This Week's Finds in Mathematical Physics (Week 256)

August 27, 2007

This Week’s Finds in Mathematical Physics (Week 256)

Posted by John Baez

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In week256, learn a bit of what happened at a conference on Poisson sigma models and Lie algebroids at the Erwin Schrödinger Institute, run by Anton Alekseev, Henrique Bursztyn and Thomas Strobl. Higher categories are finding their way into classical mechanics! Then, hear more of the Tale of Groupoidification: how to turn a span of groupoids into an operator between vector spaces.

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Here are Henrique Burstyn, Pavel Mnev, Dmitry Roytenberg and Thomas Strobl at a Heuriger near Vienna:

We reached this place at the end of a picturesque hike from the town of Baden bei Wien, through vineyards, past this church:

Posted at August 27, 2007 11:41 PM UTC

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Who’s behind that final quote? Re: This Week’s Finds in Mathematical Physics (Week 256)

I usually believe (depends on mood and setting and company present) that Scientists are studying a single real universe, by various imperfect means.

Mathematicians, on the other hand, are doing something. But Philosphers of mathematics have not been able to answer the question cited by Corfield: “How do mathematicians steer their careers?”

That’s a puzzle, because they do not have feedback from “nature” the way scientists believe they have.

Here’s an intriguing quotation at the ned of the Week 2^8 blog by John Baez:

“Viewed superficially, mathematics is the result of centuries of effort by thousands of largely unconnected individuals scattered across continents, centuries and millennia. However the internal logic of its development much more closely resembles the work of a single intellect developing its thought in a continuous and systematics way - much as in an orchestra playing a symphony written by some composer the theme moves from one instrument to another, so that as soon as one performer is forced to cut short his part, it is taken up by another player, who continues with due attention to the score.”
- I. R. Shavarevich

Only who is that? Is it an alternate spelling, as Google hints to me?

In mathematics, the Golod-Shafarevich Theorem, named after the two Russian mathematicians Evgeny Golod and Igor Shafarevich, who proved it on 1964 is an important theorem in combinatorial group theory. In its most basic form, it states that if G is a finite p-group with minimal number of generators d and has r relators in a given presentation, then

r > (d^2)/4.

References

1. Johnson, D.L. (1980). Topics in the Theory of Group Presentations (1st ed.). Cambridge University Press. ISBN 0-521-23108-6. See chapter VI.

Or is it a Socialist mathematician and writer (as the strange valorization of the collective over the individual suggests) as Wikipedia begins:

Igor Rostislavovich Shafarevich (Russian: Игорь Ростиславович Шафаревич, born June 3, 1923 in Zhytomyr) is a Russian mathematician, founder of the major school of algebraic number theory and algebraic geometry in the USSR, and a political writer. He was also an important dissident figure under the Soviet regime, a public supporter of Andrei Sakharov’s Human Rights Committee from 1970. He supported the criticisms of Alexandr Solzhenitsyn of both Soviet communism and liberal proposals for the future of Russia.

Shafarevich’s 1970s book The Socialist Phenomenon was widely circulated in the West. After the Cold War, he attacked those he called “small people,” who deny the “historical achievements” of Russia, saying his homeland must have “sound democratic statehood, based on the will of the people.” His critics call him a radical, anti-Semitic, Christian nationalist.

Shafarevich’s contribution to mathematics include the theory of the Tate-Shafarevich group (usually called ‘Sha’, written ‘Ш’, his Cyrillic initial) in Galois cohomology, and the Golod-Shafarevich theorem on class field towers. He initiated a Moscow seminar on classification of algebraic surfaces that updated around 1960 the treatment of birational geometry, and was largely responsible for the early introduction of the scheme theory approach to algebraic geometry in the Soviet school.

Shafarevich was a student of Boris Delone, and his students included Evgeny Golod, S.Y. Arakelov, I.A. Kostrikin and Yuri Manin. In view of later accusations of anti-Semitism on his part, it can be noted that his research students included some identified as Jewish, and that later, during his most serious troubles in the 1970s with the Soviet authorities, he did major work in collaboration with Ilya Piatetski-Shapiro on K3 surfaces. He is a member of the Serbian Academy of Sciences and Arts in department of Mathematics, Physics and Geo Sciences.

On his 80th birthday, Russian President Vladimir Putin hailed his “fundamental research” in mathematics and his creation of “a large scientific school that is known both in Russia and abroad.

Hmmm. I’m not a big fan of Putin, or anti-Semites. However, I have read and enjoyed a lot by Sakharov and Solzhenitsyn. And the Tate-Shafarevich group is cool, and I more than half understood it after hours of effort…

Posted by: Jonathan Vos Post on August 28, 2007 3:40 AM | Permalink | Reply to this

Re: Who’s behind that final quote? Re: This Week’s Finds in Mathematical Physics (Week 256)

Sorry, “Shavarevich” was a typo for the famous algebraic geometer Shafarevich. I’ll fix that now.

Posted by: John Baez on August 28, 2007 11:11 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Re groupoidification: Very nice. I am truly enjoying this.

Two tiny quick comments:

You write:

we’ll call this vector $[V]$

Wouldn’t it be nicer to call this vector $[v]$ ? And in fact to call $\array{ V \\ \downarrow^p \\ X }$ instead $\array{ v \\ \downarrow^p \\ X } \,.$ Seems to me that would guide the eye a little better.

Actually, thinking about it, what really deserves to be called $v$ is the morphism $p$ .

Then, towards the end, here is a tiny, tiny typo:

as defined that paper

should probably read “as defined in that paper”.

Posted by: Urs Schreiber on August 28, 2007 11:05 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Glad you’re enjoying it, and thanks for the suggestions/corrections!

I guess a lower-case $v$ would be nicer, so we could write $[Sv]$ as the result of applying the span of groupoids $S$ to $v$ , and get

$[Sv] = [S] [v]$

It’s also true that what really matters is the morphism $p: v \to X$ , not $v$ itself — but this notation gets a bit tricky when we apply it to a span, or matrix, which has one object and two morphisms.

Probably Jim Dolan’s approach is best: we should think of each morphism as an ‘index’, in a categorified version of Penrose’s abstract index notation. Then a groupoid over $X$ :

$i: v \to X$

can be written as a vector with one index:

$v_i$

Similarly, a span of groupoids:

$i: S \to X, \qquad j: S \to Y$

can be written as a matrix with two indices:

$S_{i j}$

And so on for higher-rank tensors, as David Corfield pointed out.

Summing over repeated indices is then our notation for taking weak pullbacks! And we don’t need to distinguish between upper and lower indices, due to the sneaky properties of groupoidification.

More abstractly, we can think of $v$ as an object in $[groupoids over X]$ and drop the index. Similarly, we can think of $S$ as an object in $[groupoids over X \times Y]$ and drop the indices.

So, we have both the usual mathematicians’ notation and the usual physicists’ notation available to us! Everything will look quite ordinary, but it’s all been groupoidified!

Posted by: John Baez on August 28, 2007 11:29 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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It might also be a good idea to emphasise the point that $V \to X$ , thought of as $* \leftarrow V \to X$ , is a generalised morphism from the point to $X$ - just as we think of an element as a map from the terminal object via its name. Is that where Urs pulled the $\lfloor v\rfloor$ from?

Posted by: David Roberts on August 30, 2007 4:22 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Is that where Urs pulled the $\lfloor v \rfloor$ from?

I was simply following John’s notation, where square brackets $[\cdot]$ denoted the morphism from the (3?-)category of spans of groupoids to that of vector spaces.

But I perfectly agree that the best way to think of those “combinatorial vectors” here is as generalized objects (you meant generalized objects, not generalized morphisms, right?).

What I don’t quite recall: is $\lfloor \cdot \rfloor$ the standard notation for generalized elements?

If so, then the vector we are talking about should really be denoted $[\lfloor v \rfloor]$ ;-)

Posted by: Urs Schreiber on August 30, 2007 11:24 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Urs wrote:

What I don’t quite recall: is $\lfloor \cdot \rfloor$ the standard notation for generalized elements?

No, it’s the standard notation for the ‘name’ of a morphism in a closed category. In such a category, any morphism

$f : X \to Y$

has a ‘name’

$\lfloor f \rfloor : 1 \to Hom(X,Y)$

where $Hom(X,Y)$ is the internal hom. Since a morphism from $1$ is called an ‘element’, we can say that the name of $f$ is an element of $Hom(X,Y)$ .

Clearly you skipped some classes. I’ll have to report you to the Wizard! He may punish you by making you translate Through the Looking Glass into the language of closed categories:

‘You are sad,’ the Knight said in an anxious tone: ‘let me sing you a song to comfort you.’

‘Is it very long?’ Alice asked, for she had heard a good deal of poetry that day.

‘It’s long,’ said the Knight, ‘but very, VERY beautiful. Everybody that hears me sing it–either it brings the TEARS into their eyes, or else–’

‘Or else what?’ said Alice, for the Knight had made a sudden pause.

‘Or else it doesn’t, you know. The name of the song is called “HADDOCKS’ EYES.”’

‘Oh, that’s the name of the song, is it?’ Alice said, trying to feel interested.

‘No, you don’t understand,’ the Knight said, looking a little vexed. ‘That’s what the name is CALLED. The name really IS “THE AGED AGED MAN.”’

‘Then I ought to have said “That’s what the SONG is called”?’ Alice corrected herself.

‘No, you oughtn’t: that’s quite another thing! The SONG is called “WAYS AND MEANS”: but that’s only what it’s CALLED, you know!’

‘Well, what IS the song, then?’ said Alice, who was by this time completely bewildered.

‘I was coming to that,’ the Knight said. ‘The song really IS “A-SITTING ON A GATE”: and the tune’s my own invention.’

Posted by: John Baez on August 30, 2007 1:11 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Clearly you skipped some classes. I’ll have to report you to the Wizard!

Hold on, I didn’t quite skip them. I knew somebody who was friends with someone who took notes. And I did glance over these notes.

I do remeber the concept of a name of a morphism. But I didn’t remember the floor notation.

That part from the looking glass you quote is really great. I do remember that, and my chuckling.

Posted by: Urs Schreiber on August 30, 2007 10:08 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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David wrote:

It might also be a good idea to emphasise the point that $V \to X$ , thought of as $* \leftarrow V \to X$ , is a generalised morphism from the point to $X$ …

Right! In other words: what I was calling “silly spans”

                       v
                      / \
                     /   \
                    /     \
                   v       v
                  1         X

actually give operators

$[v]: [1] \to [X]$

but since there’s a god-given isomorphism

$[1] \cong \mathbb{C}$

these are the same as vectors $[v] \in [X]$ . I needed to get ahold of those vectors first, in order to define how any span gives an operator. Afterwards, we can see those vectors as a special case of operators.

So, it’s a case of developing something from scratch, and then looking back at what you’ve done and being able to see it in a more sophisticated way.

… just as we think of an element as a map from the terminal object via its name. Is that where Urs pulled the $\lfloor v\rfloor$ from?

That last sentence is a joke, right? I was desperately groping for some notation for the ‘degroupoidification’ functor, when I realized that given an object $x$ in a groupoid $X$ , the isomorphism class $[x]$ gave a vector in the vector space associated to $X$ … which therefore deserves to be called $[X]$ !

So, I decided to use the symbol $[ \quad ]$ for degroupoidification, and Urs did the same. No $\lfloor v\rfloor$ .

Posted by: John Baez on August 30, 2007 1:11 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

Wow, that’s bizzare - I think there was some kind of rendering fluke and the [ ] looked like the floor symbol yesterday, but today it’s fine.

Posted by: David Roberts on August 31, 2007 2:59 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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the cardinality of the groupoid of finite sets is $e = 2.718281828\dots$

Since we are thinking about groupoids over other groupoids here:

I found it a useful insight that the cardinality of the groupoid of sets over the $n$ -element set is $e^n \,.$

That looks like it might be a useful fact to remember in this business. (At least it is for me. That’s why I am recalling it here, since I had trouble finding the above link.)

Posted by: Urs Schreiber on August 28, 2007 12:44 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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In my 2004 course notes I mention not only the fact you cite — that the groupoid of $n$ -colored finite sets has cardinality $e^n$ — but also that the groupoid of $\frac{1}{2}$ -colored finite sets has cardinality $e^{1/2}$ , and that the groupoid of finite-set-colored finite sets has cardinality $e^e \approx 15.154$ .

These are cute examples of a more general theory, explained in those notes.

Posted by: John Baez on August 28, 2007 7:33 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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In my 2004 course notes I mention not only the fact you cite

Ah, thanks, wasn’t aware of that.

Posted by: Urs Schreiber on August 28, 2007 7:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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So Lie $n$ -groupoids come in symplectic flavour. Do they come in other flavours à la Arnold?

Posted by: David Corfield on August 28, 2007 2:25 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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So Lie $n$ -groupoids come in symplectic flavour.

While I feel quite unsure about symplectic groupoids, I do understand symplectic Lie $n$ -algebroids. These are just “NQP”-manifolds, i.e. graded manifolds with a nilpotent odd derivative and with a compatible graded non-degenerate Poisson structure.

We know that without the P, these NQ manifolds are nothing but Lie $n$ -algebroids with weak Jacobi identity (partly because we really know, partly by definition).

I am still hoping that adding the “P” to these is nothing but also weakening the skew symmetry.

But maybe I should explain in more detail why I think so:

recall why it is that a Courant algebroid comes from a symplectic structure on a Lie algebroid:

consider the simple example where we live over a point, have just an ordinary Lie algebra $g$ in degree 1, and $\mathbb{R}$ -woth of 2-ismorphisms on each such 1-morphism $\wedge^\bullet (s g^* \oplus ss \mathbb{R}^*) \,.$

Let $\{t^a\}$ be a basis of $s g^*$ . Then a symplectic structure on this graded manifold is $\omega = \omega_{ab} d t^a \wedge d t^b \,,$ where $d t^a$ denotes the graded exterior differential of the graded “coordinate” $t^a$ .

The point is that with $t^a$ odd, $d t^a$ is even and hence this $\omega$ defines a symmetric bilinear structure on $g$ !

Also, $\omega$ is required to be compatible with the differential (the “Q”). This makes it precisely an invariant degree 2 polynomial on the Lie algebra.

This way we find that the Courant algebroid over the point comes from a Lie algebra together with a bilinear invariant form on it.

In fact, in general, for $g_{(n)}$ any Lie $n$ -algebroid, the symplectic structure should be nothing but a closed bilinear element in $\mathrm{inn}(g_{(n)})^*$ , otherwise known as the odd tangent bundle.

So we have a construction precisely as for Chern Lie $(2n+1)$ -algebras, only that there this bilinear piece is regarded as a higher coherence of the Jacobiator.

I expect we can just reinterpret this Jacobi failure as a skew failure. Notice that the skew-symmetrizator $S : [x,y] \stackrel{\sim}{\to} - [y,x]$ does send two objects to an isomorphism starting at them.

So, in the dual picture, this could come from a degree -2 map $S^*$ which maps the canonical basis $\{b\}$ of $s s \mathbb{R}^*$ to the “symplectic form” $S^* : b \mapsto \omega_{ab} d t^a \wedge d t ^b \,.$

Indeed, that symplectic form is usually required to be of degree -2, in the BV context.

Well, sorry if that sounds incoherent. If only I were not busy with something else, I would sit down and try to clear this up. But maybe with these hints somebody else can see the Fata Morgana that I am talking about

Posted by: Urs Schreiber on August 28, 2007 3:11 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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So, I am thinking a weak Lie $n$ -algebra, with both Jacobi and skew-symmetry weakend, should be something like this, dually:

A free graded commutative algebra $g^* = \wedge^\bullet ( s V^*)$ generated in degrees 1 to $n$ , together with

- a degree +1 graded differential $d$ squaring to 0, $d^2 = 0$

(that’s the known part, applying to strict skew symmetry)

- together with a degree +2 differential $p$ on $\mathrm{inn}(g)^* = (\wedge^\bullet (s V^* \oplus s s V^*), d_{\mathrm{inn}(g)} )$ which commutes with $d_{\mathrm{inn}(g)}$ : $[p,d_{\mathrm{inn}(g)}] = 0 \,.$

The basic example would be a skeletal Lie 2-algebra whose space of objects is just a Lie algebra $g$ and which has a 1-dimensional space of automorphisms on each object. Everything is strict, and we have $[x,y] = - [y,x]$ but still there is a nontrivial skew-symmetrizor automorphism $S_{x,y} : [x,y] \stackrel{\sim}{\to} - [y,x]= [x,y] \,.$

In a basis $\{t_a\}$ of $g$ this $S$ has components $k_{a b}$ mapping the second power of the space of objects to the (1-dimensional) space of morphisms starting at the Lie bracket of these objects.

There will be some coherence conditions, which should force this $k_{a b}$ to be

a) symmetric

b) ad-invariant .

More concretly, dually, the graded-commutative algebra is $\wedge^\bullet (s g^* \oplus s s \mathbb{R}^*)$ and the dual incarnation of $S$ is $p : b \mapsto k_{a b} d t^a \wedge d t^b \,,$ which, recall, is supposed to take place in $\mathrm{inn}(\cdot)$ of our Lie 2-algebra.

So, for $\{b\}$ the canonical basis of $s s \mathbb{R}$ , if we take $d b = 0 \,,$ which corresponds to the statement that the Jacobiator vanishes, then we find that $[p,d_{\mathrm{inn}(\cdot)}] = 0$ implies that $k_{a b} d t^a \wedge d t^b$ is $d_{\mathrm{inn}(\cdot)}$ -closed, which says precisely that $k_{a b}$ are the components of a degree 2 invariant polynomial on $g$ (see slides 117-140 of my talk for what’s going on here).

So, this Lie 2-algebra with strict Jacobi and weak skew symmetry happens to define precisely the same information as the Chern Lie 3-algebra (slides 151-155), which is a Lie 3-algebra with strict skew-symmetry but nontrivial coherence for the Jacobi identity in top degree.

Since the invariant polynomial $k_{a b}$ is necessarily transgressive, in turn the same information (this phrase “the same information” is the main thing that needs to be made precise, I think) as the corresponding weak Baez-Crans Lie 2-algebra called $g_k$ (slides 146-150) which has strict skew symmetry but where the Jacobiator is the Lie algebra cocycle built from $k_{a b}$ by transgression.

So, I think this goes in the right direction. But be careful, this is just a suggestion.

Posted by: Urs Schreiber on August 28, 2007 4:20 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

What relations should $k_{ab}$ satisfy?

Try calling it something other than weak - since it is still strictly skew-symmetric.

Posted by: jim stasheff on August 31, 2007 2:23 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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What relations should $k_{ab}$ satisfy?

Precisely that it be ad-invariant.

This is supposed to be the implication of imposing the obvious coherence condition which involves both the skew-symmetrizator and the Jacobiator.

This is Dmitry Roytenberg’s observation.

Unfortunately, he didn’t really go into any details in his talk in Vienna. But that’s the idea: like the weakening of the Jacobi identity gives higher antisymmetric brackets, the weakening of the skew symmetry gives higher symmetric brackets.

Try calling it something other than weak - since it is still strictly skew-symmetric.

Okay, I can try. But I am not sure what good established terminology would be.

This is related to the weakening of the Jacobi identity for the Baez-Crans type Lie $n$ -algebras:

all of them come from Lie algebras which perfectly satisfy the Jacobi identity. Still, there is a nontrivial Jacobiator around (for $n=2$ ) or even a trivial Jacobiator, but with a nontrivial coherence law somewhere in degree $n-1$ (for $n \gt 2$ ).

Same here: the bracket itself is skew symmetric, and still there may be a nontrivial “skew-symmetrizator” around.

I am not sure what the best words to describe such a situation are.

Posted by: Urs Schreiber on August 31, 2007 3:01 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

I hope Dmitry is keeping up with this.
In terms of relations, I was wondering about
k_ab k_bc
but also as you indicate,
even if k_ab = \pm k_ba
there is a higher order symmetrizer up to homtopy?

Posted by: jim stasheff on September 1, 2007 2:39 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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but also as you indicate, even if $k_{ab} = \pm k_{ba}$

I should emphasize the following:

in the setup which I am thinking of here, we’d still work

- in the $L_{\infty}$ picture with the ordinary graded-commutative coalgebra $S^c (s V)$

- or dually in the qDGCA picture with the ordinary graded commutative algebra $\wedge^\bullet (s V^*)$

and hence these objects like $k_{a b}$ etc. would always be graded symmetric.

What is no longer strictly graded symmetric is just the Lie $n$ -algebra encoded by the qDGCA equipped with a “graded symplectic form”, i.e. the linear $n$ -category $L$ and its bracket $n$ -functor $[\cdot, \cdot] : L \times L \to L \,.$

That was at least my idea (which may be wrong): that an $n$ -vector space $L$ (an $n$ -term chain complex) together with a bracket functor $[\cdot,\cdot]$ which satisfies a Joacobi identity up to coherent isomorphism and which is skew symmetric up to coherent ismorphism is equivalencly encoded in an “NQP”-manifold over a point.

So I am imagining the standard setup which we are thinking of all along, i.e. a graded commutative (co)algebra with (co)differential, but with one additional piece of data (the graded symplectic form “P”) on top of the odd differential.

Posted by: Urs Schreiber on September 3, 2007 4:39 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

NPQ - not the most charming terminology
and shall we go on to RST? ;-)
oh, right - we already have BRST

Posted by: jim stasheff on August 31, 2007 2:19 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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NPQ - not the most charming terminology

In this context, people say, with a perfectly straight face, things like

The AKSZ-BV formalism solves the BV master constraint given the data of two NPQ manifolds $\mathrm{par}$ and $\mathrm{tar}$ .

By the way, I expect in the end this should be just the differential version of

We may quantize the $n$ -particle modeled by an $n$ -groupoid $\mathrm{Par}$ and propagating on an $m$ -groupoid $\mathrm{Tar}$ .

Despite your warning (which, admittedly, I didn’t really understand, maybe you could expand on it?) I am still hoping, for reasons described here, that the “P” in “NPQ” may simply be absorbed in a weakening of the skew symnmetry of Lie $n$ -algebroid, like the “Q” is absorbed in the weakening of the Jacobi identity.

If true (I might be very wrong, but won’t give up on this idea until somebody disabuses me of it :-), this would essentially say that the kinetic part of the action is already encoded in the $n$ -groupoid structure we put on target space.

That would nicely match with the general idea I had expressed here, here and here.

But possibly I am just hallucinating. I will try to think more about this.

Posted by: Urs Schreiber on August 31, 2007 2:50 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

At some point won’t you need to find an analogue of phased sets for groupoids? Or has this been done already?

Then you could have things like spans of phased groupoids as unitary operators.

Posted by: David Corfield on September 4, 2007 12:26 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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At some point won’t you need to find an analogue of phased sets for groupoids? Or has this been done already?

I was thinking that a $G$ -phased groupoid is a groupoid over $\mathrm{INN}(G)$ .

First of all, a discrete groupoid over $\mathrm{INN}(G)$ is a $G$ -phased set. Then, if two objects in the groupoid are related by the action of $g \in G$ , then their $G$ -phase should differ by that $g$ . That’s exactly what happens for groupoids over $\mathrm{INN}(G)$ .

I am enjoying the following combination of things I have been thinking about in the context of tangent categories and $n$ -curvature, and the Tale of Groupoidification:

we may start with a $G$ -bundle with connection, presented by a functor $\mathrm{tra} : P_1(X) \to \Sigma G \,.$ But we then realize that we rather ought to be looking at its curvature $\delta \mathrm{tra} : \Pi_2(X) \to \mathrm{Grp}\downarrow \Sigma G \,.$ This sends each point $x$ in $X$ to the groupoid $\mathrm{INN}(G)$ regarded as a groupoid over $\Sigma G$ . By the Tale, I may think of this as describing a connection on a (trivial) bundle whose fibers are like a certain vector space with a $G$ -action.

(Think for instance about $G = \mathbb{Z}_n$ as an approximation to $G = U(1)$ ).

What’s interesting now is that a section of $\delta \mathrm{tra}$ thought of as a morphism into $\delta \mathrm{tra}$ is

- over each point $x\in X$ a groupoid over $\mathrm{INN}(G)$ , hence a “ $G$ -phase groupoid”.

Again, if I translate this to the Tale using $G = \mathbb{Z}_n$ it seems to say that over each point the section is (the approximation to) a complex number!

So it looks like by combining the curvature technique with the tale, we automatically turn principal $U(1)$ -bundles into line bundles, whose sections are, locally, complex numbers over each point.

Posted by: Urs Schreiber on September 4, 2007 12:53 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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So how does the counting up go? In the case of phased sets, we add up the phases. This I guess corresponds to mapping the phased set to {{*}, 0}, and taking the cardinality of the phased groupoid (in this case phased set) in the fibre above {*}. What if the target phased set were {{*}, c} for nonzero $c$ ?

What now happens with a phased groupoid? How do we add up the phases in, say, a $U(1)$ -phased groupoid?

Posted by: David Corfield on September 4, 2007 9:02 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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David Corfield wrote:

At some point won’t you need to find an analogue of phased sets for groupoids? Or has this been done already?

Jeffrey Morton introduced phased groupoids under the name of ‘ $\mathrm{U}(1)$ –groupoids’ in his paper Categorified algebra and quantum mechanics — see Section 6.1.2, page 46.

Then you could have things like spans of phased groupoids as unitary operators.

Exactly!

As you know, Jim Dolan and I introduced a theory of ‘stuff types’ generalizing Joyal’s ‘species’ so we could categorify the quantum harmonic oscillator and the theory of Feynman diagrams. The big problem with our work was that it didn’t include the phases needed to describe the unitary time evolution of the free harmonic oscillator!

This is what Jeffrey did in his paper, using ‘ $\mathrm{U}(1)$ -stuff types’ and ‘ $\mathrm{U}(1)$ -stuff operators’.

Everyone should read this paper! It explains this business very gently, with lots of pictures. I’ll give an ultra-terse summary now, but only to tempt people into reading the real story.

Now: having decided that the term ‘ $\mathrm{U}(1)$ -set’ is a really lousy term for a set equipped with a map to $\mathrm{U}(1)$ , and having switched to calling such a thing a phased set, let’s be consistent in adopting this new style of terminology.

So: we hereby define a phased groupoid to be a groupoid equipped with a functor to the groupoid with $\mathrm{U}(1)$ as its set of objects and only identity morphisms.

In other words: a phased groupoid is a groupoid such that each object is labelled by a phase (an element of $\mathrm{U}(1)$ ), and isomorphic objects are labelled by the same phase.

A phased set is then a phased groupoid with only identity morphisms.

A phased stuff type is a phased groupoid whose underlying groupoid is equipped with a functor to the groupoid of finite sets.

Jeffrey shows how every phased stuff type gives a (possibly non-normalizable) state of the quantum harmonic oscillator.

A phased stuff operator is a phased groupoid whose underlying groupoid G is part of a span:

                       G
                      / \
                     /   \
                    /     \
                   v       v
                  B         B

where B is the groupoid of finite sets.

Jeffrey shows how every phased stuff operator gives a (possibly unbounded) operator on the Hilbert space for the quantum harmonic oscillator.

And using this, he shows how to categorify the theory of Feynman diagrams as a tool for computing time evolution of the perturbed quantum harmonic oscillator!

I hope you see that this is all part of the Tale of Groupoidification…

So how does the counting up go? In the case of phased sets, we add up the phases.

Right — you’ll find this idea nicely explained in Feynman’s popular book QED. A set of possible ways for a quantum system to get from here to there is really a phased set. To compute the amplitude for it to get from here to there, we just compute the cardinality of the phased set — that is, add up the phases. The cardinality of a finite phased set can be any complex number.

But, this generalizes nicely to phased groupoids.

How do we add up the phases in, say, a $\mathrm{U}(1)$ -phased groupoid?

I’m sure you can guess! Or, read the material in Jeff’s paper leading up to the answer in Section 6.2.1, page 50.

Posted by: John Baez on September 4, 2007 11:08 PM | Permalink | Reply to this

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I hadn’t noticed that part of Jeff’s paper. So what are we to make of the difference between your and Urs’ answer to my question about phased groupoids? Are they just constructions for different purposes.

Another thing, and I ought to work this out myself, but I’m not quite clear on the idea of a vector [V] in [X]. Let’s pick some examples and denote by n the cyclic group of order $n$ , and look at some maps from n to m. Then there’s only one object in the codomain, so we’re just looking for a single coefficient.

a) 2 $\to$ 2, identity map.

b) 2 $\to$ 2, trivial map.

c) 1 $\to$ 2.

I should do the exercise you left:

define the morphisms in the essential preimage.

You even asked us twice to do this.

Posted by: David Corfield on September 5, 2007 11:49 AM | Permalink | Reply to this

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So what are we to make of the difference between your and Urs’ answer

John said: “groupoids over $\mathrm{Disc}(G)$ ”, whereas I talked about “groupoids over $G // G$ ”.

To avoid misunderstandings let me emphasize that my reply was partly wishful thinking: I said that if a $G$ -phased groupoid is a groupoid over $G // G$ and if $U(1)$ -phased groupoids are a good substitute for complex numbers, then some puzzle pieces I am holding in my hands would beautifully fit together.

Now, John’s reply doesn’t quite support this hope. So next I am hoping that there is a subtlety, yet to be unravelled, which does justify both points of view. :-)

Posted by: Urs Schreiber on September 5, 2007 12:30 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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David wrote:

I hadn’t noticed that part of Jeff’s paper.

That’s the only part that contains brand new ideas! The rest is a detailed working-out of material presented in the Fall 2003, Winter 2004, and Spring 2004 seminars at UCR.

If you didn’t notice the new stuff, I wonder if anyone else did. I know Urs did. So, the paper had at least 4 readers: Jeff, me, Urs, and the referee.

His paper is quite long, and all the truly new stuff — introducing phases into categorified quantum mechanics — comes after page 43. Maybe it should have been split into two papers.

On the other hand, even the old stuff had never been worked out in detail or published anywhere, except for a rudimentary sketch where Feynman diagrams appear briefly on the last page. So, maybe it was just your familiarity with the old stuff that made you give up before getting to the new stuff?

Writing papers is a tricky business if you actually want people to read them.

Another thing, and I ought to work this out myself, but I’m not quite clear on the idea of a vector $[V]$ in $[X]$ .

Oh? You used to love this idea back when $X$ was the groupoid of finite sets.

In that case, a groupoid $V$ over $X$ was what we called a stuff type. The vector space $[X]$ was the algebra of polynomials

$\{ \sum_{n \ge 0} a_n z^n \; : \; a_n \ge 0 \}$

since it has one basis element $z^n$ for each isomorphism class of finite set. The vector $[V] \in [X]$ was called the generating function of the stuff type $V$ .

(Pesky technical note: we should complete $[X]$ a bit, to get the algebra of formal power series, if we want it to include the generating function of $V$ when $V$ is not a finite groupoid.)

So, unless I succeeded in baffling you by moving up to a higher level of generality, you should certainly still appreciate this special case!

You should also like the case where $X$ is a symmetric group $S_n$ viewed as a one-object groupoid. Then up to equivalence we can think of $X$ as the groupoid of $n$ -element sets. A groupoid $V$ over $X$ should be thought as a groupoid of $n$ -element sets ‘equipped with extra stuff’.

In this case $[X]$ is 1-dimensional, and the size of $[V]$ is proportional to the cardinality of the groupoid $V$ .

Let’s pick some examples and denote by $\mathbf{n}$ the cyclic group of order $n$ ,…

Okay — now you’re taking the groupoid $X = \mathbf{n}$ to be the cyclic group of order $n$ , viewed as a one-object groupoid. You could equivalently think of $X$ as the groupoid of cyclically ordered $n$ -element sets.

A groupoid $V$ over $X$ is then a groupoid of cyclically ordered $n$ -element sets ‘equipped with extra stuff’. You give three examples of these:

a) $\mathbf{2} \to \mathbf{2}$ identity map.

b) $\mathbf{2}\to \mathbf{2}$ , trivial map.

c) $\mathbf{1} \to \mathbf{2}$

Here’s a little puzzle just to make sure our brains are oiled and running smoothly: in each case, what extra stuff do we have on our cyclically ordered 2-element set? I’ll do case a). In this case, there’s no extra stuff at all! I.e. we’re dealing with the wimpiest possible case of extra stuff, a ‘tautologously true property’.

I’ll let you — or other volunteers! — do the other two.

By the way: what’s a bit funny about ‘cyclically ordered $n$ -element sets’ when $n = 2$ ?

Posted by: John Baez on September 5, 2007 12:45 PM | Permalink | Reply to this

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Here’s a little puzzle […]

I’ll let you – or other volunteers! – do the other two.

Let me see if I understand. For case b):

First, I’ll try to recall the definitions: we say that the functor $\mathbf{2} \to \mathbf{2}$ forgets at most stuff, I hope, if it is essentially surjective and full, but not necessarily faithful. The idea being that morphisms upstairs are like morphisms downstairs plus some extra data which tells us how the extra stuff transforms.

So, then, I am now supposed to be looking at the trivial functor $\mathbf{2} \to \mathbf{2} \,.$ Personally I like to denote this $\Sigma \mathbb{Z}_2 \to \Sigma \mathbb{Z}_2 \,,$ but that’s my problem, not yours ;-).

Anyway, since it is supposed to be trivial, I should concentrate on understanding $\mathbf{2} \to \{\bullet\}$ first. That’s clearly full. The morphisms which are being forgotten are the automorphisms of the 2-element set, or keeping the question

what’s a bit funny about ‘cyclically ordered $n$ -element sets’ when $n=2$ ?

in mind, equivalently the automorphisms of the cyclically ordered 2-element set.

So I conclude: the trivial functor $\mathbf{2} \to \mathbf{2}$ describes the one-element set equipped with a two element set. The extra stuff is the two element set.

So it describes a funny shift in perspective: instead of regarding $\{a,b\}$ as a two-element set, we regard it as the one-element set equipped with extra data, where the extra data is a 2-element set!

Hope that’s right.

As for c): this doesn’t forget anything, I think, hence in particular describes no extra stuff.

Posted by: Urs Schreiber on September 5, 2007 1:34 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Oh, good — a volunteer. Note how David hangs back while Urs blithely steps forward.

Urs wrote:

First, I’ll try to recall the definitions: we say that the functor $\mathbf{2} \to \mathbf{2}$ forgets at most stuff, I hope, if it is essentially surjective and full, but not necessarily faithful.

No, alas!

Every functor between categories forgets at most stuff. For mere 1-categories, stuff is the most serious thing there is to forget!

We say a functor:

forgets nothing if it’s faithful, full and essentially surjective;
forgets at most properties if it’s faithful and full;
forgets at most structure if it’s faithful;
forget at most stuff in every case.

In case a), we had the identity functor

$1: \mathbf{2} \to \mathbf{2}$

This is an equivalence — so, it’s faithful, full and essentially surjective. So, it forgets nothing.

That should seem reasonable. If we think of $\mathbf{2}$ as the groupoid of 2-element sets, this functor takes a 2-element and set and does nothing to it.

The idea being that morphisms upstairs are like morphisms downstairs plus some extra data which tells us how the extra stuff transforms.

Right. The morphisms upstairs really involve ‘extra data’ compared to the morphisms downstairs when our functor fails to be faithful. Otherwise, if our functor is faithful, we say it forgets at most structure.

Example: the forgetful functor from groups to sets forgets at most structure. But the the forgetful functor from pairs of sets to sets, which throws out the second set in the pair, forgets stuff. What stuff? The second set!

Now for example b):

So, then, I am now supposed to be looking at the trivial functor $triv: \mathbf{2} \to \mathbf{2} \,.$

Okay.

Anyway, since it is supposed to be trivial, I should concentrate on understanding $\mathbf{2} \to \mathbf{1}$ first. That’s clearly full.

I’ve taken the liberty of renaming the trivial groupoid $\mathbf{1}$ here, since that’s what it is.

This functor is full and essentially surjective, but not faithful. So, it forgets stuff. What stuff does it forget? Everything, basically.

So I conclude: the trivial functor $triv: \mathbf{2} \to \mathbf{2}$ describes the one-element set equipped with a two element set. The extra stuff is the two element set.

I don’t like this sentence at all! In the forgetful functor game, when you’re given a functor $F: X \to Y$ , you’re suppose to say “an object of $X$ is an object of $Y$ equipped with ?????, which $F$ forgets”. The ????? is some extra property, structure or stuff.

So for this functor:

$triv: \mathbf{2} \to \mathbf{2}$

you’re supposed to say “a 2-element set is a 2-element set equipped with ?????, which $triv$ forgets”.

I think maybe you were thinking about this functor:

$\mathbf{2} \to \mathbf{1} \,$

the unique functor from $\mathbf{2}$ to $\mathbf{1}$ . Then it would make sense to say “a 2-element set is a 1-element set equipped with a 2-element set, which this functor forgets”.

This sounds bizarre, but it’s correct! The point is, having a 1-element set equipped with a 2-element set is just the same as having a 3-element set with one element colored black and two colored red. But, the groupoid of these is equivalent to the groupoid of 2-element sets — since the black element doesn’t have any symmetries.

It sounds less bizarre to consider

$\mathbf{2} \to \mathbf{0} \,$

and say “a 2-element set is a 0-element set equipped with a 2-element set, which this functor forgets”. But, the groupoid of all 0-element sets is equivalent to the groupoid of 1-element sets! So, if this paragraph is true, so are the previous two.

(Answer to previous puzzle: what’s funny about a cyclically ordered $n$ -element set for $n \le 2$ is it’s just the same as a $n$ -element set. So, let’s not worry about the cyclic ordering in this discussion — at least not until we count up to $\mathbf{3}$ .)

Posted by: John Baez on September 5, 2007 4:48 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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I should concentrate on understanding $\mathbf{2} \to \mathbf{1}$ first.

[…]

I think maybe you were thinking about this functor: $\mathbf{2} \to \mathbf{1}$

Yes! :-)

I thought I could get away with understanding just this one.

I didn’t recall that

In the forgetful functor game

(which I had trouble googling the rules for – thanks for the link!)

when you’re given a functor $F : X \to Y$ , you’re suppose to say “an object of $X$ is an object of $Y$ equipped with ?????, which $F$ forgets”.

Okay, I get it. Hence I am

supposed to say “a 2-element set is a 2-element set equipped with ?????, which triv forgets”.

I see.

Hm, the trivial $F$ forgets that I can permute the elements of my 2-element set, since it regards all automorphisms of the 2-element set as identities.

So, somehow the functor forgets that there is more than one two-element set around.

I’ll try this statement:

A two-element set is a two-element set equipped with information about which elements it contains, which the functor $\mathbf{2} \to \mathbf{1} \to \mathbf{2}$ forgets.

Posted by: Urs Schreiber on September 5, 2007 5:59 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Urs wrote:

A two-element set is a two-element set equipped with information about which elements it contains, which the functor $\mathbf{2} \to \mathbf{1} \to \mathbf{2}$ forgets.

That’s pretty close, but I’m not completely happy. Lest you despair, let me reassure you: this is a fairly pathological degenerate case of a forgetful functor! So, the story we need to tell will sound rather odd.

Let’s tell the story one stage at a time — that’s a good strategy. First stage: $\mathbf{2} \to \mathbf{1}$ Someone hands you a 2-element set. You completely forget everything about it and pull a 1-element set out of your pocket.

Second stage: $\mathbf{1} \to \mathbf{2}$ Someone hands you a 1-element set. You completely forget everything about it and pull a 2-element set out of your pocket.

So, composing these: $\mathbf{2} \to \mathbf{1} \to \mathbf{2}$ Someone hands you a 2-element set. You throw it over your shoulder, pull a 1-element set out of your pocket, say “Whoops! Wrong pocket!” , throw it out, and pull a 2-element set of your other pocket.

Now let’s work backwards. I want you to try to describe all these functors as ‘forgetful functors’ $F : X \to Y$ , where a $X$ -object is a $Y$ -object equipped with some extra stuff (or maybe just structure (or maybe just properties (or maybe just nothing at all))), and $F$ forgets this extra stuff.

We did it already for the first stage: $F: \mathbf{2} \to \mathbf{1}$ Here we agreed how to say it:

A 2-element set is the same as a 1-element set equipped with an extra 2-element set with points colored red. The functor $F$ throws out the red points.

Again, this is supposed to sound weird! I’m saying a 2-element set “is the same as” a 3-element set with 1 ordinary black point and 2 red points. What the hell do I mean!? Don’t worry, I’m not crazy: I mean the groupoid of 2-element sets and bijections is equivalent to 3-element sets colored this way, and color-preserving bijections.

Next, the second stage: $G: \mathbf{1} \to \mathbf{2}$ Here’s how I’d say it:

A 1-element set is the same as a 2-element set equipped with a labelling of its points and an extra 1-element set with point colored pink. The functor $G$ throws out the pink point.

Note the labelling is extra structure that ‘nails down’ the points of our 2-element set, reducing its symmetry group to the trivial group. The groupoid of 2-element labelled sets is equivalent to the trivial groupoid.

Okay! Doing $F$ and then $G$ , we get

$G F : \mathbf{2} \to \mathbf{1} \to \mathbf{2}$

Can someone fill in the blanks here?

A 2-element set is a 2-element set equipped with ?????. The functor $G F$ does ?????.

I think I’ve done most of the hard work… except for the hard work of understanding what I just said.

Posted by: John Baez on September 5, 2007 8:54 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Amazing how much fun one can have with 2-element sets.

I’ll try again:

G F(2-element set)

a 2-element set is a G(1-element set) equipped with a 2-element set

a 2-element set is a (2-element set equipped with a labelling) equipped with a 2-element set .

The functor GF forgets the 2-element set and just remembers the labelled 2-element set.

Posted by: Urs Schreiber on September 5, 2007 9:43 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Urs wrote:

a 2-element set is a (2-element set equipped with a labelling) equipped with a 2-element set.

The functor GF forgets the 2-element set and just remembers the labelled 2-element set.

Right! As you can see, this particular example is one of those things that’s trivial in such a convoluted way that takes a Ph.D. to understand it — or more precisely, to give a damn about it.

There are many more interesting functors from the groupoid of $n$ -element sets to the groupoid of $m$ -element sets, which have more interesting interpretations as ‘forgetting extra stuff’.

Posted by: John Baez on September 5, 2007 10:40 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Writing papers is a tricky business if you actually want people to read them.

On the other hand – I learned this the hard way – one cannot expect to say something as visionary as Jeffrey does in his paper, which supposedly tackles the very roots of people’s thinking about a well-known subject like quantum mechanics, and get lots of people interested in it. Good exposition can go a long way, but this is quite a hurdle.

We have this problem already among our little $n$ -Café-community here: based on Jeff’s ideas I came up with this which, if it proves to be on the right track, looks like it has the chance of being the deepest observation I’ll ever come up with in my life. Despite my fondness of this idea, and its predecessors, it is among those with the least comments here on the $n$ -Café.

Now, I am not the expositor you are, and didn’t take the time to try my best. But with ideas of this kind it takes a while. I guess.

In case you are wondering what I am talking about: I would like to see a theory of $U(1)$ -bundles with connection in terms of functors to $\Sigma U(1)$ , or 2-functors to $\Sigma \mathrm{INN}(U(1))$ whose natural notion of section automatically leads to the fact that fibers are combinatorial vector spaces of the kind appearing in the Tale, with a section being a combinatorial vector inside each of these.

That’s why I keep going on about groupoids over $\mathrm{INN}(U(1))$ . Because these I can nicely see arise as sections of $U(1)$ -bundles.

But I might be missing something. This requires further thinking.

Posted by: Urs Schreiber on September 5, 2007 2:05 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

maybe it was just your familiarity with the old stuff that made you give up before getting to the new stuff?

Yes, that’s right. That’s not obvious how to signal where the new stuff begins, when the old stuff is being written up as a paper for the first time. I suppose this only affects those who knew about the old stuff from the seminar, but then that may be a substantial part of the audience.

Posted by: David Corfield on September 5, 2007 2:34 PM | Permalink | Reply to this

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I suppose Jeff could say in his abstract: and in the final, visionary new part of this paper, we introduce PHASES into the categorified version of quantum mechanics!

But that would merely rack up points on the crackpot index.

Posted by: John Baez on September 5, 2007 4:03 PM | Permalink | Reply to this

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So, the paper had at least 4 readers: Jeff, me, Urs, and the referee.

I read it! I really enjoyed it, and I might not be doing what I’m doing now if I hadn’t read it. I did find it hard to tell what was original stuff and what wasn’t, but that didn’t really bother me, since it was all new to me.

Posted by: Jamie Vicary on September 8, 2007 10:26 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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I am wondering about that business of integration of Lie $n$ -algebroids.

For the record, I’ll first recall how it works for $n=1$ . Then I remark on the higher $n$ -case (in the form of a question).

Integrating Lie 1-Algebroids

I haven’t tried to check who first came up with this, but here is the standard way to integrate a Lie 1-algebroid to a Lie groupoid:

Let $I = [0,1]$ be the interval, let $T I$ denote the Lie algebroid of its pair groupoid (the tangent algebroid) and let $A$ denote the Lie 1-algebroid in question to be integrated.

Then, we first build a Lie 2-groupoid $G$ as follows:

its space of objects is the space underlying $A$ .

a morphism in $G$ is a morphism of Lie algebroids $\gamma : T I \to A$ composition is the obvious one induced by the (by any) diffeomorphism $[0,1] \to [0,1] \cup [0,1]$

(this is associative only up to 2-morphisms!)

for $I^2/\sim$ the standard bigon (square with vertical edges shrunk to a point) a 2-isomorphism in $G$ between two paths as above exists if there is an algebroid morphism $T I^2/\sim \to A$ which restricts to the given paths on top and bottom, respectively.

Source and target are the obvious ones. Composition in various directions comes from the obvious composition of the standard bigon.

The point then is, which John is alluding to in his TWF above:

we can in general imagine diving out 2-isomorphisms in $G$ , in order to be left with a mere 1-groupoid. But in some cases it may happen that this quotient is no loger sufficiently well behaved (not a manifold, in particular). So in some cases this leaves us with a 2-groupoid. Or a stacky 1-groupoid. Or things like that.

Notice how this method sneakily captures the ordinary integration of a Lie algebra to a Lie group:

let $A$ be a Lie 1-algebroid over a point, hence a Lie algebra $g$ . Then a Lie algebroid morphism $T I \to A$ is nothing but a $g$ -valued 1-form on $I$ .

When I first learned about this, I was confused by the fact that this morphism itself is now already regarded as a representative of a group element. No path “ordered exponential” is involved!

The trick is this: by the nonabelian Stokes’ theorem, the identitfication of two such elements via an interpolating morphism $T I^2/\sim \to A$ says precisely that two such 1-forms on the interval are identified if there is a flat $g$ -connection on the disk interpolating between them. But Stokes’s theorem says that precisely in this case do the path ordered exponentials of the given 1-forms coincide!

So these exponentials don’t appear explicitly, but the equivalence relation is exactly such that they would be respected had we computed them.

In a way this method is very close to my heart: it says in effect that every Lie groupoid is best thought of as the image of a parallel transport.

Generalization to higher $n$

I know what Lie $n$ -algebroids are. I know what morphisms of them are. I know how to generalize all the above constructions accordingly. It’s pretty obvious.

It seems that the only issue one runs into is: one needs to fix some notion of weak $(n+1)$ -groupoid.

(If we restrict to Lie $n$ -algebroids coming from “NQ”-manifolds then it even suffices to assume that we have strict inverses in the Lie $n$ -groupoid, I think. Only associativity will be weakened.)

For instance, I’d be surprised if for any Lie 2-algebroid, the obvious generalization of the above procedure wouldn’t yield a tricategory. Of course one would have to check.

In any cas,e it seems that the globular model is the best suited one. If I were to ask about a nice globular model for weak $n$ -groupoids with strict inverses. What would you point me to?

$\mathrm{inn}$ again

And then, I am of course wondering about this: since I am claiming that instead of worrying about any old Lie $n$ -algebroid $A$ , we really mostly just want to be looking at the Lie $(n+1)$ -algebroid $\mathrm{inn}(A)$ :

is there any chance that in this case we never run into the issue of quotients failing to be smooth?

That might actually be an “explanation” of that funny shift in dimension.

Posted by: Urs Schreiber on September 4, 2007 2:06 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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And, by the way, concerning the smooth structure when integrating Lie 1-algebroids:

there is the standard Chen smooth structure on paths $\gamma : T I \to A \,.$ And any quotient of a Chen-smooth space has a canonical Chen-smooth structure: a plot is a map that factors through a plot of the larger space.

Composition in the resulting structure is also manifestly smooth in this sense: this is essentially just the same kind of consideration as in the construction of the path groupoid $P_1(X)$ .

Doesn’t that seem to indicate that every Lie 1-algebroid integrates to a Chen-smooth 1-groupoid?

Posted by: Urs Schreiber on September 4, 2007 2:27 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

Urs wrote:

Doesn’t that seem to indicate that every Lie 1-algebroid integrates to a Chen-smooth 1-groupoid?

I asked Chenchang Zhu and she seemed to think so. I gave her a bunch of reading material on Chen’s smooth spaces, and she seemed pretty interested in it. So, I’m hoping she’ll work this stuff out.

By ‘work this stuff out’, I don’t mean just getting a Chen-smooth 1-groupoid from a Lie 1-algebroid, which seems likely to be pretty easy. The hard part is checking that this is ‘the right answer’ to the problem of integrating Lie algebroids to get smooth groupids. And the problem is figuring out what counts as ‘the right answer’.

I had wanted to let Chenchang work on this without the pressure of publicity. That’s why you need to read between the lines of ‘week256’ to see that I already discussed this issue:

But getting from Lie algebroids to Lie groupoids is harder… in fact, according to the standard definitions, it’s often impossible!

Luckily, one of the people who really understands this stuff was at this conference in Vienna - Chenchang Zhu.

I’m optimistic that the patterns will be very beautiful when we fully understand them.

(By the ‘standard definitions’ I mainly meant the definition of Lie groupoids using manifolds instead of Chen-smooth spaces.)

But now that you’ve broached the problem publicly, I should publicly say that I hope Chenchang solves this problem!

Posted by: John Baez on September 4, 2007 11:38 PM | Permalink | Reply to this

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I had wanted to let Chenchang work on this without the pressure of publicity.

[…]

But now that you’ve broached the problem publicly, I should publicly say that I hope Chenchang solves this problem!

I have to apologize here for being so insensitive. Sorry. I didn’t mean to give something away which was supposed to be secret.

What happened was that I got excited about the idea (which only struck me yesterday) that the method of integrating a Lie 1-algebroid by mapping paths into it has a straightforward generalization to Lie $n$ -algebroids.

For instance it seems to be clear how to proceed with the skeletal String Lie 2-algebra this way.

That’s exciting, because, the way the construction works, it seems to allow to characterize weak parallel transport 2-functors with values in a weak Lie 2-groupoid integrating the skeletal Lie 2-algebra: by the way the integration process works, we’d seem to get the statement that these 2-functors come from a 1-form $A \in \Omega^1(X,g)$ and a 2-form $B \in \Omega^2(X)$ such that $d B = \langle A \wedge [A\wedge A]\rangle$ essentially tautologously for free! (Since the 2-groupoid is essentially already defined as the image of a parallel transport)

That’s really the point I wanted to mention. Then I thought I’d put this into context by quickly reviewing the construction for $n=1$ , to a good deal just recalling what we talked about in Vienna, plus adding these simple observations about Chen smoothness.

Anyway. I hope, too, that Chenchang figures out if the “obvious” Chen-smooth structure is actually the “right” one. All my best wishes to her!

I believe I will gamble that the answers will turn out to be “Yes!” and go ahead looking at weak parallel $n$ -transport this way. I have been hoping to be able to do this for quite a while now!

As I mentioned, there is one aspect which I need to better understand: it seems that this way of integrating Lie $n$ -algebroids nicely harmonized with globular $n$ -categories. Which is nice. Only problem for me is that I am not sure I can handle weak globular $n$ -groupoids far beyond $n=2$ . That’s why I was asking for advice.

(On the other hand, $n=2$ , is already sufficient here for some interesting constructions.)

Posted by: Urs Schreiber on September 5, 2007 3:34 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

It’s too bad so many replies get lumped here without distinction as to sub threads.

Urs writes: the 2-groupoid is essentially already defined as the image of a parallel transport.

Think about this for n=1. Integrating a Lie algebra to a Lie groupoid via parallel transport? How did the Lie algebra give a parallel transport?

Posted by: jim stasheff on September 7, 2007 12:20 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Jim Stasheff wrote:

Think about this for n=1. Integrating a Lie algebra to a Lie groupoid via parallel transport?

No. Integrating a Lie algebroid to a Lie groupoid. The pictures in the introduction to Severa’s paper explain the idea more clearly than a pile of formulas would.

Posted by: John Baez on September 7, 2007 2:29 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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I assume Jim wants to see the ordinary notion of integration of a Lie algebra to a Lie group see reproduced here.

And I think it is very helpful to wonder how the description in terms of algebroid paths has a chance of producing the group one expexts to obtain by more familiar means.

I was wondering about this a lot. Then I understood it. As I wrote above

When I first learned about this, I was confused by the fact that this morphism itself is now already regarded as a representative of a group element. No “path ordered exponential” is involved!

The trick is this: by the nonabelian Stokes’ theorem, the identitfication of two such elements via an interpolating morphism $T I^2/\sim \to A$ says precisely that two such 1-forms on the interval are identified if there is a flat $g$ -connection on the disk interpolating between them. But Stokes’s theorem says that precisely in this case do the path ordered exponentials of the given 1-forms coincide!

So these exponentials don’t appear explicitly, but the equivalence relation is exactly such that they would be respected had we computed them.

For me this is a very important point for understanding what’s going on here.

Posted by: Urs Schreiber on September 7, 2007 2:58 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Urs wrote:

I have to apologize here for being so insensitive. Sorry. I didn’t mean to give something away which was supposed to be secret.

It’s probably no big deal, and if I really wanted it to stay secret I wouldn’t have replied to your comment — it’s not as if many people out there are going to say Yes! Integrating Lie algebroids to get Chen-smooth Lie groupoids! Why didn’t I think of that?!?

What happened was that I got excited about the idea (which only struck me yesterday) that the method of integrating a Lie 1-algebroid by mapping paths into it has a straightforward generalization to Lie $n$ -algebroids.

Interesting.

By the way, this is discussed in the introduction to Severa’s paper — where he also mentions the basic problem of modding out a manifold by a non-closed equivalence relation and getting something that’s not a manifold. This is the problem that Chen-smooth spaces solve. Again, the interesting question is whether this is always the ‘right’ solution.

Posted by: John Baez on September 5, 2007 8:59 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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By the way, this is discussed in the introduction to Severa’s paper

Oh, stupid me. Thanks for pointing that out. He even discusses the String Lie 2-algebra example – quite tersely – on p. 6.

Posted by: Urs Schreiber on September 5, 2007 9:16 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

hi, I indeed think integrate a Lie algebroid one will get a Chen’s Lie 1-groupoid since all the problem comes when one does quotient. Then we want more than this, namely we want the 1-1 correspondence. If we begin with a Chen’s Lie 1-groupoid, we hope to find some sort of Lie 1-albebroid. Therefore, as far as I see, one should also create a Chen’s Lie 1-algebroid. I read some part of John and Urs’s note that John gave to me in Vienna.So I assume a lot of things, for example, vector spaces, vector bundles, tangent spaces, cotangent spaces, etc. can be adapted to Chen’s smooth spaces (am I alright?)

Then it seems that to adapt the definition of Lie algebroid for Chen’s smooth space setting is possible. No locally R^n involved. The next little problem comes when we want to set up the path space and homotopies of path space. Then the usual description of path space and the homotopies involve ODE and PDE which usually demands local R^n. Therefore, we should really just interpret paths as a map from C(A) (the commutative graded ring associated to a (Chen) Lie algebroid A) to C(TI), where TI is the tangent Lie algebroid of the interval I. Similarly we can define homotopies of paths.

Then it’s something John and Urs know better: Are there NQ Chen manifold? Namely does Roytenberg’s super manifolds can base on Chern’s smooth space? (All these coordinates descriptions make me hesitate…, or maybe it’s not that important?)

Posted by: chenchang Zhu on September 6, 2007 2:54 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Hi Chenchang!

So I assume a lot of things, for example, vector spaces, vector bundles, tangent spaces, cotangent spaces, etc. can be adapted to Chen’s smooth spaces (am I alright?)

In principle there is usually a pretty obvious way to generalize all of differential geomety to the Chen-smooth context: you do whatever you want to do on the domain of each plot, and then demand the obvious gluing conditions.

For instance, a differential form on a Chen-smooth space should be a collection of differential forms, one on each domain for each plot of the space, such that whenever one plot is the pullback of another along a smooth map $\mathbb{R}^n \to \mathbb{R}^m$ , the corresponding differential forms are related by this pullback.

While it is rather obvious how to do this in every single case, I am (but that might be just me) not aware of a piece of literature which would spell this out in detail.

But Patrick Iglesias-Zemmour’s work comes pretty close to what one would hope for: if you haven’t yet, take a look at his monograph on

Diffeology

This “diffeology” is – I think only up to minor details about what one allows as domains of plots – essentially the same as the theory of Chen-smooth spaces.

I am not entirely sure what the finalization status of this document is. Apparently it is going to become a book (or already is? I am not sure).

Posted by: Urs Schreiber on September 6, 2007 4:57 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Urs wrote:

Patrick Iglesias-Zemmour’s work comes pretty close to what one would hope for…

So close, and yet so far! There are lots of good ideas in this book, but also an annoying problem.

This “diffeology” is — I think only up to minor details about what one allows as domains of plots — essentially the same as the theory of Chen-smooth spaces.

Alas, while Iglesias-Zemmour proves lots of nice abstract theorems about diffeology, these “minor details” make the theory hard to use in lots of interesting examples.

The problem is that he takes the domains of his “plots” to be copies of $\mathbb{R}^n$ , instead of convex subsets of $\mathbb{R}^n$ as Chen does.

This makes it difficult to equip the closed interval $[0,1]$ with a diffeology such that a smooth function

$f : [0,1] \to \mathbb{R}^n$

is what we normally consider to be a smooth function!

Note: I said “difficult”, not “impossible”. I don’t even know if it’s possible! We discussed this problem before, but we never resolved it. Iglesias-Zemmour’s book doesn’t seem to address this issue. There’s an obvious diffeology on $[0,1]$ , which he does discuss — but I can’t tell if the function

$\sqrt{} : [0,1] \to \mathbb{R}$

is smooth with respect to this diffeology! It’s not smooth in the usual sense, since the derivative blows up at $0$ . But, I can’t tell whether it’s smooth in Iglesias-Zemmour’s sense. It’s tricky.

And the sad thing is: none of this stuff needs to be difficult! The whole problem disappears in Chen’s approach. In Chen’s approach, there’s a smooth structure on $[0,1]$ such that the identity map $1: [0,1] \to [0,1]$ is a chart. With this smooth structure, it’s trivial to check that a function

$f : [0,1] \to \mathbb{R}^n$

is smooth in Chen’s sense iff it’s smooth in the usual sense: all the $n$ th derivatives $f^{(n)}(x)$ exist and are continuous, converging to the one-sided $n$ th derivative as $x \to 0$ or $x \to 1$ .

My remarks about $[0,1]$ also apply to all manifolds with boundary, or manifolds with corners like a cube or simplex. They’re all easy to deal with in Chen’s approach, but hard using diffeology.

I wish Iglesias-Zemmour could rewrite his manuscript to address this issue. Otherwise someone else will need to tackle it someday. The world needs a good theory of smooth spaces, with all the basic theorems written up in a nice textbook.

Posted by: John Baez on September 6, 2007 6:35 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Thanks for emphasizing this big implication of the “minor” difference in the definition of the plots.

On the other hand, the difference is minor in the following sense:

whenever you wonder how gadget X from differential geometry (vector, form, bundle, etc…) would be definined in the context of Chen-smooth spaces, simply go to the Diffeology book, and see how X is done there. Copy-and-paste the definition and insert your own macro for the definition of domains of plots. Done.

But then, as I said, it should in all these cases also already be easy to figure out the right answer directly: simply take the usual local description of X over open subsets, then replace open subsets by plots.

Hm, I am always writing comments that are too long! All I want to say is, I guess:

there is no literature on general Chen-smooth differential geometry, but it is a straightforward exercise to invent everything yourself!

:-)

(In fact, I do have old notes on this. But – for a change – I don’t want to share them here. )

Posted by: Urs Schreiber on September 6, 2007 7:17 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Urs wrote:

there is no literature on general Chen-smooth differential geometry, but it is a straightforward exercise to invent everything yourself!

Right. It’s just sad that there already exists a book of theorems one could simply quote, if they’d been based on slightly different definitions!

Posted by: John Baez on September 6, 2007 9:22 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Did you ever get into contact with Patrick Iglesias-Zemmour?

Maybe he would appreciate learning that there are a bunch of people here who’d happily buy and promote his product if only it had a little knob exchanged.

Posted by: Urs Schreiber on September 6, 2007 10:08 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

No, I’ve never contacted Iglesias-Zemmour. I’ll try it now.

Posted by: John Baez on September 7, 2007 1:06 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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I got a nice email from Anders Kock in which he explains that with the diffeology that $[0,\infty)$ inherits from being a subset of $\mathbb{R}$ , a diffeologically smooth function

$f: [0,\infty) \to \mathbb{R}^n$

is the same as what we normally consider to be a smooth function. This makes me vastly more optimistic that the same holds if we replace $[0,\infty)$ by $[0,1]$ , as in my comments above. But, I haven’t proved this yet.

It seems more challenging to show the same sort of thing for subsets of $\mathbb{R}^n$ like cubes and simplices. If we could show the same sort of thing for arbitrary convex subsets of $\mathbb{R}^n$ , I believe it would imply that the whole theory of Chen spaces is equivalent to the theory of diffeological spaces. That would be an excellent thing if true.

For details, go here. Also see the comment by Andrew Stacey.

Posted by: John Baez on September 13, 2007 10:21 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

I’ll ask again. Did he ever respond? I
once met a Patrick Iglesias (without the Zemmour)
Wonder if it’s the same?

jim

Posted by: jim stasheff on January 4, 2008 1:18 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

Urs wrote:

For instance, a differential form on a Chen-smooth space should be a collection of differential forms, one on each domain for each plot of the space, such that whenever one plot is the pullback of another along a smooth map, the corresponding differential forms are related by this pullback.

Perhaps something like that is already in Chen. After all, he invented Chen connections to handle the space of loops on a manifold.

There is a literature years ago on the analog for simplicial manifolds {M_n}
with face and degeneracy maps M_n M_{n \pm 1}

Posted by: jim stasheff on September 6, 2007 6:36 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Are there NQ Chen manifold?

I haven’t thought about this.

But let’s see

All these coordinates descriptions make me hesitate

I’d think the the local coordinate description of NQ-manifolds makes it actually easier to adopt them to the Chen-smooth context than a global description would.

I’d maybe try this:

a Chen-smooth NQ space should be a Chen-smooth space $X$ together with, on the domain $U_\phi \subset \mathbb{R}^n$ of each of its plots $U \stackrel{\phi}{\to} X$ a non-negatively graded commutative differential extension $C^\infty_\phi := (C^\infty(X) \otimes \wedge^\bullet \Gamma(E)^*, Q)$ of the algebra of functions on that domain, where $E \to U_\phi$ is a non-negatively graded vector bundle on $U_\phi$ , such that whenever two plots $\phi : U_\phi \to X$ and $\phi' : U_{\phi'} \to X$ are related by pullback along $f : U_\phi \to U_{\phi'}$ as $\phi = f^* \phi' := \phi' \circ f$ we have that the corresponding NQ algebras $C^\infty_{\phi}$ and $C^\infty_{\phi'}$ are related by pullback: $C^\infty_\phi \simeq f^* C^\infty_{\phi'} \,,$ where we think of $f$ as extended to a vector bundle isomorphism of the corresponding graded vector bundles.

Actually, this definition should differ from the ordinary definition of an NQ manifold only in that it mentions plots and their pullbacks where the ordinary definition mentions open sets and their inclusions.

So I think it should go through straightforwardly. But please note that the above is only meant as a rough suggestion. There are some subtleties here related to those choices of algebra isomorphisms and choices of vector bundle isomorphisms. This can get tricky. But I’d guess that if one just follows the standard definition closely, everything should work out.

Posted by: Urs Schreiber on September 6, 2007 5:20 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

I see. I agree that it seems a good definition. It also seems that if the odd coordinates are sections of a vector bundle over a chen manifold X, then they are still coordinate in R^n. (I assume even Chen’s vector space should be just R^n, is it right?) Then the definition (worry-free) dearer to Dimitry’s and Pavel’s heart is to define it via the algebra of such a thing, (i.e. a graded locally free sheaf of O_X-modules). See Section 2 of this paper or I just copied it here:

‘A graded manifold is a (super)manifold with an additional grading in the structure
sheaf. Specifically, it is a manifold M that possesses a coordinate atlas (of socalled
affine charts) in which each local coordinate is assigned a degree (or weight ),
and the coordinate transformations are required to preserve the total weight.’

Indeed your coherence conditions seem to be the sheaf condition for the presheaf F: {plots} —>{graded rings} to be a sheaf.

But I still have to think whether {convex sets in R^n, inclusions} has Grothendieck topology… Do you already know?

By the way, do you know whether one can get alert emails if one’s message has been replied? I see here people really act fast!!

Posted by: chenchang Zhu on September 8, 2007 5:47 PM | Permalink | Reply to this

feed aggregators

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By the way, do you know whether one can get alert emails if one’s message has been replied?

You can use what is called a feed aggregator. This is a piece of software which allows you to follow the discussion on a blog in much the same way as you read your email.

Many such aggregators can be freely and quickly downloaded. You could try for instance the web-based Bloglines.

To follow the $n$ -Café discussion, go to the Bloglines start page, then click on Feeds and then on Add. Then in the box labeled “Blog or Feed URL” insert the URL

http://golem.ph.utexas.edu/category/atom10.xml

Then hit “subscribe”. This lets you read all the entries at the $n$ -Café. To read the comments, follow the same steps but enter instead the URL

http://golem.ph.utexas.edu/category/comments.atom

The two URLs, and similar URLs on other blogs, can also be found on the start page of the $n$ -Category Café: look at the right column and scroll down almost to the end of it. There is a section titled “Syndicate”. It contains two “buttons”, labeled “full content” and “comments”, respectively. Clicking with the right mouse button on these and choosing “copy link location” copies the two URLs I gave above.

On the other hand, if you have a sufficiently new browser version, the browser itself will offer you, somewhere in the top right corner of its window, usually, a button that will automatically subscribe you to the feeds of whichever website you are currently visiting.

Posted by: Urs Schreiber on September 11, 2007 10:15 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

Even though it’s postumous, I’m very pleased to see K-T Chen recently get much acknowledgement, not only for his approach to smoothness (whihc I find a viable alternative to stacks) but also for his iterated integrals and his mathematicization of path ordered exponentials.

Posted by: jim stasheff on September 6, 2007 12:48 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Note how David hangs back while Urs blithely steps forward.

But Prof. Baez, I have lots of excuses, like having to amend the index for the paperback version of my health book. Tame, I know, when there’s work on 2 element sets to be done. Lucky there was such a capable volunteer.

Let’s see if I can ask another question and get away without answering it. So now we understand these groupoid vectors, and we know that phased groupoids are functors into $U(1)$ , considered discrete. So what about ‘phased vectors’?

Do I even know what conditions a functor between phased groupoids must satisfy concerning the phase? It would be a wee bit boring if the phase of the fibre had to match the image.

But if we were allowed, say, {{*}, $c$ } mapping to {{*}, $d$ }, do we count this as a vector with coefficient $c.d^{-1}$ ?

Posted by: David Corfield on September 6, 2007 11:09 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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David is asking about maps between phased groupoids, and I’ll answer in a roundabout way.

In Jeff’s paper, a ‘phased vector’ would be a phased groupoid $\psi$ with a map from its underlying groupoid to some groupoid $X$ :

$\array{ \psi \\ \downarrow \\ X \\ }$

This gives a vector $[\psi]$ in the space $[X]$ consisting of complex linear combinations of isomorphism classes of objects of $X$ .

(Physicists should imagine $\psi$ as a ‘wavefunction’ hovering in ghostly fashion above the ‘spectrum’ $X$ , assigning to each object $x \in X$ an ‘amplitude’.)

If $\psi$ were a mere groupoid instead of a phased groupoid, $[\psi]$ would always be a positive real linear combination of isomorphism classes of objects of $X$ . So, we wouldn’t get the complex aspect of quantum mechanics.

But now to the point: note the asymmetry here! $\psi$ is a phased groupoid, while $X$ is a mere groupoid. In Jeff’s work $X$ is usually the groupoid of finite sets. Then we call $\psi$ a ‘phased stuff type’.

I see no call for $X$ to be a phased groupoid! So, the only reason I want to understand maps between phased groupoids is that I want a category of phased groupoids $\psi$ hovering over my groupoid $X$ . Then I define the maps just the way Jeff did in his paper: the maps must preserve the phases labelling the objects.

Phased groupoids on top; groupoids below. This asymmetry seems formally a bit ugly, but it seems to get the job done: Jeff managed to categorify the whole theory of Feynman diagrams for the perturbed harmonic oscillator.

Perhaps there’s a way to make the setup formally prettier without breaking it. Or improve it more drastically!

Ultimately I think these ‘phased’ gadgets, where we sort of tack phases on to things, are a mere stopgap solution. Someday we’ll understand why quantum mechanics uses the complex numbers. Then the complex numbers will grow naturally out of the formalism, instead of being put in by hand.

But, I don’t want to postpone categorifying quantum mechanics until that happy day.

(If you could explain what Urs is talking about, that happy day might come a week sooner.)

Posted by: John Baez on September 6, 2007 12:24 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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If you could explain what Urs is talking about

At that link I am talking about this:

a) suppose you already know that your particle couples to a $U(1)$ -bundle with connection

b) then why do its quantum states, which are sections of that $U(1)$ -bundle, locally not really appear as $U(1)$ -valued functions, but as $U(1)$ -phased-groupoid valued functors

c) such that the path integral becomes an honest colimit.

So this would relate the appearance of $U(1)$ -phases in quantum mechanics directly to the fact that the “background” field is a $U(1)$ -gauge field.

This line of thoughts by itself doesn’t try to “explain” why the background field itself involves $U(1)$ in the first place.

But if we want, we can do at least a little better than that:

$U(1)$ -1-bundles with connection are the same as $(\mathbb{Z} \to \mathbb{R})$ -2-bundles with connection.

It’s sort of clear, but I did talk about that here.

The idea there was, following some discussion we had here, to replace $U(1)$ by some construction or other involving $\mathbb{Z}$ .

The above construction still involves the reals, which we may feel are still too “man made” compared to the almost god-given integers. But if you replace the 2-group $\mathbb{Z} \to \mathbb{R}$ with its discrete approximaxions $n\mathbb{Z} \to \mathbb{Z}$ we get the $\mathbb{Z}_n$ -bundles which, apparently the Tale will be about, anyway.

(One problem is that it scares me to work more on these ideas, since they are of precisely the sort that put off most of the tenured people which I am supposed to try to make a good impression on, if you know what I mean.)

Posted by: Urs Schreiber on September 6, 2007 12:54 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Urs wrote:

… if you know what I mean.

Yeah, I know what you mean. Do your more revolutionary work in private and reveal it after you get tenure. Get tenure soon so you still have lots of energy left over by the time you get it. To get tenure, publish lots of stuff that’s cutting-edge yet still sufficiently mainstream to make lots of people happy.

To make people happy, it’s always easier if you solve their problems rather than invent your own new problems. That’s a sad fact. You like to invent your own new problems — in fact, you’re inventing your own new language! To get people to appreciate this, you need to spend time teaching people this language. And the main thing professional mathematicians want to know is “what problems could I work on, if I knew this language?” The problems shouldn’t be open-ended: they should be very precise.

If it’s pure mathematicians you want to impress, characteristic classes of $n$ -bundles, Lie $n$ -algebra cohomology, and Lie $n$ -algebroids should be quite nice if you explain these concepts well and leave lots of interesting problems for people to work on.

Mathematical physicists are more fussy: they’ll want to see applications to ‘theories of physics’ — meaning theories of physics that they’re already fond of, nothing mind-blowingly new like an ‘ $n$ -particle’.

Posted by: John Baez on September 7, 2007 10:38 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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To make people happy, it’s always easier if you solve their problems rather than invent your own new problems.

Thanks for the thoughtful advice. I appreciate it. Got pretty much the same advice recently from somebody else.

While on the one hand this makes me nod and scold myself for being such a weirdo, on the other hand it makes me flare-up:

“But all this ‘new language’ this ‘ $n$ -particle’ business, everything, the cube, $n$ -curvature, Chern-Simons Lie $n$ -algebras, everything was not dreamed up for its own sake, but to solve one single concrete problem. This one.”

And I do think all of this does contribute to solving that problem. But it takes a little to assemble all puzzle pieces.

But I guess it’s getting better. The most critical phase was the one where I spent all my time trying to sort out the fundamental basics. Probably that’s where I lost about everybody.

Posted by: Urs Schreiber on September 7, 2007 11:46 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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But all this `new language’ this `n-particle’ business, everything, the cube, n-curvature, Chern-Simons Lie n-algebras, everything was not dreamed up for its own sake, but to solve one single concrete problem. This one.

Hi Urs, “this one” indeed contains a nice introduction which I very nearly understood! I’ll tell you where I got lost : when we hit “1st statment”. It’s kind of not clear to someone like me exactly where “FRS” stands on the cube… or indeed what some of those other things on the cube are doing. Let me quickly add, to save myself from embarassment, that I do understand the cube… or at least, I once did understand the cube… but a quick reminder would have been helpful at that point.

I think I can understand the 2nd statement. I have a few small-ish beefs with the idea that “a n-dimensional TQFT is an n-functor from nCob to nVect” - it worries me that both “nCob” and “nVect” are very difficult to define in the current language of higher categories, and that even a 2d TQFT doesn’t quite fit this description, as I think comes out of Jeffrey Morton’s work - but I can happily put aside those beefs and continue.

And I can understand the 3rd statement.

So all-in-all a very nice article, perhaps a bit more motivation in certain sections and it serves as good introduction to your ideas?

Posted by: Bruce Bartlett on September 7, 2007 2:24 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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not clear to someone like me exactly where “FRS” stands on the cube

Ah, thanks for saying so.

The claim is: you arrive at FRS by following all

of the three edges. It’s a “2-dimensional state sum model”.

The proposed statement is:

the FRS decoration prescription arises after expressing the value of a 2-functor on a surface in terms of local data obtained from locally trivializing that 2-functor. This is supposed to explain the 2-dimensional aspects of FRS.

And the refined proposed statement is:

this 2-functor itself is actually the component map of a transformation of a TFT 3-functor. This explains the 3-dimensional aspects of FRS.

The easier claim underlying this is: the Fukuma-Hosono-Kawai presription for obtaining a 2-dimensional TFT is really the data of a locally trivialized 2-functor.

This is what got me started on this problem:

Topological Strings from 2-Transport.

So the idea is that by obtaining a more intrinsic understanding where the FRS prescription comes from, if we’d just start with, say, the information that we have a “membrane propagating on B G” (that is: Chern-Simons theory), we’d be able to understand the urgent next questions that FRS’s “complete solution of rational CFT” raises:

- How does it generalize to non-rational CFT? To SCFT?

That’s the big problem.

Posted by: Urs Schreiber on September 7, 2007 2:49 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

Urse writes:
And I do think all of this does contribute to solving that problem. But it takes a little to assemble all puzzle pieces.

Even more crucial after assembling the pieces is the exposition. As Dan Kan remarked to me aftyer my first lecture at MIT

Just because you’ve written a formula, doesn’t mean you’ve communicated with your audience. ;-)

Posted by: jim stasheff on September 7, 2007 3:02 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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John wrote:

To make people happy, it’s always easier if you solve their problems rather than invent your own new problems.

Thanks for the thoughtful advice. I appreciate it. Got pretty much the same advice recently from somebody else.

Note I wasn’t actually advising you to make people happy: I said if you want to make people happy…

Presumably you should make them just happy enough to stay happy yourself.

But I guess it’s getting better. The most critical phase was the one where I spent all my time trying to sort out the fundamental basics. Probably that’s where I lost about everybody.

Yup.

So, when you write papers, you’ll need to return to the original concrete problem and explain how you’re solving it… in a way that you could have understood and enjoyed even before you did all this work.

In serious research, by the time you’ve solved a problem, you’ve become very different from the guy who originally set out to solve it. The new you would have a lot of trouble talking to the old you. But the new you needs to remember the old you to communicate with other people, because those other people are more like the old you.

Posted by: John Baez on September 7, 2007 1:05 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

Amen! I switched to ‘cohomological physics’ after I had tenure. An NSF reviewer once remarked that it was sad that only someone of my stature could afford to switch like that.

Posted by: jim stasheff on September 7, 2007 2:58 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Picking up on the theme for another thread about the importance of vector spaces having duals, if these groupoid vectors of yours were like ordinary vectors, we might hope for something like

$Hom(Gpd/ X, Gpd/ Y) \cong Gpd/(X \times Y) \cong Hom(Gpd/ X, Gpd) \times Gpd/Y.$

But I guess those spans in the middle term aren’t sufficient to capture each member of the first term.

What if we took phased groupoids/vectors?

Also to get at the reals rather than the complex numbers, couldn’t you choose ‘phases’ to be in $\{ +1, -1\}$ ?

Posted by: David Corfield on September 7, 2007 8:37 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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David wrote:

$Hom(Gpd/X,Gpd/Y)$

I don’t know what this means — how you’re planning to define a map from ‘groupoids over $X$ ’ to ‘groupoids over $Y$ ’. But I don’t think we need to know what this means.

I think you wrote this expression because you haven’t fully accepted the philosophy of groupoidification. When we groupoidify,

THE ANALOGUE OF A VECTOR SPACE IS A GROUPOID

and:

THE ANALOGUE OF A LINEAR OPERATOR IS A SPAN OF GROUPOIDS

Rather curiously,

THE ANALOGUE OF A VECTOR IN A VECTOR SPACE IS A GROUPOID OVER A GROUPOID

But, we don’t need to think about ‘maps from $Gpd/X$ to $Gpd/Y$ ’. It’s enough to think about maps from $X$ to $Y$ — that is, spans from $X$ to $Y$ .

There’s a lot more to say about this, because this particular slogan:

THE ANALOGUE OF A VECTOR IN A VECTOR SPACE IS A GROUPOID OVER A GROUPOID

is actually strange and disturbing if you think about it carefully. It feels like a kind of level slip, since we have ‘vector’ and ‘vector space’ on the one hand, but ‘groupoid’ and ‘groupoid’ on the other.

Nonetheless, it’s true, and it’s a consequence of the general slogan:

THE ANALOGUE OF A RANK- $N$ TENSOR $T_{ijk...}$ IS AN $N$ -LEGGED SPAN OF GROUPOIDS, WITH EACH INDEX CORRESPONDING TO A LEG

in the special case $N = 1$ .

Anyway: your question about the importance of duals is very wise. But there’s a big surprise here! In the groupoidification game, every groupoid is its own dual — it comes equipped with the groupoidified analogue of a ‘nondegenerate inner product’ $g_{ij}$ .

(You may enjoy guessing what this is, using the last slogan I just mentioned.)

So, groupoids are actually more like Hilbert spaces than vector spaces.

This is why we can easily turn a span from $X$ to $Y$ into a span from $Y$ to $X$ , just by turning it around. This is the sort of thing we can do with operators between Hilbert spaces, but not operators between vector spaces.

Another way to put it is that we don’t need to distinguished by ‘raised and lowered indices’ in the index notation for spans.

Also to get at the reals rather than the complex numbers, couldn’t you choose ‘phases’ to be in {+1,-1}?

Hey! For some reason I’d never thought of that — mainly because I was so eager to get the complex numbers. But yeah! You’re right! $\mathrm{O}(1)$ instead of $\mathrm{U}(1)$ !

Posted by: John Baez on September 7, 2007 11:12 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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THE ANALOGUE OF A VECTOR IN A VECTOR SPACE IS A GROUPOID OVER A GROUPOID

That’s why I was thinking of gathering the beasts together.

So we want a 2-legged span of groupoids to groupoidify the inner product. I guess it’s got to be $X \leftarrow X \to X$ .

Posted by: David Corfield on September 7, 2007 1:29 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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David wrote:

That’s why I was thinking of gathering the beasts together.

Right, but it turns out that in this context the correct ‘thing of all groupoids over $X$ ’ is just $X$ .

That’s supposed to be shocking…

So we want a 2-legged span of groupoids to groupoidify the inner product. I guess it’s got to be $X \leftarrow X \to X$ .

Right! The diagonal

$\Delta: X \to X \times X$

yields the span that gives $X$ its canonical ‘inner product’. It’s the groupoidified version of the ‘Kronecker delta’ $\delta_{ij}$ .

Posted by: John Baez on September 7, 2007 2:27 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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That’s supposed to be shocking.

I’m shocked. So shocked I don’t know what to say.

But perhaps I’ll burble on a bit.

Do we see this kind of behaviour elsewhere, like the collection of all sets over $X$ is kinda somehow $X$ ? At least I could imagine $X$ being extractable from that collection.

Is it important that the category of groupoids is cartesian closed?

But that might be better seen as a 2-category. Is there some (weak) cartesian closure type property there?

Posted by: David Corfield on September 7, 2007 5:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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David wrote:

I’m shocked. So shocked I don’t know what to say.

Good! I’ve succeeded in communicating the strangeness of it all.

But perhaps I’ll burble on a bit.

Good! We’re academics, after all; we speak in paragraphs, and we can even deliver a nice paragraph explaining how something leaves us speechless.

I hope you see how this business is an inevitable spinoff of the groupoidification progam, where vector spaces get promoted to groupoids, and rank- $n$ tensors get promoted to $n$ -legged spans of groupoids. It follows inevitably that rank- $0$ tensors, aka vectors in our given vector space, get promoted to groupoids over our given groupoid! But, the ‘thing of all such vectors’ got promoted to the groupoid itself.

So: the thing of all things over our given thing is — in some funny way — itself that thing.

It took Jim and I a long time to realize some of the bizarre and wonderful consequences of this strange level slip. I, for one, am still just beginning to grapple with it.

Do we see this kind of behaviour elsewhere, like the collection of all sets over $X$ is kinda somehow $X$ ? At least I could imagine $X$ being extractable from that collection.

Of course in any category, any object $X$ becomes an object over itself in the obvious way: $X \stackrel{1}{\to} X$ And, it’s then the ‘universal’ object over itself: any object over $X$ factors through this one!

But, I’m not sure this triviality really illuminates what’s going on in the example of groupoidification. It might; I just don’t know.

In fact, I don’t have the guts to talk about this anymore — at least not at such an abstract level. I’d rather explain (someday, not now) how this strange phenomenon manifests itself in concrete examples, like Hecke algebras. Somehow I want that concreteness to add ‘ballast’ to the discussion, so we don’t float off into the stratosphere.

Posted by: John Baez on September 7, 2007 8:07 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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I know you don’t want to talk about it now, but this caught my eye:

a subcategory $A$ of a larger category is adequate if every object $X$ of the larger category is canonically the colimit of the category $A/X$ of objects of $A$ equipped with structural maps to $X$ ; John [Isbell]’s equivalent definition was that the truncated Yoneda embedding of the whole category into the category of set-valued contravariant functors on $A$ is actually a full embedding. The following language is suggestive: (a) The objects of $A$ are figure-types, (b) the objects of $A/X$ are particular figures in $X$ , and (c) the morphisms of $A/X$ (commutative triangles in the big category) are incidence relations between figures. Thus adequacy means that the large category in question consists of objects entirely determined by their $A$ -figures and incidence relations, and that (d) the morphisms in the whole category are nothing but the “geometrically continuous” ones in the sense that they map figures to figures without tearing the incidence relations.

It’s written by Lawvere.

Posted by: David Corfield on September 10, 2007 3:43 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

Adequacy is given in terms of a dense class of generators (section 2.7) by Anders Kock. So the class consisting of 1 is a dense class of generators for Set.

Now it can’t just be that you’re saying that the class of all groupoids is a dense class of generators for Groupoid.

Does anyone know if there’s something like an ‘essential adequacy’?

Posted by: David Corfield on September 11, 2007 12:07 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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David recalled Lawvere’s notion of an ‘adequate’ subcategory $A$ of a category $X$ :

John [Isbell]’s equivalent definition was that the truncated Yoneda embedding of the whole category into the category of set-valued contravariant functors on $A$ is actually a full embedding.

I would have been terrified of this sentence before the age of 35 or so. Before then, I wasn’t on good terms with the Yoneda lemma.

Maybe I would have appreciated this earlier:

In rough but plain English: a subcategory $A$ of a category $X$ is ‘adequate’ if any morphism $f: x \to y$ in $X$ is determined by its composites with all morphisms from objects of $A$ to $x$ .

More precisely: we say $A$ is adequate if the functor

$F: X \to Set^{A^{op}}$

sending each object $x \in X$ to

$F(x) = hom(--, x) : A^{op} \to Set$

is full and faithful.

The Yoneda lemma says precisely that $X$ is an adequate subcategory of itself. So, the fun part is seeing if you can get away with smaller subcategories!

For example, consider the category of directed graphs. Here the full subcategory consisting of this graph:

$vertex = \bullet$

and this one:

$edge = \bullet \to \bullet$

is adequate.

Why? Because you know a map between directed graphs if you know what it does to all vertices and edges!

In other words, you know a map of directed graphs $f : x \to y$ if you know its composites with all possible maps

$g: vertex \to x$

and

$h: edge \to x$

Similarly, in the category of simplicial sets, the full subcategory consisting of simplicial sets that look just like $n$ -simplices is adequate.

Why? Because you know a map between simplicial sets if you know what it does to all the simplices!

(For would-be know-it-alls: the simplicial set that looks just like an $n$ -simplex is called the simplicial $n$ -simplex. The category of directed graphs and the category of simplicial sets are examples of presheaf categories, and unless I’m mistaken, the adequacy results I stated are special cases of a general result for presheaf categories: the full subcategory of representable presheaves is always adequate.)

David wrote:

Now it can’t just be that you’re saying that the class of all groupoids is a dense class of generators for Groupoid.

Just so people know: Kock says a collection $A$ of objects in a category $X$ is a dense class of generators when given any two morphisms $f,g: x \to y$ in $X$ , they’re equal when $fh = gh$ for every morphism $h: a \to x$ with $a \in A$ .

This is very similar to the notion of adequacy, but I’m too lazy to figure out the theorem(s) relating them. Maybe someone else feels like it?

Anyway, I don’t think I’m deliberately saying anything quite so trivial as ‘the class of all groupoids is a dense class of generators in the category of groupoids’. For any category, the collection of all objects is a dense class of generators — right? For the category Groupoid, I think we can choose a much smaller dense class of generators, namely this one:

$object \; = \; \bullet$

and this one:

$morphism \;= \; \bullet \to \bullet$

I’m not sure exactly what this rather unshocking stuff has to do with the supposedly shocking statement I made…

Posted by: John Baez on September 11, 2007 2:18 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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I’m sure I ought just to wait for your nice concrete example of a Hecke algebra in the next TWF, but I was trying to make sense of your:

it turns out that in this context the correct ‘thing of all groupoids over $X$ ’ is just $X$ .

and saw, or thought I saw, a similarity to

every object $X$ of the larger category is canonically the colimit of the category $A/X$ of objects of $A$ equipped with structural maps to $X$ .

An instance of Lawvere’s statement is that if $A$ and the ‘larger’ category are chosen to be Groupoid, that every groupoid, $X$ , is canonically the colimit of the category Groupoid/ $X$ of groupoids equipped with structural maps to $X$ .

Something tells me I still haven’t got your statement yet.

Posted by: David Corfield on September 11, 2007 3:14 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Isn’t it a problem that the category ‘morphism’ isn’t a groupoid?

Posted by: Jamie Vicary on September 11, 2007 9:43 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Jamie Vicary wrote:

Isn’t it a problem that the category ‘morphism’ isn’t a groupoid?

Yeah, sorry — I should have said ‘isomorphism’. So, I’m claiming that the groupoids

$object \; = \; \bullet$

and

$isomorphism \; = \; \bullet\stackrel{\sim}{\to} \bullet$

are a dense class of generators in $Groupoid$ .

Posted by: John Baez on September 12, 2007 10:05 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Just so people know: Kock says a collection A of objects in a category $X$ is a dense class of generators when given any two morphisms $f,g:x \to y in X$ , they’re equal when $f h=g h$ for every morphism $h:a \to x$ with $a \in A$ .

This is very similar to the notion of adequacy, but I’m too lazy to figure out the theorem(s) relating them. Maybe someone else feels like it?

The terminology can be a little confusing here:

“Adequate” is a pretty old-fashioned term, I believe; most people nowadays refer to a (full) subcategory $i: A \to X$ instead as a dense subcategory if the Kan functor

$K = \hom(i-, ?): X \to Set^{A^{op}}$

$x \mapsto (a \mapsto \hom(a, x))$

is full and faithful (see for instance Categories for the Working Mathematician, p. 243 [1st Edition]). But this differs with the notion of “dense class of generators” as used by Kock, which just means this functor is faithful. (I think a lot of people would just say “ $A$ generates $X$ ” here.)

For to say $K$ is faithful means that for any pair $f, g: x \to y$ , that $f$ and $g$ coincide whenever $K(f)$ and $K(g)$ coincide. The latter condition occurs when $K(f)_a = K(g)_a$ for every object $a$ of $A$ , where this equality in turn means that $K(f)_a(h) = K(g)_a(h)$ for every $h: a \to x$ . This gives Kock’s condition.

The typical situation where adequacy or density is considered is where the “small” category $A$ is, er, (essentially) small, and the “big” category $X$ is cocomplete, and in this case I sometimes find it more illuminating to say it in one of two (manifestly equivalent) ways:

That for every object $x$ of $X$ , the small cone given by the comma category $A/x$ is a colimit cone.
That if $L: Set^{A^{op}} \to X$ is the left adjoint to $K: X \to Set^{A^{op}}$ , then the counit $L K x \to x$ is an isomorphism for all $x$ .

(It’s an elementary observation that $K$ is fully faithful iff the counit of $L \dashv K$ is an isomorphism.)

Some examples: for a site $(C, J)$ where every representable is a sheaf (a subcanonical site), the inclusion $C \to Sh(C, J)$ is dense or adequate, because the category of sheaves is by definition a full subcategory of the category of presheaves. (Equivalently, a sheaf is its own sheafification.)

Ordinals are dense or adequate in categories, because every category $C$ is built as a canonical colimit of the relevant “figures” in $C$ (nodes, arrows, commutative triangles). If we omit commutative triangles, then we only get a faithful functor

$Cat \to Set^{\bullet \stackrel{\to}{\to} \bullet}$

inasmuch as a category is a directed graph plus some extra structure. A similar circumstance holds for groupoids, as John points out.

There are of course tons of other examples.

Posted by: Todd Trimble on September 11, 2007 10:29 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 256)

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Ordinals are dense or adequate in categories

Bah. I meant “finite ordinals” of course. Of course my real point was that we only need ordinals with three or fewer elements.

Posted by: Todd Trimble on September 13, 2007 7:23 PM | Permalink | Reply to this

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