Quantization and Cohomology (Week 27)

June 8, 2007

Quantization and Cohomology (Week 27)

Posted by John Baez

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In our final class on Quantization and Cohomology I gave a summary of the trip we’d been on, and a brief description of where we could have gone next — if only there’d been time:

Week 27 (June 5) - Review and prospectus. Classical and quantum mechanics from categories equipped with action or phase functors. Subtleties: path integrals and anafunctors. Geometric quantization of symplectic manifolds. Categorification, to obtain the classical and quantum mechanics of strings. Categorifying the theory of connections as anafunctors.

Supplementary reading:
- John Baez and Urs Schreiber, Higher gauge theory.
- John Baez, Higher gauge theory, higher categories.
- Urs Schreiber, Talks at "Higher Categories and their Applications".

Last week’s notes are here.

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It was an exhausting quarter, and I didn’t cover nearly as much as I’d hoped to in this class — in part because I never had enough time to prepare.

Still, Derek Wise took very nice notes for the whole year’s course. Apoorva Khare is turning them all into LaTeX, and Christine Dantas has been preparing figures in electronic form. My new student Alex Hoffnung hopes to polish this material into a readable paper or book. Doing this will take a lot of work, since right now the notes are rambling — I never reached various mountain-tops I’d been aiming for. But, since this material is the subject of Alex’s thesis, I hope he’ll be motivated to put in the necessary work!

Posted at June 8, 2007 6:37 PM UTC

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Re: Quantization and Cohomology (Week 27)

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One thing I need to still better understand:

The way you conceive the situation, the inner product always involves the path integral (as described in week 18). I am thinking that this amounts to adopting the point of view of quantum mechanics of constrained systems from the outset, where the path integral acts as the projector onto the physical Hilbert space, i.e. where the physical inner product

$\langle \phi,\psi \rangle_{\mathrm{phys}}$ arises from an inner product $\langle \cdot,\cdot \rangle$ on an auxiliary kinematical Hilbert space from something like a “group averaging procedure” $\langle \phi,\psi \rangle_{\mathrm{phys}} = \int_G \langle \phi, U(g)\psi \rangle \; d\mu_G \,,$ where $G$ is some gauge group and $U(g)$ a representation of it on our kinematical Hilbert space.

For systems like the relativistic particle and the like, the group in question is that of diffeomorphisms of the 1-dimensional parameter space, and we get an expression roughly of the form $\langle \phi,\psi \rangle_{\mathrm{phys}} = \int_0^\infty \langle \phi, e^{-t H}\psi \rangle \; d t \,,$ where $H$ is the Hamiltonian constraint. This, in turn, should be, then, thought of as coming from just the path integral with boundary conditions specified by $\psi$ and $\phi$ .

I have the impression that it is this kind of physical inner product for constrained systems which you have in mind when you consider the inner product as involving the path integral. Is that right?

Recently I understood, from this comment of yours, that this relation between the path integral and the inner product is the way you think of the $\mathrm{Hom}$ as generalizing the inner product.

On the other hand, I had been thinking of the inner product as the pairing $(\psi : \mathbb{C} \to H, \phi: \mathbb{C} \to H ) \mapsto \mathbb{C} \stackrel{\psi}{\to} H \stackrel{\phi^*}{\to} \mathbb{C} := \langle \phi, \psi \rangle \,,$ which I think of as the image of $\mathrm{Id}_{H}$ under $\mathrm{Hom}(\phi^*,\psi) \,.$

I am guessing that in the end the relation between these two points of views comes down to pretty much the relation between ordinary unconstrained quantum theory and that of constrained systems. But I would want to understand this more cleanly.

Posted by: urs on June 10, 2007 6:51 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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Urs wrote:

The way you conceive the situation, the inner product always involves the path integral (as described in week 18). I am thinking that this amounts to adopting the point of view of quantum mechanics of constrained systems from the outset, where the path integral acts as the projector onto the physical Hilbert space […]

I have the impression that it is this kind of physical inner product for constrained systems which you have in mind when you consider the inner product as involving the path integral. Is that right?

That’s exactly right. Maybe I should explain this a bit to the rest of the café regulars.

Without meaning to at first, in this class I somehow wound up exploring an approach to physics where we start with a category $C$ of ‘configurations’ and ‘processes’, together with a rule for assigning each process

$f: x \to y$

an action

$S(f) \in \mathbb{R}$

and thus a phase

$e^{iS(f)} \in \mathrm{U}(1).$

To apply this to, say, particle mechanics, it seems we need to let the objects of $C$ be the ‘extended configuration space’ — that is, spacetime — whose points say not only where a particle is but when it’s there. Then the morphisms will be certain paths in spacetime: trajectories of particles.

This is nice in some ways. It’s more manifestly invariant than the usual business of chopping spacetime into space and time. But, if we form $L^2$ of the space of objects of $C$ , we get not the physical Hilbert space but a bigger ‘kinematical’ Hilbert space. This kinematical Hilbert space naturally gets a degenerate inner product, where we compute the amplitude for a particle to go from one point in spacetime to another using a path integral:

$\langle \delta_y, \delta_x \rangle = \int_{f: x \to y} e^{iS(f)} D f$

To recover the physical Hilbert space, we take the quotient of this kinematical Hilbert space by the space of ‘null vectors’: vectors whose physical inner product with every other vector vanishes.

As you note, these are standard ideas among people who quantize constrained systems. I was just trying to see how they arise from the ‘category of configurations and processes’ philosophy.

Recently I understood, from this comment of yours, that this relation between the path integral and the inner product is the way you think of the Hom as generalizing the inner product.

Well, I’ve thought about inner products as decategorified homsets for a long time. There are lots of ways one could try to make this analogy precise, and I wouldn’t want to tie myself down too much.

But, one thing I’ve been wondering about a lot is how inner products get to be complex. When applying the groupoidification program to the quantum harmonic oscillator, Jeff and I were led to the notion of a U(1)-set — a set with elements labelled by phases. U(1)-sets naturally have complex cardinalities. The cardinality $|S|$ of a U(1)-set $S$ is just the sum of the phases of the points in $S$ .

(See, we’re building Feynman’s ideas of path integration right into the definition of cardinality — if you read his pop book QED, you’ll practically see this idea staring you in the face!)

$U(1)$ -sets form a monoidal category in a nice way, consistent with this cardinality operation:

$|S \times T| = |S| \times |T|$

So, it makes sense to consider categories enriched over U(1)-sets: that is, categories where given two objects $x$ and $y$ , the set $hom(x,y)$ is actually a $\mathrm{U}(1)$ -set, and composition

$\circ : hom(x,y) \times hom(y,z) \to hom(x,z)$

is a map of $\mathrm{U}(1)$ sets, and so on.

But, we get a category enriched over $\mathrm{U}(1)$ -sets from an ordinary category $C$ equipped with a functor

$e^{iS}: C \to \mathrm{U}(1)$

So, I wanted to explore this line of thinking a bit…

Posted by: John Baez on June 10, 2007 8:18 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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As I wondered somewhere else, is there any place for the cyclic category here instead of U(1)? Those cycling morphisms in $hom_{\Lambda}([n], [n])$ would seem a lot like (discrete) phase differences.

Posted by: David Corfield on June 11, 2007 7:53 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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The following may or may not be relevant.

Unless I am mixed up, $G$ -sets may be thought of as living in the over category $\mathrm{Set}_{G}$ whose objects are sets with a morphism to the set underlying $G$ $S \stackrel{s}{\to} G$ and whose morphism are morphisms of sets commuting with these anchors $\array{ S &&\stackrel{f}{\to}&& S' \\ & {}_s\searrow && \swarrow_{s'} \\ && G } \,.$

(I am being so explicit in case other readers are wondering what I am talking about.)

Now, as a couple of people have recently tried to get into my head, such over categories are a another way to achieve something like internalization, thereby equipping things with extra structure.

For instance, when Stephan Stolz and Peter Teichner talk about smooth categories and smooth functors between them, they consider the over (2-)category $\mathrm{Cat}_{\mathrm{Manifolds}} \,.$

This is closely related to looking at stacks on manifolds, and to looking at categories internal to sheaves on manifolds (but I am confused as to the exact passage between these points of view).

Anyway, what I just wanted to say here is that I think, analogously, we might maybe want to think of a $G$ -set as a set internal to sheaves on $G$ . Here I am thinking of $G$ as equipped with the discrete topology. Then a sheaf on $G$ is nothing but a set for each element of $G$ , hence a $G$ -set.

Okay, as I said, I don’t know if this has any relevance. But maybe something like this might one day help explain from first principles why we need to consider $G$ -sets.

Posted by: urs on June 11, 2007 1:11 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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Just to get this straight, when you say $G$ -sets are

sets with a morphism to the set underlying $G$ ,

this is not the kind of $G$ -set found in a group action?

Posted by: David Corfield on June 11, 2007 2:52 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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Oh, right. No, here a $G$ -set is just a set, all whose elements are equipped with a label, which is taken from $G$ .

It’s more like a $G$ -graded set, if you like.

Posted by: urs on June 11, 2007 3:03 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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I’m sorry, folks! I forgot to mention that my use of the term ‘U(1)-sets’ was very confusing… a bad choice of terminology.

Normally an M-set means a set equipped with an action of the monoid $M$ . This is especially true when $M$ is actually a group.

But, in Jeff’s paper, an M-set means a set ‘over’ the monoid $M$ . In other words, a set equipped with a function to the monoid $M$ . This is the unorthodox sense in which I was using the term ‘ $\mathrm{U}(1)$ -set’.

To reduce confusion, I should stop calling these things $\mathrm{U}(1)$ -sets. But if the idea turns out to be useful, we may want a snappier term than ‘sets over $\mathrm{U}(1)$ ’. Maybe something like ‘phase sets’ or ‘phased sets’.

Let’s try it — in this comment, I’ll call sets over $\mathrm{U}(1)$ phased sets.

So: Urs is completely right that phased sets are the category of interest here, and that this category is a topos. But, in his paper, Jeff shows how to make phased sets into a symmetric monoidal category with a different, noncartesian tensor product, using the group structure on $\mathrm{U}(1)$ . And this is what we need for quantum mechanics.

The idea here is that if we have one set of points $\{i\}$ labelled by phases $\alpha_i$ , and another set of points $\{j\}$ labelled by phases $\beta_j$ , their product should be the set of pairs $\{(i,j)\}$ labelled by phases $\alpha_i \beta_j$ .

This product works well if we define the cardinality of a phased set to be the sum of the phases of its elements. Then, the cardinality of a product of phased sets is the product of their cardinalities!

David’s remark on the cyclic category had occurred to me too — but perhaps like him I was confused by my terminology! Cyclic sets are like topological spaces with a $\mathrm{U}(1)$ action. These are rather different than phased sets, which are sets over $\mathrm{U}(1)$ .

The idea David raises could still be important here — maybe after some kind of change of viewpoint, or level shift. In particular, Dwyer Hopkins and Kan note that cyclic sets are like spaces with a $\mathrm{U}(1)$ -action or spaces over $BU(1)$ . There’s a general pattern here, so that spaces with a $\mathbb{Z}$ -action should be like spaces over $B\mathbb{Z} = \mathrm{U}(1)$ … that is, ‘phased spaces’.

To see what I mean, consider the Möbius strip as a space over $\mathrm{U}(1)$ . What space with $\mathbb{Z}$ -action does this correspond to?

Going from ‘phased sets’ to ‘phased spaces’ is actually nice for other reasons, too: it really amounts to replacing sets with $\infty$ -groupoids.

So, there could be some interesting stuff to do here. We’ll just have to keep straight the difference between ‘phased spaces’ and ‘phase spaces’… which are actually quite related, as this course showed!

Posted by: John Baez on June 11, 2007 6:03 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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There’s a general pattern here, so that spaces with a $\mathbb{Z}$ -action should be like spaces over $B\mathbb{Z} = U(1)$ … that is, ‘phased spaces’.

Ah, interesting! I see. So maybe to understand this, we really need to solve your old puzzle, which asked how it comes that $\mathbb{Z}$ $U(1) \simeq B\mathbb{Z}$ $\mathbb{C}P^\infty \simeq B B\mathbb{Z}$ all are so very relevant for quantum physics.

So is there a way in which I can think of a phased set as a set with a $\mathbb{Z}$ -action? How?

Posted by: urs on June 11, 2007 6:35 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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Out of interest, has any further progress been made on understanding John’s question from TWF149:

How about $K(\mathbb{Z},3)$ ?

beyond what he told us towards the end of TWF151.

Is this $BBB\mathbb{Z}$ also proving important for physics?

Posted by: David Corfield on June 12, 2007 8:25 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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David Corfield wrote:

Is this $BBB\mathbb{Z}$ also proving important for physics?

On the one hand, definitely yes!

$BBB\mathbb{Z}$ is the classifying space for $\U(1)$ gerbes, or what I might prefer to call $\U(1)[1]$ -2-bundles. These are becoming ever more popular in string theory, so their classifying space is implicitly and sometimes explicitly present in a lot of work these days.

On the other hand, I still haven’t seen the geometry of $K(\mathbb{Z},3)$ cropping up in work on mathematical physics as often as I’d like.

And, I’m still quite dissatisfied with my geometrical intuition of this space. I still haven’t gotten much further than I had back in week151, where I gave two nice pictures of $K(\mathbb{Z},3)$ :

$K(\mathbb{Z},3)$ is the space of all configurations of particles and antiparticles on the 3-sphere — that is, finite collections of points of $S^3$ labelled by integers, given a topology such that when points collide, their integers add. This kind of construction generalizes easily to any $K(A,n)$ , but I haven’t managed to exploit it get any insights special to $K(\mathbb{Z},3)$ .
Since $S^3$ is the unit quaternions, we get an interesting noncommutative multiplication on $K(\mathbb{Z},3)$ from this picture, and also some interesting actions of $\SU(2)$ — coming from left multiplication, right multiplication, and conjugation, with only the last fixing the based point and preserving the noncommutative product.

But, I don’t know about anything exciting to do with this.
This gang of Australians found a construction of $K(\mathbb{Z},3)$ with clearer ties to physics and more applications (at least so far):

Alan L. Carey, Diarmuid Crowley and Michael K. Murray, Principal bundles and the Dixmier-Douady class, Comm. Math. Physics 193 (1998) 171-196, available as hep-th/9702147.

Here’s how it goes, at least in part.

Suppose $H$ is a countable-dimensional Hilbert space. Then the unitary group $\mathrm{U}(H)$ is contractible, and the group $\mathrm{U}(1)$ acts freely via multiplying by phases. So, the quotient — the projective unitary group $PU(H)$ — is the same as

$BU(1) \simeq K(\mathbb{Z},2).$

Next, we say a linear operator

$A: H \to H$

is Hilbert-Schmidt if the trace of $A A^*$ is finite. The space of Hilbert-Schmidt operators is a Hilbert space in its own right, with this inner product:

$\langle A, B\rangle = tr(A B^*)$

Let’s call this Hilbert space $X$ . The unitary group $\mathrm{U}(H)$ acts on $X$ by conjugation, and this gives an action of the projective unitary group $PU(H)$ on $X$ , because phases commute with everything. This in turn gives an action of $PU(H)$ on $\mathrm{U}(X)$ !

It turns out that this action is free, so it makes $\mathrm{U}(X)$ into the total space of a principal $PU(H)$ -bundle:

$PU(H) \to \mathrm{U}(X) \to \mathrm{U}(X)/PU(H)$

But $X$ is a countable-dimensional Hilbert space, so $\mathrm{U}(X)$ is contractible, so this is the universal principal $PU(H)$ -bundle.

Since

$PU(H) \simeq BU(1) \simeq K(\mathbb{Z},2),$

this means

$\mathrm{U}(X)/PU(H) \simeq BBU(1) \simeq K(\mathbb{Z},3) .$

While the second construction may seem complicated, it’s actually very nice! It’s related to lots of structures that are important in quantum physics — unitary operators, trace-class operators, etcetera — and it gives a model of $K(\mathbb{Z},3)$ as a highly symmetrical infinite-dimensional manifold.

So, there should be lots one can do with it… but I haven’t seen this stuff. In part, I just haven’t had time to keep up with what the Australians have been doing!

(These are the gerbey Australians, as opposed to the $n$ -categorical Australians. Danny Stevenson, here at Riverside, comes from the former tradition — and I’ve have plenty of trouble finding time just to keep up with what he’s doing! Luckily Danny will be moving to Hamburg and working with Urs next year, so then it will be Urs who has to keep up with Danny’s creativity. )

Finally, here’s a little footnote for those who don’t already know and love $\mathrm{U}(1)[1]$ . This 2-group is like $\mathrm{U}(1)[1]$ shifted up one notch. It’s the 2-group with one object and $\U(1)$ ’s worth of morphisms. We can do this ‘shifting’ trick to any abelian group, and we can iterate it. So, given an abelian group $A$ , there’s an $(n+1)$ -group $A[n]$ with one object, one morphism, … , and $A$ ’s worth of $n$ -morphisms. Since this is a strict abelian $(n+1)$ -group, we can profitably regard it as a chain complex with $A$ as $n$ -chains and all the other chain groups vanishing. Indeed, folks have long enjoyed taking a chain complex $C$ and forming a shifted version $C[n]$ . We’re just reinterpreting it as a construction involving $n$ -categories!

Indeed, we can go further in this direction, by thinking of $A[n]$ as the $n$ -group of $A[n-1]$ -torsors. That makes the connection to gerbes more obvious.

Posted by: John Baez on June 12, 2007 5:30 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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I am trying to see if maybe by slightly shifting the point of view we can see clearer.

Here is a random attempt, possibly irrelevant:

In this entry I tried to extract the essence of the difference between the Segal picture of QFT (functoriality) with the Atiyah picture (sewing) by the following construction:

given any functor $F : C \to D$ with $D$ $\mathrm{Vect}$ -enriched, we may equivalently think of it as follows.

From $C$ we cook up the category $\tilde C$ whose

- object are collections of morphisms of $C$

- morphisms go from one collection to another, which is obtained by composing a couple of morphisms in the former.

We have two canonical monoidal functors $\tilde C \to \mathrm{Vect} \,,$ one of them, $I$ , sends everything to the identity morphism on tensor unit object, the other, $H$ , sends any morphism $a \stackrel{f}{\to} b$ to $\mathrm{Hom}_D(F(a),F(b))$ .

The original functor $F$ is then encoded in a natural transformation $\tilde F : I \to F \,.$ This natural transformation encodes the “sewing constraints”, which are equivalent to the functoriality of $F$ .

This is a trivial game with category theory, the point of it being that it clarifies some things, conceptually, that people do in quantum field theory.

Anyway, my observation here is this:

maybe we want to think of a phased set rather as an object in $\tilde C$ , where $C = \Sigma U(1)$ is the group $U(1)$ , regarded as a category.

This is just a random idea.

Posted by: urs on June 11, 2007 6:59 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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Another random idea. If a phased set is a map from a set, $X$ , to the underlying set of $U(1)$ , then it also corresponds to a group homomorphism from the free group on $X$ , $F(X)$ , to $U(1)$ .

Posted by: David Corfield on June 12, 2007 8:35 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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To see what I mean, consider the Möbius strip as a space over $U(1)$ . What space with $\mathbb{Z}$ -action does this correspond to?

Should correspond to the two point space with the $\mathbb{Z}$ -action coming from regarding it as $\mathbb{Z}_2$ .

In general, quoting diectly from John’s lectures, slide number 6:

Covering spaces of $B$ with fiber $F$ are classified by actions of $\pi_1(B)$ on $F$ .

In our case here $B = U(1)$ and $\pi_1(B) = \mathbb{Z}$ .

Posted by: urs on June 11, 2007 7:13 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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Urs wrote:

To see what I mean, consider the Möbius strip as a space over U(1). What space with $\mathbb{Z}$ -action does this correspond to?

Should correspond to the two point space with the $\mathbb{Z}$ -action coming from regarding it as $\mathbb{Z}/2$ .

Right! And as you note, this is an instance of a general fact:

Covering spaces of $B$ with fiber $F$ are classified by actions of $\pi_1(B)$ on $F$ .

Taking $B = K(G,1)$ to be the classifying space of a discrete group $G$ , so $\pi_1(B) = G$ , we get:

For any discrete group $G$ , covering spaces of $K(G,1)$ are classified by actions of $G$ on discrete spaces.

Note this correspondence ‘shifts dimension by 1’: a discrete $G$ -space is $0$ -dimensional from the viewpoint of homotopy groups, while a covering space of a $K(G,1)$ is 1-dimensional. This is that shift of dimension you’re familiar with elsewhere, e.g. Schreier theory.

This dimension shift is why I’m not optimistic about this question unless it’s fixed a bit:

So is there a way in which I can think of a phased set as a set with a $\mathbb{Z}$ -action? How?

Sets with $\mathbb{Z}$ -action, should correspond to certain phased spaces, namely covering spaces of $\mathbb{U}(1)$ — not mere phased sets. The topology is crucial.

Posted by: John Baez on June 12, 2007 6:30 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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The topology is crucial.

That’s what I thought. So can we use spaces over $U(1)$ as a sensible replacement for sets over $U(1)$ for the particular application that we have in mind here? In particular, if we care about these spaces only up to homotopy (as we would if we regarded them in terms of their fibers and the $\mathbb{Z}$ -action on these), then how would $U(1)$ -cardinality survive?

Posted by: urs on June 12, 2007 11:00 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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Urs wrote:

So can we use spaces over U(1) as a sensible replacement for sets over U(1) for the particular application that we have in mind here?

I don’t know — I’d have to think about this for a while.

In particular, if we care about these spaces only up to homotopy (as we would if we regarded them in terms of their fibers and the $\mathbb{Z}$ -action on these), then how would U(1)-cardinality survive?

Yeah, that sounds like a serious problem.

By the way, if we switch from sets to Hilbert spaces, everything works very nicely:

Hilbert spaces with unitary $\mathbb{Z}$ -action are the same as $\mathrm{U}(1)$ -graded Hilbert spaces.

The reason is that we can take any unitary representation of $\mathbb{Z}$ and decompose it into eigenspaces of the unitary operator that generates the action of $\mathbb{Z}$ . Each eigenspace corresponds to some phase or other, so we get a $\mathrm{U}(1)$ -graded Hilbert space.

We can also go back.

In fact, this is just a special case of Pontryagin duality. Every locally compact abelian group $A$ has a ‘dual group’ $A^*$ , and

Hilbert spaces with strongly continuous unitary $A$ -action are the same as $A^*$ -graded Hilbert spaces.

It seems to be just an amazing fluke of nature that we have both

$\mathrm{B}\mathbb{Z} = \mathrm{U}(1)$

and

$\mathbb{Z}^* = \mathrm{U}(1).$

The former fact tells us that:

Spaces with $\mathbb{Z}$ -action are the same as fibrations over $\mathrm{U}(1)$ .

The latter fact tells us that:

Hilbert spaces with unitary $\mathbb{Z}$ -action are the same as $\mathrm{U}(1)$ -graded Hilbert spaces.

I don’t know what this means. I just realized the strange similarity of these facts, and thought I should mention it!

Posted by: John Baez on June 12, 2007 6:01 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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Yeah, that sounds like a serious problem.

By the way, if we switch from sets to Hilbert spaces, everything works very nicely:

Hey, Mr. Wizard, that spell seems to go in the wrong direction!

I was hoping you would instead prepare to cast a spell of the following kind:

Yeah, that sounds like a serious problem.

By the way, if we switch from sets to groupoids, everything works very nicely, as to be explained in the Tale of Groupoidification

Posted by: urs on June 12, 2007 9:09 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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Alas, while the Wizard is struggling in his back room laboratory trying to get work phases into the groupoidification program in a beautiful natural way, he’s been unable to so far. Listen in while he putters with his test tubes and alembics:

Blast! I turned up the heat but it still didn’t dissolve! I’ll have to redo the whole experiment. $\mathrm{U}(1)$ , $\mathrm{U}(1)$ , there has to be a way to get it in there… Maybe take the nerve of the cyclic category? Maybe Egan was on to something… where is that story? Oh, here:

He took Cordelia’s hands and they waltzed across the scape together. “The central mystery of quantum mechanics has always been: why can’t you just count the ways things can happen? Why do you have to assign each alternative a phase, so they can cancel as well as reinforce each other? We knew the rules for doing it, we knew the consequences — but we had no idea what phases were, or where they came from.”

Hmm… does he say where they come from?…

Posted by: John Baez on June 14, 2007 5:36 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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Blast! I turned up the heat but it still didn’t dissolve!

Hm, let’s see. There on the wizard’s cauldron I see three knobs, labeled “heat”, “volume” and “pressure”.

What if we’d play also with the second one?

But not without first having another look at the almanac, chapter CCLII.

Let’s see, there it says that I am supposed to think of

- a vector space with a group $G$ represented on it

- as a groupoid with a faithful functor to $G$

and

- of a linear equivariant map of vector spaces on which $G$ is represented

- as a span of groupoids over $G$ .

Okay. Now let me try to recall exactly what it is we are trying to conjure:

We want to see complex vector spaces materialize.

To this lowly apprentice here, this looks like we ought to be looking for vector spaces with a $U(1)$ -action on them.

(Unitary vector spaces with a $\mathbb{Z}$ -action on them would give us a $U(1)$ -grading on these. But the riddle here is to figure out how to get a unitary vector space in the first place! – I think.)

Hence it seems that the complex vector spaces which we are looking for are first of all best to be thought of as groupoids over $U(1)$ .

$\mathrm{Gr} \to \Sigma U(1)$

An therefore it is this particular occurence of the group $U(1)$ which deserves further explanation. It seems to me.

So: can we understand a groupoid over $U(1)$ as something involving not $U(1)$ but $\mathbb{Z}$ ?

Another idea is needed.

Here in this old book titled Random general purpose ideas for the hard working wizard it says on p. 5483, 21st paragraph:

To get rid of $B$ s, increase $n$ .

Er. Hm. Wait. Mr. Wizard:

What is a 2-groupoid over $\Sigma \Sigma \mathbb{Z}$ ?

(Where these symbols denote the strict 2-groupoid with a single object, a single 1-morphism, with one 2-morphism per integer and with horizontal and vertical composition of these being addition of integers.)

I mean, suppose we had such a beast and felt the inclination to regard 2-groupoids as categories enriched over simplicial sets. Then we’d form the nerves of all Hom-categories in sight. When we don’t have the nerve to consider nerves, we’d geometrically realize them. This turns our 2-groupoid over $\Sigma^2 \mathbb{Z}$ into a 1-groupoid over $\Sigma U(1)$ , I suppose. Hence into a complex vector space.

——————————

Can I refrain from making the following remark? No I cannot. Here it goes:

Suppose the above is a good idea (perhaps not). If so, we find ourselves in the situation that our action pseudo-functors ought to take values in the 3-category of spans in 2-groupoids over $\Sigma^2 \mathbb{Z}$ . Something like that.

Since this is a 3-category, it would imply that the domain ought to be a 2-category! That’s the case for the 2-particle.

Posted by: urs on June 15, 2007 12:40 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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I’m just one of the wannabe-apprentice-wizards watching through the window. With all the experiments going on in there the window is smoking up and I can’t make out the three knobs on the cauldron! But I do at least have a magical childhood wish for where I’d love to see the wizard work $U(1)$ into the mix : I’d like to see it enter in the “gerbey” way. That is, it would be really cool if the wizard were to add to his basic ingredients, not just groupoids over $G$ , but “central extensions” of these groupoids, i.e. groupoids over $G$ whose hom-sets are $U(1)$ -torsors.

Posted by: Bruce Bartlett on June 15, 2007 1:02 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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It would be really cool if the wizard were to add to his basic ingredients, not just groupoids over G, but “central extensions” of these groupoids, i.e. groupoids over G whose hom-sets are U(1)-torsors.

That would be cool if it worked — but I would still be a bit disappointed. Surely the big hope is that everything will be fundamentally combinatorial. Groupoid cardinality lets us get the positive reals out combinatorially, so why give up the hope that we’ll somehow get the complexes out combinatorially?

Posted by: Jamie Vicary on June 15, 2007 1:13 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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That would be cool if it worked – but I would still be a bit disappointed.

Yes, I agree with that. It seems to me that we need to carefully distinguish different roles in which one and the same actor, $U(1)$ , appears throughout our lives.

Well, maybe in the end we’ll see connections undreamed of.

Im mean, Bruce is asking for gerbes where I have argued a minute ago that we should be able to get

complex vector spaces

$\simeq$

vector spaces with $U(1)$ representation

“ $\simeq$ ”

groupoids over $\Sigma U(1)$

from

2-groupoids over $\Sigma^2 \mathbb{Z}$

“ $\simeq$ ”

2-vector spaces with 2-representation of the 2-group $\Sigma \mathbb{Z}$ .

(I hope somebody disabuses me of that idea soon, if necessary, because I am about to like it a whole lot…)

Posted by: urs on June 15, 2007 1:23 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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That would be cool if it worked - but I would still be a bit disappointed.

Ah, you’re right, I see your point.

Posted by: Bruce Bartlett on June 15, 2007 2:18 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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John had this to say about the idea of not putting in $U(1)$ by hand in response to a question I had posed.

There’s even a nice offer for someone to work with him at the end. I’d jump at it.

Posted by: David Corfield on June 15, 2007 3:05 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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John had this to say about the idea of not putting in $U(1)$ by hand in

Since we are messing with the very fundaments of the universe here (I hope that’s clear to everybody: if the Wizard really gets this spell to work, everybody better duck and cover!) it pays to be really careful with distinguishing our concepts.

As the message David linked to also indicates, there are actually two different approaches here. Sorry for being pedantic about that, but it strikes me that we were discussing one of them when, for instance, Jamie chimed in with a remark about the other.

On the one hand we have:

Leinster cardinalities of categories (having groupoid cardinality as a special case). This shows up when considering colimits, i.e. when “integrating functors over categories”. Should play a role in the path integral.

On the other hand we have

Combinatoric replacements for representation theory (Tale of Groupoidification): instead of looking at representations of groups on vector spaces, we look at representations of groups on mere sets. Nothing is lost this way (or at least that’s the idea) when the morphisms of these set-representations are suitably generalized.

This might be relevant for understanding the true nature of “Hilbert spaces of states” in quantum theory.

Clearly, both these apsects ought to be closely related and should even turn out to be parts of one big picture. But before we arrive there, let’s carefully distinguish between the two.

Posted by: urs on June 15, 2007 3:55 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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Sorry for carelessly chiming in! :# Actually, I’m having trouble following the earlier parts of this discussion — can someone point me to a definition of $B \mathbb{Z}$ ?

Posted by: Jamie Vicary on June 15, 2007 4:48 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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Sorry for carelessly chiming in! :#

I am glad you did!

Hope you don’t mind my reply. I just thought it would be important to point out that you had something slightly different in mind than we were discussing in the previous messages.

can someone point me to a definition of $B \mathbb{Z}$

That’s just the classifying space of $\mathbb{Z}$ -bundles.

$B \mathbb{Z} \simeq U(1) \,.$

I believe, in the present context, the best way to think about it here is that

$B G$

is the realization of the nerve of $\Sigma G$

$B G := |\Sigma G| \,,$

where I follow my habit of regarding a group $G$ as a monoid and then writing $\Sigma G$ for the corresponding 1-object category.

Posted by: urs on June 15, 2007 4:57 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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I can highly recommend:

John Baez: Classifying Spaces Made Easy.

Posted by: urs on June 15, 2007 5:03 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

That link took me to
Generalized Cohomology Theories and Eilenberg-MacLane Spaces
which told me more than I think can be called Classifying spaces made easy.
Can you suggest a more succinct and easier link?

I’ve finally caught on to what a pseudofunctor is, but generalizing to pseudo 2-functors seems to me to have some options. Is there a commonly accepted
defintion?

Posted by: jim stasheff on June 15, 2007 5:57 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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Here is a secret message from one apprentice to the other, spoken in low voice such as not to disturb the wizard.

Hey Bruce, let me invite you to think about 2-groupoids over $\Sigma^2 \mathbb{Z}$ . I am getting the suspicion that meditating over this leads to enlightment of the second kind.

Remember that I was arguing that a complex vector space is really a 2-groupoid fibered over $\Sigma^2 \mathbb{Z}$ .

Let’s accept that for a moment and see what it would imply.

By general abstract nonsense (I know that Igor Bakovich has worked on a theorem showing this in detail) such 2-groupoids fibered over $\Sigma^2 \mathbb{Z}$ are equivalent to pseudo 2-functors

$F : \Sigma^2 \mathbb{Z} \to 2\mathrm{Cat}$

(“second Grothendieck construction”).

Now, it just occured to me that this is something of utmost interest. (I cannot quite exclude the possibility that you mentioned something like the following to me before which I forgot and now rediscovered. But in any case:)

What is a pseudo 2-functor from $\Sigma^2 \mathbb{Z}$ to $2\mathrm{Cat}$ ?

Right, what is it?

Well, basically it’s a 2-category together with a chosen endomorphism $F(1)$ on the identity 2-functor on it.

So it’s an object in a braided monoidal category.

Moreover…

Posted by: urs on June 15, 2007 4:11 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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I realize I might better explain the terminology I used.

For me here, a “pseuo $n$ -functor” is something that goes from an $n$ -category to an $(n+1)$ -category. So it’s the same as an $(n+1)$ -functor, but restricted to $(n+1)$ -categorical domains which are “discrete at top level” (i.e. which are really just $n$ -categories).

And I am using the general fact that

fibered $n$ -categories

$\array{ P_n \\ \downarrow \\ B_n }$ over a fixed base $n$ -category $B_n$ are equivalent to pseudo $n$ -functors

$\array{ n\mathrm{Cat} \\ \uparrow \\ B_n } \,.$

For instance a mere bundle (of sets, say, let’s not worry about extra structure for the moment)

$\array{ P_0 \\ \downarrow \\ B_0 }$

is the same as the “pseudo 0-functor”

$\array{ \mathrm{Set} \\ \uparrow \\ B_0 }$

which sends every point in $B_0$ to the set of points lying over it.

Here I am talking about the $(n=2)$ -version of this:

2-groupoids fibered over the 2-groupoid $\Sigma^2 \mathbb{Z}$

$\array{ \mathrm{Gr} \\ \downarrow \\ \Sigma^2 \mathbb{Z} }$

should be equivalent (there might be quite a couple of technical qualifications needed here to make this precise) to “pseudo 2-functors”

$\array{ 2\mathrm{Grpd} \\ \uparrow \\ \Sigma^2 \mathbb{Z} } \,.$

Posted by: urs on June 15, 2007 4:52 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 27)

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Going from ‘phased sets’ to ‘phased spaces’ is actually nice for other reasons, too: it really amounts to replacing sets with $\infty$ -groupoids.

Right now I don’t see how this passage from sets to spaces wouldn’t lose what we are trying to achieve: a categorification of addition and multiplication of complex numbers.

Say I replace all my spaces over $U(1)$ with representatives of their fibers $F$ , remembering just the holonomy as I go around the $U(1)$ circle.

Taking disjoint union of spaces over $U(1)$ should still amount to taking disjoint union of these fibers. But how is this to remember addition of unimodular numbers?

The source of the problem seems to be that a space over $U(1)$ is not so different from a space over $S^1$ …

Or am I missing something?

Posted by: urs on June 11, 2007 7:37 PM | Permalink | Reply to this

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June 8, 2007