vector spaces serves as a foundation for a theory of enriched categories parallel to and even richer than ordinary category theory, due to marvelous hom-tensor structures on those categories not shared by the category of groups generally. This in part accounts for the ‘unreasonable effectiveness’ of abelian groups in the study of group representations and many other things.

Posted by: Todd Trimble on May 30, 2007 5:06 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Can you give me an example of such interesting category (linear for example), but where the definition of objects does not depend on abelian property?

And could you please give me some weblinks to articles related to closed categories with interrelation to linearity (or abelianity)? Thank you!

Posted by: osman on May 30, 2007 6:08 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Buried within my post was a remark whose hidden point is that the category of abelian groups is a closed category

[…]

This in part accounts for the ‘unreasonable effectiveness’ of abelian groups in the study of group representations and many other things.

Can this be turned around to a sensible statement along the lines that every closed category is a category of “abelian objects” in some meaningful sense?

Posted by: urs on May 30, 2007 6:38 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Yes, there is a theorem by Fred Linton, in “Autonomous equational categories”, J. Math. Mech. 15, 637-642, a version of which you can find in Section 3.10 of Borceux’s Handbook of categorical algebra, vol. 2.

Let’s note $\mathbf{T}$ for an algebraic theory. We call $\mathbf{T}$ commutative if each operation commutes with each other operation. This is equivalent to the fact that each n-ary operation of the theory induces a morphism of algebras $A^n\rightarrow A$ for each $\mathbf{T}$-algebra $A$. Then $\mathbf{T}$ is commutative if and only if for each $\mathbf{T}$-algebras $A,B$, the set $\mathbf{T}\text{-}\mathrm{Alg}(A,B)$ of morphisms of algebras is a subalgebra of $B^{UA}$ (where $U$ is the forgetful functor to Set). This makes $\mathbf{T}\text{-}\mathrm{Alg}$ a closed category.

There is a generalization of this for monads : Anders Kock (Closed categories generated by commutative monads, J. Austral. Math. Soc. 12 (1971) 405-424) defined commutative monads and showed that a commutative monad is exactly a monad in the 2-category of symmetrical monoidal categories. There is a 2-categorical version : “Pseudo-commutative monads and pseudo-closed 2-categories”, Martin Hyland and John Power, JPAA 175 (2002), 141-185.

Posted by: Mathieu Dupont on May 30, 2007 9:16 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

I asked, following Todd Trimble’s interesting remark that abelianness is related to closedness hence to the possibility to enrich over:

Can this be turned around to a sensible statement along the lines that every closed category is a category of “abelian objects” in some meaningful sense?

Mathieu Dupont said:

Yes

Now that’s interesting! Thanks for sharing this.

Posted by: urs on May 30, 2007 9:29 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

But I understood the importance of your question and Mathieu’s answer just now. Anyway it’s good, thank you. :)

Posted by: osman on May 6, 2008 4:29 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

And recently I found one more interesting direction: mathematical notion of causality. Let’s say, following this article, that non-autonomous system is presheaf over total order (like order of $\mathbb{R}$) and autonomous system is presheaf over archimedean group (like $\mathbb{R}$ itself). But any archimedean totally ordered group is abelian. So if we start from non-autonomous system and try to describe the notion of causality and autonomity, then we possible can obtain naturally archimedean totally ordered group and, consequently, abelianity. So, we will have the sequence:

mathematical causality over non-autonomous system -> autonomous system -> archimedean totally ordered group -> abelian group

Posted by: osman on May 30, 2007 4:47 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Todd wrote:

I think I can understand what you mean when you say that abelian-ness is unnatural from certain points of view (as we have learned from quantum mechanics for instance, but also just from contemplating the very simplest symmetry groups and categories generally). But from other points of view, abelian-ness is of course highly natural and expected.

When I first glanced at this, I momentarily thought you said “abelian-ness is natural from certain points of view, as we have learned from quantum mechanics for instance,…” And this would be correct too.

After all, one big lesson of quantum mechanics is the superposition principle. This says that, mysteriously, the space of states of a physical system is not a mere set, but a vector space: an object in an abelian category!

So, while quantum mechanics is famously noncommutative, it’s also famously abelian!

This abelian-ness has been on my mind a lot lately — since as you know, the Tale of Groupoidification will seek to explain the seeming appearance of vector spaces and linear operators in quantum physics as really just a decategorified shadow of what’s really going on: groupoids and spans of groupoids!

Posted by: John Baez on May 31, 2007 9:49 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

John Baez wrote:

the Tale of Groupoidification will seek to explain the seeming appearance of vector spaces and linear operators in quantum physics as really just a decategorified shadow of what’s really going on: groupoids and spans of groupoids!

Are you going to categorify all constituents of abelian-group-ness together? I mean not just categorify abelian monoid, but additive group? Or this is somehow unimportant here?

Posted by: osman on June 1, 2007 6:30 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Osman wrote:

Are you going to categorify all constituents of abelian-group-ness together? I mean not just categorify abelian monoid, but additive group? Or this is somehow unimportant here?

It’s a lot less important than one might think. For example, a huge amount of group representation theory can be described in a different way where we never mention numbers, vector spaces, linear operators, etcetera. Instead of vector spaces, we use groupoids. Instead of linear operators, we use spans of groupoids.

The basic idea is sketched in week247. Jeffrey Morton used this idea to redescribe the quantum harmonic oscillator and Feynman diagrams here. I’ll explain more details in future episodes! It’s a long but fun story.

Posted by: John Baez on June 1, 2007 5:19 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

John Baez wrote:

Instead of vector spaces, we use groupoids. Instead of linear operators, we use spans of groupoids.

“What a nice surprise”…

Excuse me for one more naive and maybe premature question, but it’s very intriguing: is it possible to use groupoids instead of vector spaces in case of differential geometry?

Posted by: osman on June 1, 2007 7:39 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Osman writes:

is it possible to use groupoids instead of vector spaces in case of differential geometry?

Don’t know! Haven’t even tried!

Posted by: John Baez on June 1, 2007 11:37 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

but the concept “differentiation” and everything derived from it - like Lie algebras of Lie groups - is by definition about approximating things by the abelian groups $\mathbb{R}^n$ (“linear approximation”).

And what about Differential Geometry over General Base Fields and Rings?

Posted by: osman on May 30, 2007 11:11 AM | Permalink | Reply to this

### A linearity fallacy?

And what about Differential Geometry over General Base Fields and Rings?

Right. As I tried to say, one can generalize ordinary differential geometry in many ways.

In fact, the very first sentence of the document you link to reads:

Classical, real finite-dimensional differential geometry and Lie theory can be
generalized in several directions.

All I was trying to say is that if $X$ is a rich theory that took a long while to develop and investigate, then any generalizations of $X$ will tend to be much richer still, and take correspondingly much more time to investigate.

In particular when applications are scarce: certainly the great popularity of $\mathbb{R}^n$ is not unrelated to the physical fact that real vector spaces are a very good model for the very geometry we perceive in the world.

This vector space model is certainly just a first approximation of real geometry, of course. A while ago it was noticed that we may have to allow for Riemannian spaces, too, if we want to model reality. And indeed, differential geometry is a huge subject nowadays.

Now, people are expecting that all kinds of weird non-commutative or non-whatnot geometry will become relevant if and when our microscopes become much better than they currently are.

I am sure that once this is actually confirmed, and many people will need to work with “non-abelian” geometry in their everyday life, much more activity still will take place in this subject.

So what I want to say is simply this: you are right that eventually it makes good sense to look into “non-abelian” generalizations of everything in sight. Just like it would make sense to generalize everything ever done to the world of $\infty$-categories. But it can’t happen overnight.

Maybe you would enjoy what somebody I recently met had to say about the repeated linearity fallacy of mankind.

P.S. On an administrative note: I fixed the hyperlink to the document you gave. You had given it without the http:// prefix. That makes the blog software interpret the link as relative to the domain golem.ph.utexas.edu, instead of as intended.

Posted by: urs on May 30, 2007 11:44 AM | Permalink | Reply to this

### Re: A linearity fallacy?

In particular when applications are scarce: certainly the great popularity of $\mathbb{R}^n$ is not unrelated to the physical fact that real vector spaces are a very good model for the very geometry we perceive in the world.

As I told already physical reasons are not so interesting for me. Moreover, thanks to several contemporary notions, I hope that the peculiar role of abelian groups can be explained mathematically. 1. Coproducts - decategorification can give us abelian semigroups. 2. Combinatorial species - decategorification of analytic functors gives us analytic functions and derivatives. 3. There are attempts to categorify rational numbers - but I don’t know how high-quality does it have.

Maybe I forget something.

Posted by: osman on May 30, 2007 12:47 PM | Permalink | Reply to this

### Re: A linearity fallacy?

This will probably be my last reply to this issue.

As I told already physical reasons are not so interesting for me.

But they may be for other people. And not just physicists. I believe there is strong evidence that the development of subjects in pure mathematics over the centuries has been greatly influenced by relations to the physical world. Be it because of applications, or be it because of the inner workings of human minds.

Why you (I mean cafe) try to categorify everything using additive categories?

and I gave what I think is part of the answer (in as far as it is really true that “we” try to categorify everything using additive categories).

Posted by: urs on May 30, 2007 1:26 PM | Permalink | Reply to this

### Re: A linearity fallacy?

(in as far as it is really true that we try to categorify everything using additive categories)

Surely it was an exaggeration. Anyway thanks a lot for your answers.

Posted by: osman on May 30, 2007 1:35 PM | Permalink | Reply to this

### Re: A linearity fallacy?

And I don’t think that notions of linearity and abelianity are so stronly interrelated. I’d say that general linear operator is morphism in some category of topological universal algebras with idempotents (for example topological groups).

Posted by: osman on May 30, 2007 1:10 PM | Permalink | Reply to this

### Re: A linearity fallacy?

Maybe you would enjoy what somebody I recently met had to say about the repeated linearity fallacy of mankind.

This text is too difficult for me now. And I don’t know what is “linear category”. Since I learn mathematics using Web and nothing else, could somebody give me good online article about linear categories?

Posted by: osman on May 30, 2007 4:03 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Exactly, all comes from differentiation. There is a huge world living from this (in particular all differential geometry and big parts of functional analysis).

Meanwhile by using other way of differentiating (see explanation below) some mathematicians obtained impressive results (like Pierre Pansu differential for Carnot groups in order to prove Mostow rigidity).

This post is far too short in order to explain this beautiful subject. The ideea is that in the definition of the differential enter a passage to the limit
in some expressions involving dilatations (aka contractions, or homotheties, pick your word). For each point in the space
and each positive coefficient you have such a dilatation. Just change this field
of dilatations and you get a new notion of
differentiation.

What about commutativity? The space itself (it has to be a metric space for first) rescales with respect to the one parameter group of dilatations centerd around one fixed point (precisely you look through the microscope given by dilatations to smaller and smaller balls) and in the limit it becomes a kind of tangent space (in a metric sense) at the point.

Is this tangent space a vector space? In general not! In some cases is a (non commutative) contraction group.

There are results in group theory showing that in some cases such contraction groups have to be Carnot groups (like the Heisenberg group), in other cases they can be p-adic nilpotent groups, products of these, etc .

So this apparent innocuous change of dilatations really leads to some spectacular changes.

Posted by: marius on May 30, 2007 12:06 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

See http://xxx.arxiv.org/abs/0705.1440 for a definition of linearity in this context.

Nice article, thank you! I have the idea of obtaining “vector spaces” from abstract path-spaces through construction of “non-numeric length space”. I think, it’s pretty close to your ideas, but maybe it’s my mistake since I’m not professional. Briefly, I suggest the following steps:

1. Since abstract path-space (APS) is small free category over some graph, we can obtain a notion of homomorphism of APS - this is simply a functor.
2. Then we should introduce some equivalence relation between paths, as a copy of intuitive relation “these paths have the same length”. Of course we should not use the notion of length itself, quite the contrary, equivalence classes of our relation form the “semigroup of lengths” through concatenation of paths.
3. Now we have the “measured abstract path-space” or better “abstract path-space with lengths”. And we have the new groups of automorphisms of our space, as automorphisms introduced in step 1 and simultaneously automorphisms of our equivalence relation introduced in the step 2.
4. Since our equivalence classes will be totally ordered with the minimal element (as length of one-point-path), we have the “length space” (wich is particular case of metric space), and we have the notion of automorphism of this space, and then (maybe) we can introduce the notion of movement group, dilatations and finally “vector spaces”. Everything “whithout numbers”.

But I have not enough mathematical education to finish this properly. Anyway, your article gives me some hope. So my question is following: is it possible to propagate your theory to lengths-spaces with lengths in arbitrary archimedean group?

Posted by: osman on May 31, 2007 9:56 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Thanks. I am not quite sure why, but your description of APS recalled me a puzzling characterization of a line segment and of a curve in terms of IFS (iterated function systems), from the basic paper by John Hutchinson “Fractals and self-similarity” (see Remark 3.4, page 14, “in response to a query of B. Mandelbrot concerning characterisations of the line”, and Section 3.5, page 15, “Parametrised curves”).

Is there a categorification of these remark and section?

Posted by: marius on May 31, 2007 12:43 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Urs wrote:

It’s just that geometry modeled on $\mathbb{R}^n$ is already immensely rich.

I think I would be (possibly) satisfied in the case when category of abelian-group-ness-geometries would be characterized in some independent terms. But in what terms? - I still don’t know…

Posted by: osman on June 4, 2007 2:38 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Osman said, “why everything is abelian?”

Some people have argued that any upper bounds on the amount of non-abelianness around us should be pretty weak. But let me try to actually give some upper bounds.

The basic idea is that objects which are non-abelian often fail to admit higher operations. For example, abelian groups can admit ring structures, but a bilinear map on a general group factors through its abelianization.

Bergman and Hausknecht’s book “Cogroups and co-rings in categories of associative rings” is a pretty extensive reference for things like this. Part of it is based on the work of Tall–Wraith Let me try to say a few words about what they do, as far as I understand it.

In what sense is a (possibly non-commutative) ring naturally the kind of object that knows how to act on an abelian group? The answer is that is represents a comonad on abelian groups. More precisely, we say an endofunctor on the category of abelian groups is representable if its composition with the forgetful functor to sets is representable. If $R$ is an abelian group, then the functor $Hom(R,-)$ naturally takes values in the category of abelian groups, and a ring structure on $R$ is the same thing as a comonad structure on the functor.

The interesting thing is that Bergman and Hausknecht show that it’s hard to do similar things in non-abelian situations. For example, there is a theorem of Kan that says that the category of representable endofunctors on the category of groups is equivalent to the category of pointed sets. It follows that the category of representable comonads on the category of groups is the same thing as a monoid in the category of pointed sets, which is the same thing as a monoid. In other words, the only thing that naturally knows how to act on groups is just a monoid. Whereas in the category of abelian groups, we went up to the next level and got rings.

A similar thing happens at the next level. What are the representable comonads on the category of rings? What about the category of commutative rings? Here things are a little more interesting, because there are objects that know how to act on rings, but not by ring endomorphisms! For instance, a Lie algebra knows how to act on a commutative ring. Such a thing is just a homomorphism from the given Lie algebra to the Lie algebra of derivations of the ring. In fact, more generally, any cocommutative bialgebra knows how to act on a commutative ring. In the case of Lie algebras, you get the commutative ring representing the functor by taking the symmetric algebra of the enveloping algebra. In the case of a group, you take the symmetric algebra of its group algebra.

The great thing is that in the case of commutative rings, there are even more examples, ones that don’t come from bialgebras. For example, in positive characteristic, the Frobenius map knows how to act on a commutative ring. There is a representable comonad $T$ on the category of commutative $Z/pZ$-algebras such that an action of $T$ on $R$ is nothing more than saying that the Frobenius map on $R$ is bijective. We could make another comonad that requires that the Frobenius map be the identity. Even more interesting, over $Z$, we could ask for lifts of the Frobenius map from characteristic $p$ to characteristic $0$.

But all these wonderful things can’t happen with noncommutative rings. Of course there is no non-commutative Frobenius map, at least in the straightforward sense. In fact, Bergman and Hausknecht prove that, at least when $F$ is $Q$ or $Z/pZ$, the only representable comonads on the category of $F$-algebras are ones coming from bialgebras combined with ones coming from anti-endomorphisms, ie maps from an algebra to its opposite. (Admittedly, they don’t address the situation over $Z$, which is much more important.)

Now that I’ve said all that, they also give some evidence that goes against this point of view. For example, even though the only representable comonads on the category of groups come from boring monoids, there are actually representable comonads on the category of monoids that don’t come from monoids. For example, it’s possible to endow a monoid $M$ with a “right-inverse” operator. This is an anti-endomorphism $r: M\to M$ satisfying the identity $m\cdot r(m)=1$ and such that $r$ composed with itself is the identity. You can then use this structure to make other representable comonads. In fact, they give a complete description of all the representable comonads, and they have a similar flavor involving left and right inverse operators. It would be pretty interesting to study the next level of this hierarchy.

Posted by: James on June 4, 2007 12:27 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

This is a difficult stuff for me, but I think I catched the main idea on.

Anyway it sounds for me as andecdotic story about boy who lost his coins at night time, and looked for them under street lamp, because there was the light. (Oh, my English!!!) On the other hand I’m sure that abelian-group-ness is not an anecdot.

And your examples show the great direction of thinking about abelian-group-ness, thank you.

Posted by: osman on June 4, 2007 2:20 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Point taken about the keys under the streetlight, but three things:

1. The set up of what I wrote works very generally, probably in any category equipped with a monadic functor to Sets. I put the part about rings acting on abelian groups in the beginning just to start off with an example. I don’t think it makes representable comonads any less non-abelian.

2. You can make the streetlight comment about almost anything. After all everything I know about mathematics I learned in an abelian-centric mathematical culture. So you have to give me something to grab onto, otherwise what’s the point of the question!

3. My comments were rather vague, I know. I was putting off making them for a while, but then I realized that if I waited till I could write something good, it would never get done. I’m only hoping there are no actual mistakes.

Posted by: James on June 4, 2007 2:44 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

James wrote:

You can make the streetlight comment about almost anything.

And moreover, I do not understand myself when I do such comment, unfortunately. Maybe I would be satisfied in the case when category of objects (say, geometries) that we usually study using abelian groups, would be characterized with some independent terms…

Anyway facts given in your comment has it’s internal beauty!

Posted by: osman on June 4, 2007 2:58 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

This is a composite reply to some questions raised by Osman and Urs.

Can you give me an example of such interesting category (linear for example), but where the definition of objects does not depend on abelian property?

How about the category of sets with relations as morphisms? This can be regarded as enriched in the category of (idempotent) commutative monoids: any two relations $R$, $S$ from a set $X$ to a set $Y$ can be “added” by taking the union $R \cup S$.

In some sense, examples like this are within the theme of the Tale of Groupoidification: spans between sets or between groupoids can be added too, and by decategorification they lead to some classically interesting linear operators [like Hecke operators], as John will probably be telling us in future editions of This Week’s Finds.

An abstract class of examples of categories which makes no explicit reference to linearity but where maps can be added is the class of categories with biproducts. A category has biproducts if:

• It has a ‘zero’ object: an object which is both initial and terminal.
• For any two objects $A$, $B$, there is an object $C$ which is simultaneously a product and coproduct of $A$ and $B$. To be on the safe side: if $i_A$, $i_B$ are the coproduct injections into $C$ and $p_A$, $p_B$ are the product projections out of $C$, then $p_A \circ i_A = id_A$ $p_B \circ i_B = id_B$ whereas both $p_A \circ i_B$ and $p_B \circ i_A$ factor through the zero object.

Then it is a fact that categories with biproducts are essentially the same as Comm-enriched categories with products, where Comm is the closed category of commutative monoids. In effect, morphisms between biproducts can be handled pretty much as matrices. (Cf. Robin Houston’s comment under ‘Preprints’ here that products in compact closed categories are biproducts.)

And could you please give me some weblinks to articles related to closed categories with interrelation to linearity (or abelianity)?

The definitive reference for enriched category theory is the late Max Kelly’s book. Off-hand I don’t know online references which specifically develop the linear case, but maybe someone else does or I can find out.

Urs wrote:

Can this be turned around to a sensible statement along the lines that every closed category is a category of “abelian objects” in some meaningful sense?

I was about to answer somewhat unimaginatively, “not that I can think of” [for example, I don’t know how to think of a cartesian closed category in this way]. But then it seemed to me that a somewhat more satisfying answer could be given.

Let $V$ be a symmetric monoidal closed category. If $V$ has arbitrary coproducts, then the underlying-set functor

$U = hom(I, -): V \to Set$

$F = - \otimes I: Set \to V$

which takes a set S to an S-fold coproduct of copies of $I$ ($I$ being the monoidal unit). Furthermore, the adjunction takes place within the 2-category of symmetric monoidal categories, symmetric monoidal functors, and monoidal transformations.

Thus we have a symmetric monoidal monad $M = U F$ acting on the monoidal category Set. Now, if $V$ is equivalent to the Eilenberg-Moore object of $M$ in this 2-category, then it seems to me there is a very reasonable sense in which $V$ consists of ‘abelian objects’.

More precisely, a (symmetric) monoidal monad on Set is a “commutative theory” in the sense that the operations of the theory are also homomorphisms of the theory. This was shown in work by Anders Kock in the early 70’s. (Oh, I see that Mathieu Dupont has already answered you complete with references – thanks Mathieu!)

Posted by: Todd Trimble on May 30, 2007 10:59 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Your post is full of facts very interesting for me. And moreover, the last one, where monads in monoidal categories give us commutative theories, is also very handsome. But I’d like to be more precise now.

Commutativity of monoid can give us such interesing theory as combinatorial species that is pretty close to calculus. I’d say this is categorified calculus in non-negative numbers. But mathematics is occupied not by simply commutative monoids, but by abelian groups, and this is what we call usually “geometry”.

So we should obtain naturality of all constituents together: it must be commutative and it must be group. Saying figuratively, I seems like “calculating square root” of theory of combinatorial species.

Posted by: osman on May 31, 2007 12:04 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Categorifying negative numbers is actually tricky. John Baez talked about negative sets here, and gave a number of good references.

Another thing that people have tried is to use superalgebra; for example, instead of considering linear combinatorial species $F_0$ (i.e., species valued in vector spaces), consider “virtual” species valued in $\mathbb{Z}_2$-graded vector spaces. A virtual species is given by a pair of linear species

$(F_0, F_1)$

which is to be thought of as a formal difference $F_0 - F_1$ (hence, at the decategorified level, this involves passage to a Grothendieck group). In good cases, a virtual species may also come equipped with differential graded structure, and one may consider the derived category of DG-species as another method for dealing with categorified negatives.

But none of these methods is perfect.

Posted by: Todd Trimble on May 31, 2007 7:05 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Todd Trimble wrote:

A virtual species is given by a pair of linear species

Yes, I know that (from the book “Combinatorial species and tree-like structures”) and I would like to have experts opinion regarding this attempt of rational combinatorics (i.e. categorification of rational numbers).

But none of these methods is perfect.

What I exactly feel: all of them are unnatural, artificial, handmade… So I think there must by quite another approuch, similar to the Anders Kock’s result that you and Mathieu Dupont described here.

Posted by: osman on May 31, 2007 7:37 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

But until this new approuch is not developed, I’m going to think in my directions, like abstract path-spaces and mathematical notion of causality. Both of them are attempts of abstract non-numerical analysis of physical thinking.

Posted by: osman on May 31, 2007 8:07 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

James wrote:

But of the millions of other categories in the night sky, why is the category of finite-dimensional vector spaces the one whose objects it is so useful to make finite groups act on?

There are lots of answers to this. But, I began describing a quite radical answer in week252, based on groupoidification.

While I still need to explain the details, and give lots of examples, I described the basic idea right after the imaginary conversation where Burnside was advocating actions of groups on sets, while Frobenius was pushing group representations:

The point is this. Suppose we have two $G$-sets, say $X$ and $Y$. Any $G$-set map from $X$ to $Y$ gives an intertwining operator from $\mathbb{C}[X]$ to $\mathbb{C}[Y]$. But, even after taking linear combinations of these, there are typically plenty of intertwining operators that don’t arise this way. It’s these extra intertwining operators that let us chop representations into smaller pieces — the irreducible representations.

But where do these extra intertwining operators come from? They come from invariant relations between $X$ and $Y$!

And, what are these extra intertwining operators called? In some famous special cases, like in study of modular forms, they’re called “Hecke operators”. In some other famous special cases, like in the study of symmetric groups, they form algebras called “Hecke algebras”.

A lot of people don’t even know that Hecke operators and Hecke algebras are two faces of the same idea: getting intertwining operators from invariant relations. But, we’ll see this is true, once we look at some examples.

I think I’ll save those for future episodes. But if you’ve followed the Tale so far, you can probably stand a few extra hints of where we’re going. Recall from “week250” that invariant relations between $G$-sets are just spans of groupoids equipped with some extra stuff. So, invariant relations between $G$-sets are just a warmup for the more general, and simpler, theory of spans of groupoids. I said back in “week248” that spans of groupoids give linear operators. What I’m trying to say now is that these linear operators are a massive generalization — but also a simplification - of what people call “Hecke operators”.

In other words: when we think we’re studying the category with:

• representations of a group $G$ on vector spaces as objects
• intertwining operators as morphisms

we often might as well be studying the category of:

• actions of $G$ on sets
• $G$-invariant relations between sets

or even better, the 2-category of:

• groupoids over $G$
• spans of groupoids over $G$
• maps between spans of groupoids over $G$

The point is that this 2-category gives a ‘purely combinatorial’ substitute for the theory of representations of $G$ on vector spaces!

To make this clear, I’ll need to work through a bunch of examples, starting with the permutation groups $S_n$.

Posted by: John Baez on June 1, 2007 5:09 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

In other words, when we think we’re studying [$\mathrm{Rep}(G)$] we often might as well be studying […] the 2-category [$\mathrm{Span}_G(\mathrm{Grpd})$].

The point is that this 2-category gives a ‘purely combinatorial’ substitute for the theory of representations of $G$ on vector spaces!

I am beginning to appreciate this amazingly cool fact. You are really saying: all these vector spaces we see around us – like in quantum mechanics – are an illusion. We don’t have to categorify things to get to spans of groupoids, etc, but these vector spaces themselves already should be thought as representing objects in a 2-category.

That sounds like it should lead to the answer of the puzzle I was fighting with a while ago:

namely on closer inspection, like when one tries to formulate everything arrow-theoretically, it appears as if already in ordinary quantum mechanics everything is in an annoying way “shifted in degree”: the codomains of our functors are 1-categories, where if they were instead 2-categories, everything would fall into place much more naturally.

I am suspecting that all this isn’t unrelated to that $\mathrm{INN}$-phenomenon. It must be.

Posted by: urs on June 1, 2007 6:02 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Should we then expect that distinctions applying to ordinary representations will find their equivalents in your groupoid setting? E.g., positive energy representations.

Posted by: David Corfield on June 2, 2007 11:10 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Expressing the concept of “positive energy representation” in the language of groupoids sounds tricky, since it’s rather closely wedded to the complex numbers. For example, there’s no obvious notion of a positive-energy real representation of a Lie group.

To see this, consider the primordial example: the real line! A strongly continuous unitary (complex) representation $U$ of $\mathbb{R}$ has a skew-adjoint generator

$A = {d \over d t} U(t) |_{t=0}$

which then has

$U(t) = exp(t A) .$

The spectrum of $A$ lies on the imaginary line, so it makes no sense to say it’s positive.

Using the magic of $i$, we can then define a self-adjoint generator

$H = i A$

which has

$U(t) = exp(-i t H)$

Since the spectrum of $H$ lies on the real axis, it makes sense to declare $H$ positive when its spectrum lies in $[0,+\infty)$. We then say $U$ is a positive-energy representation.

Note how the ghost of Galois hovers over the proceedings, snickering quietly. If we’d picked $-i$ instead of $i$ as our square root of $-1$, what used to be a ‘positive-energy representation’ would now be declared ‘negative-energy’, and vice versa!

If we were working over the real numbers, we wouldn’t have positive-energy representations.

See how everything changes:

A strongly continuous orthogonal (real) representation $U$ of $\mathbb{R}$ has a skew-adjoint generator

$A = {d \over d t} U(t) |_{t=0}$

which then has

$U(t) = exp(t A) .$

The spectrum of $A$ lies on the imaginary line, so it makes no sense to say it’s positive. In fact, if any number $a$ lies in the spectrum, so must its complex conjugate $\overline{a}$!

So far, groupoidification seems to work most easily for the most ‘field-independent’ aspects of linear algebra. The concept of positive energy is very special to the complex numbers. So, if we want to groupoidify the theory of positive-energy representations, we have our work cut out for us.

So far, Jim Todd and I are much closer to groupoidifying the whole theory of unitary representations of finite-dimensional compact simple Lie groups — or equivalently, finite-dimensional holomomorphic representations of complex simple Lie groups. The concept of ‘positive energy representation’ doesn’t apply here.

In short: there’s a lot of work to do, and I’ve got to tell you about it before you can see where the frontier is… but ‘positive energy representations’ are a bit beyond the frontier, right now.

By the way, I mentioned this business about skew-adjoint versus self-adjoint generators in my discussion of real, complex and quaternionic quantum mechanics in week251:

Another special way in which $\mathbb{C}$ is better than $\mathbb{H}$ or $\mathbb{R}$ is that only for a complex Hilbert space is there a correspondence between continuous 1-parameter groups of unitary operators and self-adjoint operators. We always get a skew-adjoint operator, but only in the complex case can we convert this into a self-adjoint operator by dividing by $i$.

In the case of $\mathbb{R}$ there aren’t enough square roots of -1. In the case of the quaternions, $\mathbb{H}$, there turn out to be ‘too many’ — and the problem is, they don’t commute. So, if $A$ is linear, $H = i A$ is not!

Posted by: John Baez on June 3, 2007 2:19 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Urs wrote:

You are really saying: all these vector spaces we see around us — like in quantum mechanics — are an illusion.

That’s the dream!

The big question is: precisely how much of this dream can become a reality? In the Tale of Groupoidification, I plan to say a lot about this. A lot of stuff works amazingly well. A lot of stuff doesn’t work yet.

You know the following, but I feel like repeating it to the rest of the world:

Jeffrey Morton showed how to develop a lot of Feynman diagram theory for perturbed harmonic oscillators purely combinatorially, using spans of groupoids (in this case called ‘stuff operators’) as a replacement for operators. But, to get the whole theory to work, we need something to describe the concept of ‘phase’, which is so important in quantum mechanics. Jeff got the job done by replacing sets by $\U(1)$-sets: sets whose elements are labelled by phases. This may not be the ultimate solution. But, it shows how the dream can get a bit tricky to implement, past a certain point.

it appears as if already in ordinary quantum mechanics everything is in an annoying way “shifted in degree”: the codomains of our functors are 1-categories, where if they were instead 2-categories, everything would fall into place much more naturally.

Posted by: John Baez on June 3, 2007 1:12 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

You know the following, but I feel like repeating it to the rest of the world:

Jeffrey Morton showed how to develop a lot of Feynman diagram theory for perturbed harmonic oscillators purely combinatorially,

At the beginning I had the impression that Jeffrey’s work was intended as a categorification of quantum mechanics. But around that comment of mine which I mentioned, I began wondering if maybe the natural home of all these nice constructions and observations he gives is less so a categorified quantum mechanics – something we would imagine applying to stuff more general than quantum point particles – but rather something like an unravelling of the true structure underlying ordinary quantum mechanics. Something which we should really be applying to ordinary quantum point particles if we were doing it right, instead of that linear algebra which we instead do use.

What I have grasped of the “tale of groupoidification” in fact seems to strengthen this feeling.

The idea was this: quantization of a point particle wants to be a pushforward operation: the parallel transport functor which describes its classical coupling to an electromagnetic field, say, is “pushed forward to a point”.

This woks nice on objects (and in fact on $(k \lt n)$-morphisms when we do $n$-particles). But for it to work on ($n$-)morphisms, we have to intervene by hand: choose a measure and do this and that to get the path integral we want.

That seems to indicate that something is wrong. Especially when we work with arbitrary $n$-particles: here push-forward quantization is nice and canonical for a fraction of $n/(n+1)$ of the system, but becomes awkward and “man made” for the last (and most important) $\frac{1}{n+1}$st part of the journey.

And part of the reason is clearly: at top level we are pushing forward 0-functors (functions). And there is no canonical way to do that. So if these 0-functors secretly were 1-functors, with everything else accordinly shifted by one degree, then maybe we’d have a better chance to grasp all of quantization as just one nice natural and canonical push-forward operation.

So what I tried to play around with beginning at that comment I mentioned is to see what can be improved if we assume (quite following Jeffrey Morton, but thinking about ordinary QM all along) that all the numbers we encounter are actually (to be thought of as) sets or the like.

At that point I was thinking of replacing $\mathbb{C}$-modules (vector spaces) with $\mathbb{C}\mathrm{Set}$-module categories.

It seems to me that this is pretty close to groupoidification, and maybe what I should really have considered is the latter. I need to think more about this and try to better understand this.

But in any case, it seemed to me that if we assumed that our “wave functions” are actually not 0-functors but 1-functors, which assigns sets (certainly with some extra structure, but maybe consider just plain finite sets for the moment) to points, then suddenly it does make sense to consider the path integral as an honest push-forward to a point at all levels.

I tried to indicate how this does seem to work in The Canonical 1-Particle.

The crucial information there is the following:

if we do the push-forward quantization for a 1-dimensional target space which is modelled by a graph which has from every vertex precisely two edges emanating to the two neighbour vertices, the as we compute the quantum propagator for a single time step, we have to compute the colimit of our set-valued wave function over categories of the shape

$(x-1) \leftarrow (x) \rightarrow (x+1)$

This is actually one of the simple toy examples discussed in Tom Leinster’s paper. Its Leinster measure is $1 \leftarrow -1 \rightarrow 1 \,.$ If you think about it, you see the exponentiated lattice Laplace operator here, to first order.

First one might think that this is just a coincidence. But I don’t think so. In that entry I went one step further to the path integral over two time steps by push-forward. And it does come from a colimit over a category whose Leinster-measure gives the second power of the exponentiated lattice Laplace operator, i.e. indeed the right path integral result in that order.

So for that reason I was hoping/thinking that we will find that if only we look close enough we find that all these numbers in ordinary quantum mechanics are relly sets of some sort, and all these vector spaces really modules for these kinds of sets.

Now, what you keep saying in the tale of groupoidification seems to be quite consistent with this hope. Roughly at least. Maybe I should replace modules for sets by groupoids, instead. Or something like this.

Posted by: urs on June 3, 2007 9:14 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Urs wrote:

At the beginning I had the impression that Jeffrey’s work was intended as a categorification of quantum mechanics. But around that comment of mine which I mentioned, I began wondering if maybe the natural home of all these nice constructions and observations he gives is less so a categorified quantum mechanics — something we would imagine applying to stuff more general than quantum point particles — but rather something like an unravelling of the true structure underlying ordinary quantum mechanics.

Right, that’s what his work is supposed to be. And yet it is a categorification, technically.

So, this raises an interesting general point. We’re always running around trying to categorify everything, but there are at least two things categorification can do. One is to take a mathematical structure and boost its dimension by one: for example, going from the math of particle physics to the math of string physics. Another is to take a structure that was sort of misunderstood and ‘squashed’, and understand it better by unsquashing it: for example, taking combinatorics phrased in terms of natural numbers, and phrasing it in terms of finite sets. These seem like quite different things to do — at least they seem that way to me right now! — but categorification can be good for both.

Joyal’s work on ‘species’ is a classic case of the latter sort of categorification. Joyal took the generating functions combinatorists like to use for counting things, and realized these are secretly just squashed versions of species, namely functors

$F: FinSet_0 \to Set.$

This realization then let us unsquash some of the math behind the quantum harmonic oscillator and Feynman diagrams.

But, it turns out that everything works better — the combinatorics, the harmonic oscillator and the Feynman diagrams! — if we go half a step further and consider stuff types, namely weak 2-functors

$F: FinSet_0 \to Gpd$

Note that I say ‘half a step’ further, which is related to some of your later points. Why just ‘half a step’? Isn’t going from $Set$ to $Gpd$ a full step upwards on the $n$-categorical ladder?

Well, in a way not! Species are secretly the same thing as faithful functors

$\Psi = \hat{F} : X \to FinSet_0$

where $X$ is a groupoid, while stuff types are the same as arbitrary functors

$\Psi = \hat{F} : X \to FinSet_0$

where $X$ is a groupoid.

Maybe this is a full step up the $n$-categorical ladder in some sense, but it usually feels more like ‘doing things right’. It’s sort of like the difference between standing on a ladder with one foot higher than the other, and standing on it with both feet at the same level. That’s why it feels like going up a ‘half step’.

Now I think this really is related to your idea:

The idea was this: quantization of a point particle wants to be a pushforward operation: the parallel transport functor which describes its classical coupling to an electromagnetic field, say, is “pushed forward to a point”.

This woks nice on objects (and in fact on $(k \lt n)$-morphisms when we do $n$-particles). But for it to work on ($n$-)morphisms, we have to intervene by hand: choose a measure and do this and that to get the path integral we want.

That seems to indicate that something is wrong. Especially when we work with arbitrary $n$-particles: here push-forward quantization is nice and canonical for a fraction of $n/(n+1)$ of the system, but becomes awkward and “man made” for the last (and most important) $\frac{1}{n+1}$st part of the journey.

Thanks for re-explaining this. I see much better what you mean, now.

In Jeff’s work, states of the quantum harmonic oscillator are replaced by stuff types

$\Psi : X \to FinSet_0$

and operators are replaced by stuff operators, namely spans of groupoids:

                 T
/ \
/   \
/     \
/       \
/         \
/           \
v             v
FinSet_0        FinSet_0


But, all this generalizes to situations where we replace $FinSet_0$ by other groupoids… so we’re really trying to do quantum mechanics in a new way, using spans of groupoids as a replacement for operators.

(In reality, so far we can do only a small fragment of quantum mechanics this way. But, we’re allowed to dream…)

In particular, as you note, the push-pull (or pull-push) operations involved in path integrals become nicer in this new framework, since we don’t need to specify a measure. Given a ‘state’ thought of as a functor between groupoids:

$\Psi : X \to A$

and given an ‘operator’ thought of as a span of groupoids:

                 T
/ \
/   \
/     \
/       \
/         \
/           \
v             v
A             B


we can pull back $\Psi$ to $T$ and then push it forwards to $B$, getting a state we could call

$T \Psi: T X \to B$

(In this formalism, the tricky part is the pulling back, not the pushing forwards! That’s because I’m thinking of states as functors $\Psi = \hat{F}: X \to A$ instead of weak 2-functors $F: A \to Gpd$. In the latter approach, which is equivalent, the tricky part is the pushing forwards!)

So, yeah — I think I roughly understand how your remarks connect to the Tale of Groupoidification.

Let me try to summarize in one sentence. When you see a measure space, maybe you should try to understand it better by seeing it as a squashed version of a groupoid, or category.

Posted by: John Baez on June 4, 2007 2:52 PM | Permalink | Reply to this

### measures from categories

Let me try to summarize in one sentence. When you see a measure space, maybe you should try to understand it better by seeing it as a squashed version of a groupoid, or category.

Good, thanks!

What I find fascinating about this is that this means that the measures we see around us might be encoded in the structure of a category (or a groupoid). But for measures appearing in quantum mechanics, this category must be essentially a category version of configuration space!

But, if true, this would be quite an insight: we could try to read off the category structure of configuration space from the measures we already know we have to use for them.

As a first evidence, for the simple case of Dijkgraaf-Witten theory, this is exactly what happens!

As a second evidence, it seems that the measure induced by the category coming from taking configuration space to be the graph $\array{ \cdots (x-1) \stackrel{\leftarrow}{\rightarrow} (x) \stackrel{\leftarrow}{\rightarrow} (x+1) \cdots }$ does yield the euclidean finite-time propagator for a particle on the discretized line.

If this are not just coincidences, something really relevant is going on here.

Posted by: urs on June 4, 2007 3:35 PM | Permalink | Reply to this

### Re: measures from categories

Very neat. I’m watching with enthusiasm. And, needless to say, I am always happy (and not surprised) to see n-diamonds make an appearance in a natural way.

When I was thinking about this stuff, it was always a bit mystical to me how beautiful topological “stuff” works out combinatorially. Trying to introduce a metric, Hodge star, or whatnot, always seemed like a bit of voodoo added on top of something that was otherwise beautiful. If I’m understanding you correctly (which would be a miracle), you are suggesting that metric information can also be encoded beautifully if you can find a way to “unsquash” the measure appropriately via this magical categorification stuff you guys are always talking about :)

I’d be very interested in this from a practical perspective because of the relation between “metric” and “material”. If you think about simulating (as I did for 8 years) electromagnetic waves in complex media, then you can convince yourself that dielectric and magnetic material properties can be encoded in the Hodge star, or metric, or measure. It would be fascinating, to me at least, to find a way to think about electromagnetic wave propagation through complex media as a categorification of some configuration space (or whatever). Not only is spacetime an n-category, but the material that makes up my body is as well ;)

Eric

Posted by: Eric on June 4, 2007 4:41 PM | Permalink | Reply to this

### Re: measures from categories

metric information can also be encoded beautifully if you can find a way to “unsquash” the measure appropriately via this magical categorification stuff you guys are always talking about :)

You recall my first attempt to get the category structure of spacetime by demanding its Leinster measure to reproduce the standard measure.

Here the idea was that all edges of the “spacetime category” would be interpreted as light-like. So given that and the measure, we’d also have a metric.

I haven’t yet figured out how to derive the measure that actually appears in the path integral for the relativistic particle in this setup. (Which doesn’t mean that that’s not possible.)

What I later did in the canonical 1-particle was for the non-relativistic case.

Since then, I stopped thinking about this, being busy with other things. I am hoping that with more thinking applied here, also the relativistic case could be understood. And there we have a good chance of also getting a metric on top of our measures. I think.

Posted by: urs on June 4, 2007 4:52 PM | Permalink | Reply to this

### Re: measures from categories

I haven’t yet figured out how to derive the measure that actually appears in the path integral for the relativistic particle in this setup. (Which doesn’t mean that that’s not possible.)

Nearly 4 years later, has there been in change in status regarding this comment? Just curious. I still think this is an absolutely beautiful concept.

Posted by: Eric Forgy on April 10, 2011 5:31 PM | Permalink | Reply to this

### Re: measures from categories

has there been in change in status regarding this comment?

No change of status, unfortunately. I didn’t think about this for a long time. But I do (still) agree: if there is something to the idea of generalizing $\infty$-groupoid cardinality to $(\infty,1)$-categories (or enriched categories, to start with) as Tom and Simon suggest, then the question whether a Lorentzian space, thought of as light cone structure plus volume density, can be usefully understood in these terms, would seem natural and interesting. But also I feel I am still missing something about the deeper meaning of all this. I think internally I have decided to wait five years and then check Tom and Simons’ latest articles for the latest hints.

Posted by: Urs Schreiber on April 11, 2011 10:29 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

John wrote (my paraphrase)

[fibered categories] $\Psi = \hat F : X \to \mathrm{FinSet}_0$ [are the same as pseudofunctors] $\mathrm{FinSet}_0 \to \mathrm{Gpd}$

in order to explain why when working with just 1-categories (here: a groupoid $X$ or the category if finite sets $\mathrm{FinSet}_0$), we might find ourselves using 2-categories (here: that of groupoids $\mathrm{Gpd}$).

That’s great! This is my hunch I tried to express: that funny half-a-step here is precisely the one we also encounter in Schreier theory of groupoid extensions (hence that “$\mathrm{INN}$-phenomenon” I mentioned, which shows up when applying that Schreier theory to Atiyah sequences of groupoids).

It’s in the end the Grothedieck construction, I guess, which relates fibred categories and pseudofunctors.

It looks like all the mathematical ingredients are there which would now allow a Heisenberg-like step, creating a new theory of physics, beneath quantum mechanics…

Posted by: urs on June 4, 2007 5:13 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

I see a new paper came out on the archive today, “A categorical framework for the quantum harmonic oscillator”.

Posted by: Bruce Bartlett on June 7, 2007 9:26 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Osman wrote:

I would like to have experts’ opinions regarding this attempt of rational combinatorics (i.e. categorification of rational numbers).

Hey, thanks for pointing out this paper:

It’s interesting… especially since it refers heavily to stuff I did!

It begins with an intro to categorification.

Then it describes how James Dolan and I categorified the nonnegative rational numbers using groupoids in our paper From finite sets to Feynman diagrams, back around 2000. The idea is to define the cardinality of a finite groupoid, which can be any nonnegative rational number.

(This is precisely the stuff we’re developing further now with the help of Todd Trimble — the stuff I’m starting to talk about in The Tale of Groupoidification!)

Next, the paper develops a theory of groupoid-valued species, namely functors

$F: FinSet_0 \to Gpd$

from the groupoid of finite sets and bijections to the groupoid of groupoids. I’ll have to check, but I’m afraid they’re considering strict functors instead of what they should be using: weak 2-functors. By an old result of Grothendieck, a weak 2-functor

$F: FinSet_0 \to Gpd$

is essentially the same as a functor

$p: X \to FinSet_0$

where $X$ is a groupoid. James Dolan and I call these things stuff types. We proposed these as a generalization of Joyal’s species, for the exact same reason Blandin and Diaz use groupoid-valued species. But, stuff types are slightly more general, and their formal properties are better.

Blanding and Diaz also make the mistake of limiting attention to groupoid-valued species with $F(\emptyset) = \emptyset$. One of the joys of stuff types is that this is no longer necessary!

It’s quite possible they didn’t realize that stuff types were almost the same as groupoid-valued species, only better. This is explained in Jeff Morton’s paper, and my course notes.

But never mind. The really new and interesting thing in Blandin and Diaz’s paper — at first glance — is their attempt to categorify the whole ring of rational numbers.

They categorify $\mathbb{Q}$ using pairs of groupoids — or if you prefer, $\mathbb{Z}_2$-graded groupoids. This works in the obvious way: the cardinality of a pair $(G_0,G_1)$ is the cardinality of $G_0$ minus the cardinality of $G_1$, so it can be any rational number.

So, they define a rational species to be a functor

$F: FinSet_0 \to Gpd^2$

satisfying a couple of conditions… where again, they should really leave out these conditions and let $F$ be a weak 2-functor.

This idea is clearly related to the older concept of ‘virtual species’, mentioned earlier in this discussion by Todd Trimble.

A rational species has a ‘generating function’, which can be any power series

$\sum_n a_n z^n$

with rational coefficients $a_n$. And, the really fun part is that they get rational species whose generating functions have $a_n$ being the Bernoulli numbers!!!

I’d been trying to do that for a while.

However, nowadays I’m more inclined to think of a categorified rational number as a category — not a pair of groupoids. After all, Tom Leinster’s generalization of groupoid cardinality to categories can be negative!

Getting negative cardinalities by taking a pair of things and subtracting their cardinalities is more artificial than what Tom does: namely, discover the right concept of cardinality of a category, and notice that it can be negative!

Posted by: John Baez on June 3, 2007 3:36 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

John baez wrote:

It’s interesting… especially since it refers heavily to stuff I did!

As I told, “John Baez” is magic sequence of letters when googling, at least for me.

So, they define a rational species to be a functor … satisfying a couple of conditions

On the page 11 they categorify ring of integers through ${FinSet}^2$. I do not understand how they obtain their form of cartesian product, and I can suggest another categorification of this ring using ${FinSet}^2$.

This goes from traditional algebra: instead of ${FinSet}^2$ we should look at endofunctors over this category. At least some of them should preserve coproducts and form a ring with multiplying given by composition of functors. Then $-1$ has obvious meaning: this is just interchanging sets in pairs.

Sorry for my dilettantic ideas.

namely, discover the right concept of cardinality of a category, and notice that it can be negative!

Sounds very interesting, thank you.

Posted by: osman on June 3, 2007 9:58 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

On the page 11 they categorify ring of integers through FinSet 2. I do not understand how they obtain their form of cartesian product

I think most category theorists would shy away from calling it a Cartesian product, and prefer to call it a tensor product instead. In any case, it’s obtained via a beautiful piece of theory due to Brian Day (I think originally in his PhD thesis).

What Day showed is that a tensor product structure on a category $\mathbb{C}$ induces a corresponding tensor product on the functor category $[\mathbb{C}, \mathrm{Set}]$.

The general definition is very easy to understand if (and only if) you know and love coends. But the special case here is easy to understand even if you don’t. If we have a monoid $M$, we can think of $M$ as being a discrete category with a tensor product on it. This then induces a tensor product on $\mathrm{Set}^{|M|}$. Here’s how it works:

An object of $\mathrm{Set}^{|M|}$ is an indexed collection $\langle X_i | i \in M\rangle$, and the tensor is $\langle X_i\rangle \otimes\langle Y_i\rangle = \langle \sum_{a\cdot b = i}X_a\times Y_b\rangle;$ the unit object $\langle I_i\rangle$ has $I_1 = 1$ and $I_j = \emptyset$ for $j\neq 1$.

If you want to introduce negativity, the natural monoid to consider is the set $\{+1, -1\}$ under multiplication, which is isomorphic to $(\mathbb{Z}_2, +)$. Let’s calculate the induced tensor on $\mathrm{Set}^2$: an object is a pair $(X_+, X_-)$, the unit is $(1, 0)$, and $(X_+, X_-)\otimes(Y_+, Y_-) = (X_+\times Y_+ + X_-\times Y_-, X_+\times Y_- + X_-\times Y_+),$ since $+1 = +1\times +1 = -1\times-1$, and $-1 = +1\times-1 = -1\times+1$.

So $\mathrm{Set}^2$ admits at least four different tensor products: there is the categorical product and categorical coproduct (taken elementwise), and the convolution tensors induced by the two different 2-element monoids $(\mathbb{Z}_2, +)$ and $(\mathbb{Z}_2, \times)$. (Easy exercise: give an explicit form for the tensor induced by the latter.)

Posted by: Robin on June 3, 2007 4:13 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Robin wrote:

most category theorists … prefer to call it a tensor product instead

But they call it cartesian product: I am not mathematician, so this deceives me a little. After your explanation everything is clear, thank you. Anyway I think my idea is more natural.

Posted by: osman on June 3, 2007 6:53 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

You propose to consider the monoidal category of product-preserving endofunctors on $\mathrm{FinSet}^2$.

Let us write an element of $\mathrm{FinSet}^2$ as a column vector $\left( \array{X \\ Y} \right)$, for reasons which will become clear below. If $F: \mathrm{FinSet}^2 \to \mathrm{FinSet}^2$ preserves coproducts then, up to isomorphism, it is determined by $F\left( \array{1 \\ 0} \right)$ and $F\left( \array{0 \\ 1} \right)$. Each of these is a pair of finite sets, so $F$ is essentially determined by four finite sets. Let’s write it as

(1)$\left( \array{A & B \\ C & D} \right),$

where $F\left( \array{1 \\ 0} \right) = \left( \array{A \\ C} \right)$ and $F\left( \array{0 \\ 1} \right)=\left( \array{B \\ D} \right)$. Since $F$ preserves coproducts,

(2)$F\left( \array{X \\ Y} \right) \cong \left( \array{A\times X + B\times Y \\ C\times X + D\times Y} \right).$

Now, suppose we have two such endofunctors

(3)$F_1=\left( \array{A_1 & B_1 \\ C_1 & D_1} \right)$

and

(4)$F_2=\left( \array{A_2 & B_2 \\ C_2 & D_2} \right).$

It is easy to check that $F_1F_2$ is represented by the product of the two matrices. So the objects of your category are essentially $2\times 2$ matrices of finite sets, with the tensor product being matrix multiplication. You don’t say how you propose to regard these as integers, but the obvious way to decategorify is to take the determinant. This is consistent with your suggestion that the switch map

(5)$\left( \array{0 & 1 \\ 1 & 0} \right)$

should represent $-1$.

This category generalises $\mathbb{Z}_2-\mathrm{Set}$, in the sense that $\mathbb{Z}_2-\mathrm{Set}$ is monoidally equivalent to the full subcategory on symmetric matrices. Notice, though, that this inclusion does not preserve the interpretation as integers.

This is all thoroughly reasonable, and an interesting idea. I wonder why you consider it “more natural” though?

Posted by: Robin on June 3, 2007 10:39 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Robin wrote:

I wonder why you consider it “more natural” though?

I’m afraid I cannot give you quite mathematical reason now. Both my reasons live on the epistemological bound of math.

First, I learned from the few starting pages of the book of Nathan Jacobson “The theory of rings” that we obtain rings in mathematical practice as rings of endomorphisms (of abelian groups). Trying to reinterpret this fact for $FinSet^2$ I realized this idea: we should obtain rings as decategorification of rings of endofunctors.

Second reason is fully epistemological. When people start counting things in the childhood and learn multiplication, they do not multiply sets! They multiply operations and sets. For example: take this 2 apples 3 times. Obviously “2 aplles” is set, but “taking it 3 times” is endofunctor over $FinSet$!

On the second step of abstraction we start multiplying endofunctors, but since “functorial” part of these operation is automated by brain, we forget this. And finally we learn the table of multiplication and totally forget functorial nature of multiplication.

Posted by: osman on June 4, 2007 6:19 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Robin, in your convenient terms I considered endofunctors of the form $T \left( \array {a & 0 \\ 0 & a} \right)$ where $T$ is one of $\left( \array {1 & 0 \\ 0 & 1} \right)$ and $\left( \array {0 & 1 \\ 1 & 0} \right)$ and cardinality of any $F$ is given by cardinality of $FinSet^2$-object $F \left( \array {1 \\ 0} \right)$.

Posted by: osman on June 4, 2007 1:25 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

So your assumption about symmetric matrices is quite correct, but cardinality of $F$ is given by cardinality of $F \left ( \array{1 \\ 0} \right )$ calculated as described in the discussed article. If this is fully equivalent to $\mathbb{Z}_2-set$ then it’s fine. :)

Posted by: osman on June 4, 2007 1:47 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

When reading “cartesian product” I always understand it as “categorical product”. This was a reason of my misunderstanding. Sorry.

Posted by: osman on June 3, 2007 9:55 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

A new paper by Rafael Diaz, this time with Edmundo Castillo: Rota-Baxter Categories.

Our aim in this work is to study the categorification of Rota-Baxter rings, an algebraic structure under current active research because of its capacity to unify notions coming from probability theory, combinatorics, symmetric functions, and the renormalization of Feynman integrals among others. Applications of Rota-Baxter categories in the context of renormalization of Feynman integrals will be study in forthcoming works.

Posted by: David Corfield on June 11, 2007 12:22 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

The Tale of Groupoidification is great. I can’t wait to hear more about Hecke operators and spans of groupoids. Any chance that groupoids whose hom-sets are $U(1)$-torsors (i.e. “gerbes”) will be thrown into the mix at some point?

Posted by: Bruce Bartlett on June 3, 2007 11:23 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

Bruce wrote:

I can’t wait to hear more about Hecke operators and spans of groupoids.

They’re coming up soon!

Any chance that groupoids whose hom-sets are U(1)-torsors (i.e. “gerbes”) will be thrown into the mix at some point?

They’re not on the menu, but you never know. Here’s what’s on the menu:

• Coxeter groups, buildings, and the quantization of logic
• Hecke algebras, Hecke operators and Radon transforms
• categorified quantum groups and Khovanov homology
• Kleinian singularities and the McKay correspondence
• quiver representations and Hall algebras
• intersection cohomology, perverse sheaves and $D$-modules
• Verma modules and the Kazhdan–Lusztig conjecture
• $q$-deformation and the Weil conjectures
• how all this stuff is related

This is gonna keep me busy for quite a while!

Posted by: John Baez on June 4, 2007 2:31 AM | Permalink | Reply to this
Read the post The n-Café Quantum Conjecture
Weblog: The n-Category Café
Excerpt: Why it seems that quantum mechanics ought to be the de-refinement of a refined theory which lives in one categorical degree higher than usual.
Tracked: June 8, 2007 11:42 AM
Read the post The n-Café Quantum Conjecture
Weblog: The n-Category Café
Excerpt: Why it seems that quantum mechanics ought to be the de-refinement of a refined theory which lives in one categorical degree higher than usual.
Tracked: June 8, 2007 11:42 AM

### Re: This Week’s Finds in Mathematical Physics (Week 252)

I am wondering about the following:

I know that for a class of interesting applications, where naively it looks like one needs principal $G$-spaces, one actually needs principal $\mathrm{INN}(G)$ 2-spaces, where $\mathrm{INN}(G) \subset \mathrm{AUT}(G) := \mathrm{Aut}_{\mathrm{Cat}}(\Sigma G)$ is the 2-group of inner automorphisms of $G$.

If one knew only this fact alone and be happily ignorant of everything else, it would be tempting to try to study the representation theory of $G$ by looking at the 2-category

$\mathrm{Hom}_{ 2\mathrm{Cat}_{\mathrm{Gray}}} ( \Sigma \mathrm{INN}(G) , \mathrm{Grpd} ) \,.$

Or maybe $\mathrm{Hom}_{ 2\mathrm{Cat}_{\mathrm{Gray}}} ( \Sigma \mathrm{INN}(G) , \mathrm{Cat} ) \,.$

A representation of $G$ in this sense would be

- a groupoid $\mathrm{Gr}$

- on which the group $G$ acts strictly by functorial automorphisms

- such that all these automorphisms are pairwise naturally isomorphic.

I am wondering how this would relate, if at all, to the 2-category of groupoids and spans over $\Sigma G$.

I have not yet thought this through to any resonable extent. But I thought I might mention it nevertheless.

Posted by: urs on June 18, 2007 11:56 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

As a first consistency check, notice that all ordinary actions of $G$ on sets are contained in 2-reps of $\mathrm{INN}(G)$ on groupoids:

given any set $X$ with a $G$-action on it, let $\mathrm{Gr}$ be the corresponding action groupoid.

On this action groupoid $\mathrm{INN}(G)$ is represented in the obvious way:

each $g \in G$ acts on $\mathrm{Gr}$ as the autofunctor $g : \mathrm{Gr} \to \mathrm{Gr}$ which is given by $g : \array{ x \\ \;\downarrow^h \\ h x } \;\; \mapsto \;\; \array{ g x \\ \;\;\;\downarrow^{g h g^{-1}} \\ g h x }$ for all $x \in X$ and $h \in G$.

There is indeed a unique natural isomorphism between any two such functors.

So, ordinary “representations” of the group $G$ on sets are recovered. The question would be: how much else?

Posted by: urs on June 18, 2007 12:57 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

all ordinary actions of $G$ on sets are contained in 2-reps of $\mathrm{INN}(G)$ on groupoids

Maybe that already exhausts all interesting 2-reps $\Sigma \mathrm{INN}(G) \to \mathrm{Grpd}$. Not sure yet. But let’s concentrate on these ordinary $G$-sets regarded as $\mathrm{INN}(G)$-groupoids, for the moment.

Then in any case we do have more general morphisms now.

For $X$ and $Y$ $G$-spaces (i.e. equipped with an action of the group $G$), a morphism between them is of course a map $f : X \to Y$ such that $f (g \cdot_X x) = g \cdot_Y f(x)$ for all $x \in X$.

This relation may be relaxed as we pass to morphisms of $\mathrm{INN}(G)$-groupoids. For these one finds that there have to exist natural isomorphisms $\phi_g(x) : f (g\cdot_X x) \to g \cdot_Y f(x)$ in the action groupoid corresponding to $Y$.

I wanted to see if this can be related to spans of groupoids over $\Sigma G$, but I don’t sufficiently understand the situation yet.

Posted by: urs on June 18, 2007 3:42 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

How does this compare to a map that respects the G action up to homotopy?

Are the maps φg to be coherent in some sense? e.g φgh = ?

Posted by: jim stasheff on June 19, 2007 1:38 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

There are some conditions on $G$, right, where the 2-group $INN(G)$ has $G$’s worth of objects and precisely one morphism between any two objects? Perhaps the case where the center of $G$ is trivial?

Anyway, when $INN(G)$ has the form I just mentioned, it’s equivalent to the trivial 2-group, so its representation theory, when done in a morally correct way, should be completely dull. Here by “representation theory” I mean the study of its (suitably weak) actions on anything, modulo the correct concept of equivalence.

Posted by: John Baez on June 19, 2007 5:07 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

There are some conditions on $G$

Hm, here are no conditions I think. For any group $G$, there is the crossed module $(\mathrm{Id} : G \to G)$. The corresponing 2-group is, as a groupoid, the codiscrete groupoid over $G$.

it’s equivalent to the trivial 2-group

Yes.

so its representation theory, when done in a morally correct way, should be completely dull.

On the one hand, yes, certainly, since $\mathrm{INN}(G)$ is trivializable (equivalent to the trivial 2-group) it follows that, when we are always very moral, we can’t distinguish it from the trivial 2-group.

But past experience seems to indicate that this is not what we should do! Here is the main example:

For $G$ Lie, smooth 2-functors from 2-paths to $\Sigma \mathrm{INN}(G)$ are in bijective correspondence with 1-forms with values in $\mathrm{Lie}(G)$.

Here “bijective correspondence” in a very immoral statement, in that it is not supposed to be thought of up to equivalence. Rather, given any such functor, differentiating it gives that 1-form and given that 1-form, there is a unique such 2-functor such that the 1-form is its differential.

So 2-functors with values in $\mathrm{INN}(G)$ behave just like 1-functors with values in $G$.

(This turns out to be the crucial mechanism which underlies general connections with values in Lie $n$-algebras: for $g_{(n)}$ any Lie $n$-algebra, a connection is locally a morphism into $\mathrm{inn}(g_{(n)})$.)

But the 2-functors have a more flexible notion of equivalence than the 1-functors:

among these are

- the ordinary gauge transformations of the 1-form

- transformations which add another 1-form to the former.

Only the first kind corresponds to the gauge transformations we want to allow if we think of connection 1-forms.

Using the second transformations, one finds in fact that all 2-functors with values in $\mathrm{INN}(G)$ are equivalent to the trivial such functor.

I like to think of it this way:

while every 2-functor with values in $\mathrm{INN}(G)$ is equivalent to the trivial such 2-functor, there is nontrivial information in the choice of that equivalence:

the equivalence is itself a 1-functor, which comes exactly from the 1-form and a choice of gauge transformation of that.

In formulas: with $1 : P_2(X) \to \Sigma(\mathrm{INN}(G))$ the trivial 2-functor that sends everything to the identity and with $\mathrm{curv} : P_2(X) \to \Sigma(\mathrm{INN}(G))$ any 2-functor, what is of genuine interest is the category $\mathrm{Hom}(1,\mathrm{curv}) \,.$ The fact that $\mathrm{INN}(G)$ is equivalent to the trivial 2-group ensures that this Hom-category is always non-empty. So it’s a good thing!

To move all this from the principal into the associated world, I would like to consider representations of $\mathrm{INN}(G)$. As before, I won’t be interested in these representations up to equivalence (because then nothing would be left, as you say) but rather in the choices of trivializing equivalences.

So I should put it this way:

while all 2-reps in $\mathrm{Hom}(\Sigma \mathrm{INN}(G), \mathrm{Grpd})$ are necessarily equivalent, interesting information lies in the choice of such equivalence, i.e. in the categories $\mathrm{Hom}(1,\rho)\,,$ where 1 is the trvial such 2-rep and $\rho$ any such 2-rep.

Posted by: urs on June 19, 2007 6:50 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

What’s the general message in this? That even if two $n$-categories are ($n$)-equivalent, there are situations where it isn’t ‘evil’ to consider them different, situations which involve the highest level morphisms in Homs in or out of them?

Or is it a sign that these two entities weren’t really just $n$-categories? Perhaps there was extra structure on them, so that they’re not equivalent.

If you consider the bicategory of rings $M_n(\mathbb{R})$, bimodules and bimodule homomorphisms, you see it is equivalent to a trivial bicategory. But perhaps you believe you shouldn’t treat Morita equivalent rings as the same, so you look for more structure in the bicategory. This I take it is what Mike Shulman is doing in the paper I mentioned.

Posted by: David Corfield on June 19, 2007 8:58 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 252)

What’s the general message in this?

I think the message is this:

Whenever something is trivializable, there may still be interesting information encoded in how it trivializes.

It is only the canonically trivializable things which are really uninteresting.

There is a list of examples for this after which my discussion above was modeled. (I talked about these in Sections, States, Twists and Holography.)

Usually, when some thing $T$ is trivializable it means that certain morphisms $e : I \to T$ exist, where $I$ is the trivial thing.

There may be interesting information in the $e$!

The standard example which sort of motivates all these considerations here which I am coming from is this:

an ordinar bundle gerbe trivializes in this sense once we regard it in a category of higher rank 2-vector bundles. This means we have trivializing morphisms into it. These are known as “twisted bundles” or “gerbe modules” or whatnot. They live in one categorical dimension lower. And they contain nontrivial information.

What I am talking about with respect to $\mathrm{INN}(G)$ is actually a nonabelian generalization of this phenomenon.

Posted by: urs on June 19, 2007 11:45 AM | Permalink | Reply to this

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