Higher-Dimensional Algebra VII: Groupoidification

August 23, 2009

Higher-Dimensional Algebra VII: Groupoidification

Posted by John Baez

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Check this out:

John Baez, Alex Hoffnung and Christopher Walker, Higher-dimensional algebra VII: groupoidification — latest version with all known corrections, arXiv version or published version.

This is a somewhat expanded and improved version of our paper Groupoidification made easy, more suitable for publication. It now includes material on Hall algebras — or, very loosely speaking, quantum groups — and various different recipes for turning spans of groupoids into linear operators between vector spaces.

Comments and corrections are most welcome!

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I still haven’t satisfies Urs’ request to include material about groupoidified traces. I’m not quite sure where to put it.

Here’s the abstract:

Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of `degroupoidification’: a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present three applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter $q$ is a prime power. We illustrate this with the simplest nontrivial example, coming from the $A_2$ Dynkin diagram. In this example we show that the solution of the Yang–Baxter equation built into the $A_2$ Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field $\mathbb{F}_q$ . The third application is to Hall algebras. We explain how the standard construction of the Hall algebra from the category of $\mathbb{F}_q$ representations of a simply-laced quiver can be seen as an example of degroupoidification. This in turn provides a new way to categorify—or more precisely, groupoidify—the positive part of the quantum group associated to the quiver.

Alex Hoffnung will go a lot further with Hecke algebras in his thesis, and Christopher Walker will dig into Hall algebras in his.

Posted at August 23, 2009 2:38 AM UTC

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Re: Higher-Dimensional Algebra VII: Groupoidification

Am I right in thinking that the proposition on the bottom of p9 can be broken into

1) a statement about taking the tensor product of two vectors in different vector spaces
2) a groupoidification of “R tensor V is isomorphic to V”

that would make your “here’s how to multiply a vector by a scalar” into “here’s how to tensor two vectors, and if one of them is a scalar, then you can think of the tensor product as living in the original vector space”
?

Posted by: Allen K. on August 23, 2009 3:46 AM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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Hi! Yes, you’re right! To really show off, we could say it like this: degroupoidification is a monoidal 2-functor from the weak 2-category of

groupoids,
tame spans of groupoids,
equivalence classes of maps of tame spans

to the 2-category of

real vector spaces,
linear operators,
identity 2-morphisms, thrown in just to give our category the pretensions of being a 2-category.

In fact Alex Hoffnung has already proved this as part of writing up HDA8. But we’re trying to keep things sort of lowbrow here.

However, I’m about to write some “Conclusions” where we sketch stuff we don’t actually do, including “index-juggling for groupoidified tensors”, and that might be a good place to admit the existence of this deeper stratum.

(In fact degroupoidification is really a weak symmetric monoidal 3-functor, but we’re in no position to prove that!)

(Also: to make everything I’m saying true, you either need to restrict to groupoids with finitely many isomorphism classes of objects, or use the ‘zeroth homology’ approach to degroupoidification rather than the ‘zeroth cohomology’ approach — see Section 3. But what you said is still true in the ‘zeroth cohomology’ approach we start off using.)

Posted by: John Baez on August 23, 2009 11:15 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

Oh, and please include a \tableofcontents . (There, I’ve typed it out for you; all you need to do is drop it in after \end{abstract}.) Sometimes I wish the arXiv rejected submissions that don’t have them.

Posted by: Allen K. on August 23, 2009 4:17 AM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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Well done for basically being finished with HDA VII!

I love degroupoidification, and the Hall algebra / quiver stuff you’ve gotten out of it is great for someone like me who struggles to understand the more usual “purely algebraic” way of doing it. I’m terribly looking forward to learning this stuff now from this paper.

I have a subtle “beef” with the approach you taken here, one of the only points in all of HDA where I would wish you would do it differently :-)

I would only want to tweak the program in one respect: oftentimes the groupoid has a natural line bundle sitting over it. Then to degroupoidify, one should take the sections of this line bundle: i.e. the vector space of “twisted” linear combinations of objects, rather than just linear combinations.

This matters even for finite groupoids. A line bundle $L$ over a finite groupoid $G$ is something which associates to each object $x \in G$ a complex line $L_x$ , and to each arrow in $g: x \to y$ in $G$ a linear map

$L(g) : L_x \to L_y$

such that composition is respected. If we choose a basis for each complex line $L_x$ , this line bundle (or “equivariant” line bundle) becomes precisely a 1-cocycle on the groupoid. And the space of sections of this line bundle is trivial over the components where the line bundle is nontrivial!

This is the elegant formula “Theorem 6” in Simon’s “The Drinfeld double of a finite group via gerbes and finite groupoids” paper:

Theorem 6. The dimension of the space of flat sections of the line bundle $L$ over a groupoid $G$ is given by integrating the transgression of $L$ over the loop groupoid.

The nice thing about working with sections of equivariant line bundles (instead of just formal linear combinations of isomorphism classes of objects) is of course that it then enables us to think of representations of groups geometrically, viz. as sections of line bundles over action groupoids, as we have discussed before. There is an equivalence of categories

$LBun (G) \to Rep(G)$

where $LBun(G)$ is the category of “hermitian equivariant holomorphic line bundles over compact complex manifolds equipped with a $G$ -action”, and $Rep(G)$ is the category of unitary representations of a Lie group $G$ (it need not be compact, since this theorem is at its heart a tautology!).

If one wants to avoid “continuous” groupoids and purely stick with finite ones, the analog of the above result (as I’ve just realized) would be Brauer’s theorem on induced characters, which in this language basically says:

Every representation of a finite group $G$ is, considered as an element in the Grothendieck ring $K(Rep(G))$ , an integral linear combination of representations of the form $Sections(L)$ , where $L$ is a line bundle over the action groupoid of $G$ on a finite set (viz, a 1-cocycle).

So this is my one “beef” with the degroupoidification program: I think it is ungeometric to degroupoidify by taking “formal linear combinations of isomorphism classes of objects”. I think it is geometrically more natural to take “flat sections of an equivariant line bundle over a finite groupoid”. When the line bundle is trivial, these are equivalent of course, but I would still prefer to talk about “flat sections of a line bundle” rather than “linear combinations of isomorphism classes” because it is evil (as you taught us!) to take isomorphism classes!

I even have this small difference when it comes to the basic formula for cardinality: your definition reads

$|G| = \sum_{isomorphism classes of objects [x]} 1/|Aut(x)|$

whereas I think it is a bit more natural to sum over the objects themselves, rather than over their isomorphism classes:

$|G| = \sum_{objects x} 1/ |morphisms out of x|$

This formula is a bit less applicable than the previous one again, since we need the number of elements in an isomorphism class to be finite. However, Weinstein taught us how to do it in the smooth setting, and his starting point was the second formula, not the first. Anyhow this is just a small thing!

Posted by: Bruce Bartlett on August 23, 2009 11:37 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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Bruce wrote:

Well done for basically being finished with HDA VII !

Thanks! Though comments like yours and Urs’ suggest that it’s not quite done. I guess it just spills over into too many related subjects…

I love degroupoidification, and the Hall algebra / quiver stuff you’ve gotten out of it is great for someone like me who struggles to understand the more usual “purely algebraic” way of doing it. I’m terribly looking forward to learning this stuff now from this paper.

Great. This paper just sketches the ideas: we’re writing HDA8 on Hecke algebras, and I imagine HDA9 being about Hall algebras.

I would only want to tweak the program in one respect: oftentimes the groupoid has a natural line bundle sitting over it. Then to degroupoidify, one should take the sections of this line bundle: i.e. the vector space of “twisted” linear combinations of objects, rather than just linear combinations.

Yes, that sounds like something you’d say.

And I agree, it’d be a very useful extension of the whole formalism. Particularly in HDA8 where we’ll be studying representations of finite groups… thanks to this:

There is an equivalence of categories

$LBun (G) \to Rep(G)$

where $LBun(G)$ is the category of “hermitian equivariant holomorphic line bundles over compact complex manifolds equipped with a $G$ -action”, and $Rep(G)$ is the category of unitary representations of a Lie group G.

Yes, I love this result of yours.

If one wants to avoid “continuous” groupoids and purely stick with finite ones,…

.. and we do, for now. Life is short, and it’s good to leave work for future generations.

the analog of the above result (as I’ve just realized) would be Brauer’s theorem on induced characters, which in this language basically says:

Every representation of a finite group $G$ is, considered as an element in the Grothendieck ring $K(Rep(G))$ , an integral linear combination of representations of the form $Sections(L)$ , where $L$ is a line bundle over the action groupoid of $G$ on a finite set (viz, a 1-cocycle).

Thanks for pointing that out! I’m a big fan of another theorem of Brauer’s, namely that every representation of a finite group over the complex numbers is already defined over the field you get by taking the rationals and throwing in all roots of unity. I bet this is related somehow… it means that if you’re insanely algebraic, you can use this field instead of $\mathbb{C}$ when studying finite group representations, and probably also equivariant bundles over finite groupoids.

It’s not that I’m dying to throw out $\mathbb{C}$ , but I’m dying to know where $\mathbb{C}$ comes from, and this is some sort of clue…

So this is my one “beef” with the degroupoidification program: I think it is ungeometric to degroupoidify by taking “formal linear combinations of isomorphism classes of objects”. I think it is geometrically more natural to take “flat sections of an equivariant line bundle over a finite groupoid”. When the line bundle is trivial, these are equivalent of course, but I would still prefer to talk about “flat sections of a line bundle” rather than “linear combinations of isomorphism classes” because it is evil (as you taught us!) to take isomorphism classes!

To nitpick a bit: taking linear combinations of isomorphism classes is degroupoidification, and I’m only teaching people how to commit that sin so they can avoid it by groupoidification — i.e., working with groupoids. You need to know a bit about sin to be virtuous.

I don’t think there’s anything sinful about working with groupoids that don’t have line bundles over them… it’s just less general and perhaps ultimately less interesting.

I even have this small difference when it comes to the basic formula for cardinality: your definition reads

$|G| = \sum_{isomorphism classes of objects [x]} 1/|Aut(x)|$

whereas I think it is a bit more natural to sum over the objects themselves, rather than over their isomorphism classes:

$|G| = \sum_{objects x} 1/ |morphisms out of x|$

You’ll be happy to know this is Lemma 33, and we find it very useful.

But in fact, rereading that lemma, I just noticed a little mistake and fixed it. So thanks!

Posted by: John Baez on August 23, 2009 11:48 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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Greetings from the Netherlands’ north-see Dunes, where I am spending a short vacation.

Three comments on this notion of degroupoidifying groupoids with line bundles on them:

a) a span of “groupoids with line bundles on them” (or with more general bundles on them) is what is called (for line 2-bundles) a bi-brane here. In Lurie’s Classification it is called a morphism in $Fam_1(Vect)$ (see from page 57 onwards)

b) Johan Alm has given a concrete formula for the degroupoidification of such spans of groupoids with bundles on them which does generalize the Baez-ian version and which is on objects the one indicated in section 3 of Freed-Hopkins-Lurie-Teleman. Johan is currently working on writing out the 2-version explicitly.

This and how the story continues for all $n$ is somewhat vaguely but impressively indicated in section 8 of Freed-Hopkins-Lurie-Teleman.

By reading a bit between the lines of that and going back to the discussion of $Fam_n(C)$ in “On the classification…” you can guess a grand program sketched there for obtaining quantum field theories from quantizing classical theories.

One mystery, therefore, that remains is why these four are enjoying such a productive collaboration on this subject, while so many contributors on this blog keep each fiddling around with this with their local crowd without getting an organized blog-based attack going.

Posted by: Urs Schreiber on August 26, 2009 4:20 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

Time for a Polymath project?

Posted by: David Corfield on August 26, 2009 4:46 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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Urs wrote:

One mystery, therefore, that remains is why these four are enjoying such a productive collaboration on this subject, while so many contributors on this blog keep each fiddling around with this with their local crowd without getting an organized blog-based attack going.

As for me, the reason is that I have a lot more fun talking to Jim Dolan than working collaboratively on a blog.

You may consider this ‘fiddling around’ — and indeed it’s probably not the most efficient method of developing a large theory. But I find that as soon as I start thinking of math as a business where teams of researchers try to efficiently maximize production, I also start thinking it’s… not fun anymore.

Jim and I started thinking about groupoidification around 1998 when I realized that an $n$ -element set mod the action of a $m$ -element group should give a groupoid with $n/m$ elements, and he realized this gave a new outlook on how Joyal’s species form a categorified version of the Hilbert space for a harmonic oscillator. That was a lot of fun, because these ideas seemed almost crazy, and nobody was talking about them.

Now groupoidification is becoming a small part of a much vaster program, and lots of very good people are working on this program. For example: you, Ben-Zvi, Freed, Hopkins, Lurie, and Teleman. So now I don’t feel that same sense of freedom, of exploring an unknown and unoccupied wilderness. Instead, it feels like a race where I’d need to keep up with what other people are doing. I get tired just thinking about it! So, I prefer to think about other subjects now… even though it will take a year or two to finish off HDA7, HDA8 and HDA9.

But as for you, Urs: why don’t you try to collaborate with Ben-Zvi, Freed, Hopkins, Lurie, or Teleman? I’m sure the results would be amazing!

Posted by: John Baez on August 26, 2009 8:12 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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I should add that while I personally am not interested in joining any ‘massive blog collaborations’, I think it would be great if such a collaboration developed on the $n$ -Café.

From what I’ve seen, it seems that a big project like this require a charismatic leader who works tirelessly keep it on track, but lets other people take a lot of initiative and get a lot of credit for their work.

When you think of the Polymath projects you think of Tim Gowers and Terry Tao. When you think of Wikipedia you think of Jimmy Wales, and when you think of Linux you think of Linus Torvalds.

Torvalds seems to have put forth the most thoughtful comments about running such large projects. For example:

Nobody should start to undertake a large project. You start with a small trivial project, and you should never expect it to get large. If you do, you’ll just overdesign and generally think it is more important than it likely is at that stage. Or worse, you might be scared away by the sheer size of the work you envision.

So start small, and think about the details. Don’t think about some big picture and fancy design. If it doesn’t solve some fairly immediate need, it’s almost certainly over-designed. And don’t expect people to jump in and help you. That’s not how these things work. You need to get something half-way useful first, and then others will say “hey, that almost works for me”, and they’ll get involved in the project.

And if there is anything I’ve learnt from Linux, it’s that projects have a life of their own, and you should not try to enforce your “vision” too strongly on them. Most often you’re wrong anyway, and if you’re not flexible and willing to take input from others (and willing to change direction when it turned out your vision was flawed), you’ll never get anything good done.

In other words, be willing to admit your mistakes, and don’t expect to get anywhere big in any kind of short timeframe. I’ve been doing Linux for thirteen years, and I expect to do it for quite some time still. If I had expected to do something that big, I’d never have started. It started out small and insignificant, and that’s how I thought about it.

Also:

While it turns out that most people are idiots, the corollary to that is sadly that you are one too, and that while we can all bask in the secure knowledge that we’re better than the average person (let’s face it, nobody ever believes that they’re average or below-average), we should also admit that we’re not the sharpest knife around, and there will be other people that are less of an idiot that you are.

Some people react badly to smart people. Others take advantage of them. Make sure that you, as a [manager], are in the second group. Suck up to them, because they are the people who will make your job easier. In particular, they’ll be able to make your decisions for you, which is what the game is all about.

So when you find somebody smarter than you are, just coast along. Your management responsibilities largely become ones of saying “Sounds like a good idea - go wild”, or “That sounds good, but what about xxx?”. The second version in particular is a great way to either learn something new about “xxx” or seem extra managerial by pointing out something the smarter person hadn’t thought about. In either case, you win.

One thing to look out for is to realize that greatness in one area does not necessarily translate to other areas. So you might prod people in specific directions, but let’s face it, they might be good at what they do, and suck at everything else. The good news is that people tend to naturally gravitate back to what they are good at, so it’s not like you are doing something irreversible when you do prod them in some direction, just don’t push too hard.

Posted by: John Baez on August 26, 2009 10:52 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

Why not turn the journal club into the beginnings of this blog-based attack?

I think following Torvalds’ advice and starting very small is a good idea.

I don’t think the initial goal has to be to compete with anyone. A better initial goal might be to see if we can successfully produce a piece of work.

Posted by: Alex Hoffnung on August 27, 2009 7:10 AM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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Alex wrote:

Why not turn the journal club into the beginnings of this blog-based attack?

I think that’s a great idea. If you ‘journal club’ folks tried to prove a theorem or two, it might boost your energy level a notch or two. I get the feeling you were just trying to understand David Ben–Zvi’s work. That’s great, but mathematicians always seem to get more excited when they’re trying to prove stuff. Maybe there’s an easy extension of David’s work that you guys could try, just for starters. I’m sure Urs and David are brimming with ideas; the trick will be to find a bite-sized one.

Posted by: John Baez on August 27, 2009 8:31 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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Why not turn the journal club into the beginnings of this blog-based attack?

I think that’s a great idea. If you ‘journal club’ folks tried to prove a theorem or two, it might boost your energy level a notch or two. I get the feeling you were just trying to understand David Ben–Zvi’s work. That’s great, but mathematicians always seem to get more excited when they’re trying to prove stuff. Maybe there’s an easy extension of David’s work that you guys could try, just for starters. I’m sure Urs and David are brimming with ideas; the trick will be to find a bite-sized one.

I tried to initiate this investigation here recently as an outgrowth of the Journal Club activity.

This contains some bite-sized qustions, as indicated.

In the bigger picture, this is to be merged with the other thing I was hoping we might have a collaborative discussion about, you know, that exercise in groupoidification.

Posted by: Urs Schreiber on August 29, 2009 5:38 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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I tried to initiate this investigation here recently as an outgrowth of the Journal Club activity.

Indeed, you did. I would like to pick up those discussions about geometric function objects and the exercise in groupoidification.

I think I need to spend a little time digesting what you have already begun. That should generate enough questions or ideas to get a dialog going.

Posted by: Alex Hoffnung on September 3, 2009 3:42 AM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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André Joyal formalized the idea of ‘a structure we can put on finite sets’ in terms of espèces de structures, or ‘structure types’ [6, 17, 18]. Later his work was generalized to ‘stuff types’ [4], which are a key example of groupoidication.

Now we have Joyal (and Kock) writing about graphical species. So can the analogy be completed:

structure types : stuff types :: graphical species : *** ?

Hmm, what’s the general form of a category on which species can be defined? According to this lecture by Bergeron, you want it to be

…a groupoid with a comonoidal structure on the corresponding free additive category, with nice features.

In other words, we want to have a “good” notion of dissection for objects… (p. 36)

Might we say that tensorial species (p. 21 of Bergeron lecture) are made redundant by groupoidification?

On p. 33 he wants the values of his generalized species to be a category with two monoidal structures, hence $Vect$ works, as do $G-Set$ and $Varieties over a finite field$ (p. 45). I guess you’re seeing how far you can get with $Groupoid$ .

Any answers to the above might help me understand what to make of your stuff types not being able to be ‘directly recast’ (p. 2) in terms of Fiore et al.’s generalized species of structures. You could imagine presheaves of groupoids for values.

Posted by: David Corfield on August 24, 2009 10:24 AM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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David wrote:

Any answers to the above might help me understand what to make of your stuff types not being able to be ‘directly recast’ (p. 2) in terms of Fiore et al.’s generalized species of structures.

Well if that’s what you want to understand, I can just explain it. Generalized species of structures and stuff types generalize species in two different directions. Generalized species of structures are all about taking ordinary species and seeing them as part of a larger setup. Groupoidification is about taking species and replacing letting groupoids take the place of sets.

Let me explain ‘generalized species of structure’. A plain old fashioned species is a functor

$\widehat{F} : FinSet_0 \to Set$

where $FinSet_0$ is the groupoid of sets and bijections.

But this is the same as a profunctor

$F : FinSet_0 \longrightarrow 1$

where $1$ is the category with one object and one morphism, and I’m using long arrows to stand for profunctors since I don’t know how to write slashed arrows here.

Furthermore, $FinSet_0$ is the free symmetric monoidal category on one generator. In other words,

$FinSet_0 \simeq P 1$

where $P C$ is the free symmetric monoidal category on the category $C$ .

So, a species is a profunctor

$F : P 1 \longrightarrow 1$

A generalized species of structure is a profunctor

$F : P A \longrightarrow B$

where $A$ and $B$ are arbitrary categories. It turns out that there’s a cartesian closed bicategory where the morphisms are generalized species of structure, and this fact fits species very nicely into a larger context. In particular, there’s a funny way to ‘compose’ a profunctor

$P A \longrightarrow B$

and a profunctor

$P B \longrightarrow C$

to get a profunctor

$P A \longrightarrow C$

and if we take $A = B = C = 1$ this is the usual ‘substitution’ operation on species.

Quiz question: what word beginning with K does the above funny ‘composition’ process remind you of?

Quiz question: what further generalization would naturally include both stuff types and generalized species of structure?

Posted by: John Baez on August 24, 2009 5:31 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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Quiz 1: $K$ for Kleisli.

Quiz 2: I was suggesting presheaves of groupoids above. Given that a profunctor $F: P A \to B$ is a functor $P A \times B^{op} \to Set$ , we could have a functor $P A \times B^{op} \to Groupoid$ .

Posted by: David Corfield on August 24, 2009 7:25 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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David wrote:

Quiz 1: K for Kleisli.

Right, the idea of defining a morphism from $A$ to $B$ to be a profunctor

$P A \longrightarrow B$

should remind us of the Kleisli construction, especially if we don’t worry much about which way the arrow is pointing… which is okay here, since a profunctor

$X \longrightarrow Y$

is the same as a profunctor

$Y^{op} \longrightarrow X^{op}$

And the resemblance is even clearer when we realize that $P$ is a kind of monad: the ‘free symmetric monoidal category on a category’ monad.

But there’s some serious technical work to be done here, which that paper goes ahead and does. First of all, $P$ starts out life as a categorified monad, or ‘pseudomonad’, on $Cat$ . Second of all, we need to enhance it to become a pseudomonad on $Prof$ . Third of all, we need to check that this pseudomonad on Prof really does have a kind of ‘Kleisli bicategory’.

Quiz 2: I was suggesting presheaves of groupoids above. Given that a profunctor $F: P A \to B$ is a functor $P A \times B^{op} \to Set$ , we could have a functor $P A \times B^{op} \to Groupoid$ .

Right. All this should work, in principle. If this is what you meant all along, I apologize for my elementary remarks before. But you’ll note that implementing your suggestion demands that we categorify the setup even more: instead of the 2-category $Prof$ where an object is a category and a morphism is a functor

$A \times B^{op} \to Set$

you are suggesting that we work with some sort of 3-category where an object is a category (or 2-category) and a morphism is a weak 2-functor

$A \times B^{op} \to Gpd$

So, to actually implement this idea, you’d need to take the theory of monads and Kleisli categories and push it up to the 3-category level. And this is too much work for anyone to actually do, right now!

And this is roughly why Fiore and company say one cannot ‘directly recast’ the theory of stuff types into the language of generalized species.

In fact, I bet there’s a simpler reason they said this. I bet they just mean that stuff types aren’t examples of generalized species. I thought you didn’t understand this, so I thought I had to explain generalized species to you!

Remember, these are guys who don’t idly speculate in print about things they can’t do yet.

Posted by: John Baez on August 24, 2009 8:30 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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Remember, these are guys who don’t idly speculate in print about things they can’t do yet.

Perhaps they should visit us here sometime. Then we could ask what generalized species are good for.

Already we saw Fiore’s say

…generalised species encompass and generalise various of the notions of species used in combinatorics. Furthermore, they have a rich mathematical structure (akin to models of Girard’s linear logic) that can be described purely within Lawvere’s generalised logic. Indeed, I will present and treat the cartesian closed structure, the linear structure, the differential structure, etc. of generalised species axiomatically in this mathematical framework. As an upshot, I will observe that the setting allows for interpretations of computational calculi (like the lambda calculus, both typed and untyped; the recently introduced differential lambda calculus of Ehrhard and Regnier; etc) that can be directly seen as translations into a more basic elementary calculus of interacting agents that compute by communicating and operating upon structured data.

You may also be interested to read him say

I started working in domain theory, with my thesis Axiomatic Domain Theory in Categories of Partial Maps. There are many different categories of domains in which the same results hold–importantly adequacy theorems–and I wanted to show that these followed from general abstract principles. Thus, it was natural to axiomatize and investigate domain-theoretic models in this light. Along the way, and in enjoyable collaboration with various colleagues, I applied the axiomatic approach in different computational scenarios. In particular, with Eugenio Moggi and Davide Sangiorgi we constructed a denotational model of the $\pi$ -calculus for which we established full abstraction axiomatically. Not much later, it was pointed out to me (at a PSSL meeting, if I remember correctly) that the constructor that we use for modeling bound output had been considered by André Joyal in the different setting of combinatorial species of structures as a differentiation operator. Ever since I’ve been interested in understanding and deepening the connection. Yet another piece was added to the puzzle by my work with Gordon Plotkin and Daniele Turi on algebraic theories with variable-binding operators. There an operator of context extension (again similar to that of differentiation) serves for modeling variable-binding. The first part of my contribution to the FOSSACS conference proceedings gives an outline of the unifying technical themes underlying these developments.

Posted by: David Corfield on August 24, 2009 10:18 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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David wrote:

Then we could ask what generalized species are good for.

To me they’re just obviously good.

You should think of $Set$ as a categorified version of a commutative ring, or rig.

So, you should think of a profunctor $A \longrightarrow B$ as a categorified version of a linear operator.

So, you should think of a profunctor $A \longrightarrow P B$ as a categorified version of a linear operator from a vector space to a polynomial algebra. You can also think of this as a polynomial function from one vector space to another.

(You’ll notice now I’m writing $A \longrightarrow P B$ instead of $P A \longrightarrow B$ . I keep wanting to do that — you caught me doing it in my first comment, which I then corrected. This time I think I’m right to do so, but I could still have it backwards. In any case it’s easy to fix.)

So, I think it’s just obviously good to study these ideas, just like it’s obviously good to study vector spaces and polynomials.

But thanks for quoting Fiore about his motivations! Indeed, there’s a lot of cool but mysterious stuff going on in linear logic and now the differential lambda calculus, which suggests that logic is hybridizing first with linear algebra and now calculus. And this should be very related to the paper by Fiore, Gambino, Hyland and Winskel. After all, profunctors bring linearity into category theory, and generalized species bring in derivatives.

I know that Hyland is very much involved in this line of thought: when I was last in Cambridge, a few years ago, he organized a little micro-conference where Ehrhard explained the differential lambda calculus and I explained stuff types. I talked some more to Ehrhard this summer and we realized that he really is extending linear logic, which already includes an analogue of Fock space (= polynomial algebra), to also include annihilation and creation operators (= differentiation and multiplication operators)!

Feynman diagrams are sure to follow. Unfortunately there was still enough of a communication barrier that we weren’t able to fully comprehend the significance of it all. I barely understand linear logic, while he barely understands quantum field theory.

I think there should be some conferences where they take 4 or 5 people, throw them into a dungeon, and only let them out when they figure out what’s going on.

Posted by: John Baez on August 27, 2009 8:09 AM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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I noticed a big paper on the arXiv today:

Thomas M. Fiore, Wolfgang Lück and Roman Sauer, Finiteness obstructions and Euler characteristics of categories.

Abstract. We introduce notions of finiteness obstruction, Euler characteristic, $L^2$ -Euler characteristic, and Möbius inversion for wide classes of categories. The finiteness obstruction of a category $\Gamma$ of type (FP) is a class in the projective class group $KO_0(R\Gamma)$ ); the Euler characteristic and $L^2$ -Euler characteristic are respectively its $R\Gamma$ -rank and $L^2$ -rank. We also extend the second author’s K-theoretic Moebius inversion from finite categories to quasi-finite categories. Our main example is the proper orbit category, for which these invariants are established notions in the geometry and topology of classifying spaces for proper group actions. Baez-Dolan’s groupoid cardinality and Leinster’s Euler characteristic are special cases of the $L^2$ -Euler characteristic. Some of Leinster’s results on Moebius-Rota inversion are special cases of the K-theoretic Moebius inversion.

As for how it compares with the Euler characteristic of Tom Leinster, there is a whole section detailing this. For one thing, it seems they agree when the group action (simplifying things a bit here) is free, whereas if the action isn’t free they differ.

Posted by: Bruce Bartlett on August 25, 2009 1:51 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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The latest version of HDA7 has:

A section on tensors and traces, designed to satisfy Urs and link up with the $n$ Lab page on the trace of a span.
More information on various conventions for degroupoidification and how they relate to the homology and cohomology of groupoids.

I think it’s done!

Posted by: John Baez on August 28, 2009 10:09 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

The “arXiv version” link goes to arXiv:0908.2469, “A Prehistory of n-Categorical Physics”, not arXiv:0908.4305, “Higher-Dimensional Algebra VII: Groupoidification”. Also, I keep getting “page load error” messages from the math.ucr.edu server.

Posted by: Blake Stacey on September 3, 2009 7:14 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

Sorry about that screwed-up arXiv link: it’s fixed now. The UCR website was down this morning but now it’s back up.

Posted by: John Baez on September 3, 2009 10:26 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

Great!

Posted by: Blake Stacey on September 4, 2009 6:31 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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A new paper – Weak Pullbacks of Topological Groupoids:

This paper is part of a project we are currently working on, in which we are extending groupoidification from the discrete setting to the realm of topology and measure theory. Groupoidification is a form of categorification, introduced by John Baez and James Dolan. It has been successfully applied to several structures, which include Feynman Diagrams, Hecke Algebras and Hall Algebras. An excellent account of groupoidification and its triumphs to date can be found in [2]. So far, the scope of groupoidification and its inverse process of degroupoidification has been limited to purely algebraic structures and discrete groupoids.

[2] is HDA7.

Posted by: David Corfield on January 19, 2011 9:05 AM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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A new paper

I have just glanced over it. The notion of weak pullback of discrete groupoids and of topological groupoids is traditional (usually called a homotopy pullback), the observation here is that if the topological groupoids carry suitable Haar measures, then so does their weak pullback.

By the way, except possibly for the information introduced by such a measure, the “groupoidification” operation makes sense very generally in every $\infty$ -topos, for instance for topological $\infty$ -groupoids, for smooth $\infty$ -groupoids, for algebraic $\infty$ -groupods. That’s described here.

Posted by: Urs Schreiber on January 19, 2011 6:02 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

Is there any specific (measureless) topological weak pullback (or homotopy pullback) reference that you recommend for us to cite?

In terms of the weak pullback, the step up from discrete groupoids to purely topological groupoids is indeed easy. The topology on the weak pullback is naturally induced from the topologies of the groupoids that comprise the cospan, and the homomorphisms (functors) are automatically continuous. The upgrade to what we called “Haar groupoids” is the non-trivial part, which we present in the paper.

The measure data (e.g. the Haar system) is essential for many of the great things people do with Borel and topological groupoids: integration, representation theory, von-Neumann and C*-algebras. Our attempt at degroupoidification also relies on the measure data.

There is an underlying terminology issue, which we mention at the end of the introduction to the paper. Haar systems and measures for groupoids (in both the topological context and in the purely measure theoretic context) have been extensively studied. I think George Mackey was the pioneer, over half a century ago. It would have been nice to call our groupoids “measure groupoids”, but the term is reserved in the literature for groupoids in the purely measure theoretic setting. Their topological counterparts are usually referred to as “locally compact groupoids”, and they often carry some or all of the measure data we imposed.

Posted by: Aviv Censor on January 21, 2011 8:32 AM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

Hi Aviv,

I thought I’d mention that the kind of stacks that Urs is referring to are just like the ones I talked about in your seminar last year. The only real difference is that I was working with manifolds (for the site category) rather than topological spaces. In those notes, what I called the “2-fiber product of stacks” is the pullback that Urs is talking about.
(My apologies if you knew that already.)

Posted by: Chris Rogers on January 21, 2011 5:47 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

I don’t know if there is a canonical reference to ‘weak pullback’ - which should more properly be called bipullback to avoid confusion with a weak pullback in a 1-category - but one place where it is proved that in a finitely complete category the ‘standard description’ really is the bipullback is in section 3 of this paper by Enrico Vitale (he really verifies the universal property and so on, most people leave it as an exercise or similar).

Posted by: David Roberts on January 21, 2011 10:19 PM | Permalink | Reply to this

(2,1)-limits

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I don’t know if there is a canonical reference to ‘weak pullback’

There is: 2-limit. See there references listed there.

which should more properly be called bipullback

Unfortunately the prefix conventions here differs rather widely. The “bi-” prefix is however usually reserved for the correct general notion of limit in the correct general notion of 2-categories (“bicategories”).

What is somewhat important for the discussion here, however, is that we are talking about pullbacks of topological groupoids . These happen not in the most general 2-categorical context, but in a $(2,1)$ -category . Bilimits/weak 2-limits that are actually $(2,1)$ -limits are considerably easier to handle and considerably more is known about them.

In fact, under the name homotopy limits these are well understood and classical since the 1960s.

Here “homotopy limit” refers to a special means of computing or presenting $(2,1)$ -limits and generally $(\infty,1)$ -limits in terms of 1-categorical limits.

For instance a “weak pullback” of bare groupoids, taking place in the $(2,1)$ -category $Grpd$ , may be computed as a homotopy limit in the folk model structure on the 1-category $Grpd$ , or in the Quillen model structure on simplicial sets on diagrams of 1-truncated Kan complexes. (Examples for such computations are worked out at homotopy limit.)

More generally, weak pullbacks in (2,1)-toposes – such as those in which topological groupoids, Lie groupoids, algebraic groupoids etc. live – may be computed as homotopy limits in a model structure on simplicial presheaves over the given $(2,1)$ -site on diagrams of 1-truncated Kan-complex valued presheaves.

Examples of such computations of pullbacks of topological $\infty$ -groupoids are for instance at Euclidean topological $\infty$ -groupoids where topological principal $\infty$ -bundles are discussed as weak pullbacks of the point along their classifying maps of topological $\infty$ -groupoids.

Posted by: Urs Schreiber on January 22, 2011 12:38 AM | Permalink | Reply to this

Re: (2,1)-limits

Is there a corresponding page for homotopy quotient
and is that an instance of homotopy colimit? I’m linguistically challenged.

Posted by: jim stasheff on January 22, 2011 1:30 PM | Permalink | Reply to this

Re: (2,1)-limits

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Is there a corresponding page for homotopy quotient and is that an instance of homotopy colimit?

Yes, a homotopy quotient is a special kind of homtopy colimit, just as for quotients and colimits.

Currently the $n$ Lab has a single page on both homotopy limits and colimits. But of course a homotopy limit is equivalently a homotopy colimit in the oppposite category, and vice versa.

Posted by: Urs Schreiber on January 22, 2011 11:12 PM | Permalink | Reply to this

Re: (2,1)-limits

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Is there a corresponding page for homotopy quotient and is that an instance of homotopy colimit?

There is also the page on groupoid objects in an $(\infty,1)$ -category which now has a section with remarks on the the relation to $(\infty,1)$ -quotients / homotopy quotients.

Posted by: Urs Schreiber on January 23, 2011 6:50 PM | Permalink | Reply to this

Re: (2,1)-limits

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weak pullbacks in (2,1)-toposes – such as those in which topological groupoids, Lie groupoids, algebraic groupoids etc. live – may be computed as homotopy limits in a model structure on simplicial presheaves over the given (2,1)-site on diagrams of 1-truncated Kan-complex valued presheaves.

Or, if you like to keep easy things easy, in a model structure on presheaves of groupoids, no simplicial gizmos required. (-:

In the spirit of Max Kelly (keeping terminology simple), I would tend to just call this a “pullback” – when it is understood that we’re working in a (2,1)-category and we only care about things up to equivalence, it is the only sensible notion of pullback. If your (2,1)-category happens to be strict, then you can talk about strict 2- (or (2,1)-) pullbacks, but they’re usually only interesting insofar as they may help you compute the non-strict ones (e.g. if one of the legs is a fibration).

Posted by: Mike Shulman on January 23, 2011 4:41 AM | Permalink | Reply to this

Re: (2,1)-limits

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weak pullbacks in (2,1)-toposes – such as those in which topological groupoids, Lie groupoids, algebraic groupoids etc. live – may be computed as homotopy limits in a model structure on simplicial presheaves over the given (2,1)-site on diagrams of 1-truncated Kan-complex valued presheaves.

Or, if you like to keep easy things easy, in a model structure on presheaves of groupoids, no simplicial gizmos required. (-:

Right, okay. I wrote a quick entry model structure for (2,1)-sheaves with some remarks on this (far from comprehensive, am in a rush).

An obvious thing to do is to take the natural model structure on groupoids $Grpd_{nat}$ , form the projective model structure on functors $[C^{op}, Grpd_{nat}]$ from this (not sure if it would work for the injective structure, too, no time to think about it right now) and then left Bousfield localize at the set of covering sieve inclusions with respect to some coverage on $C$ .

Another approach would be to first start with the full model structure for $(\infty,1)$ -sheaves $[C^{op}, sSet_{Quillen}]_{loc}$ and then localize this to 1-truncated objects by localizing at morphisms $\partial \Delta[n]\cdot U \to \Delta[n]\cdot U$ for $n \geq 2$ and $U \in C$ . (This forms the $(2,1)$ -topos inside the $(\infty,1)$ -topos).

There is a nice article by Sharon Hollander (her PhD thesis, as far as I know), A homotopy theory for stacks, where essentially these two approaches are considered and shown to be Quillen equivalent.

However, while nice, it is not entirely clear that this is necessarily easier than (or in practice actually much different from) using the full model structure for $(\infty,1)$ -sheaves and just concentrating on the 1-truncated objects in there. This is what Jardine does in his Stacks and the homotopy therory of simplicial presheaves.

In the spirit of Max Kelly (keeping terminology simple), I would tend to just call this [a $(2,1)$ -pullback] a “pullback” – when it is understood that we’re working in a (2,1)-category and we only care about things up to equivalence, it is the only sensible notion of pullback.

Sure, when we all agree on what are talking about we’ll just say “pullback” and drop all prefixes. But I think in the kind of discussion we are having here it is good to make qualifiers explicit.

Posted by: Urs Schreiber on January 23, 2011 1:11 PM | Permalink | Reply to this

Re: (2,1)-limits versus…

I suggest it depends on how often one would say
e.g. (2,1)-pullback in a given article/blogpost
Once or twice - OK
but if it’s going to be repeated multi-times
I prefer to follow Frank Adams:
e.g. henceforth in this article/blogpost, pullback
SHALL mean (2,1)-pullback

Posted by: jim stasheff on January 23, 2011 1:26 PM | Permalink | Reply to this

Re: (2,1)-limits

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it is not entirely clear that this is necessarily easier than (or in practice actually much different from) using the full model structure for (∞,1)-sheaves and just concentrating on the 1-truncated objects in there.

I would be willing to bet that at least some readers of this blog would find it at least conceptually easier. Also, the canonical model structure on Gpd is much easier to describe and construct than the standard model structure on sSet, and that probably goes double for presheaves thereof.

Posted by: Mike Shulman on January 23, 2011 8:01 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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Is there any specific (measureless) topological weak pullback (or homotopy pullback) reference that you recommend for us to cite?

The keyword under which this is usually discussed is topological stacks :

One embeds groupoids internal to topological spaces into the $(2,1)$ -topos of $(2,1)$ -sheaves ( = stacks) on a suitable site of all topological spaces. After that embedding the morphisms between the topological groupoids are automatically the correct Morita morphisms of topological groupoids and the equivalences are automatically the correct Morita equivalences.

The $n$ Lab lists some references at topological stack and more at geometric stack.

In the standard texts weak pullbacks are usually defined a few pages before the definition of atlas/chart of a stack.

Posted by: Urs Schreiber on January 21, 2011 2:47 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

I wish it weren’t me saying this, but reading

Groupoidification is a form of categorification, introduced by John Baez and James Dolan.

I do feel somewhat left out, since it seems to me that the groupoidification project began during a time of numerous conversations between myself and James Dolan. Of course James and John were conversing about this too, and I wasn’t part of those conversations, just as John wasn’t part of the conversations between me and James.

Certainly I am not blaming the authors of the paper David is referring to; how would they know?

Exact accreditation and attribution are close to impossible here, but I feel sure that James would acknowledge the influence of our collaboration, if asked. I might be able to rummage up specific emails to set some of the history straight.

Posted by: Todd Trimble on January 20, 2011 4:46 PM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

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I’ll try to get it dealt with. If anyone looks at HDA7, they’ll see that every time Jim Dolan is mentioned, you are too — except for his early work on ‘From Finite Sets to Feynman Diagrams’.

Posted by: John Baez on January 21, 2011 5:59 AM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

We apologize to Todd, we will have this fixed. Alex Hoffnung and John Baez have corrected us in the past, but we forgot.

Posted by: Aviv Censor on January 21, 2011 6:56 AM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

And not to mention some of the other people involved in the early days of categorification, like Yetter (even though this was a slightly different tack)…

I emailed the authors (with no reply to date) asking about the 2-categorical aspects. For example, they discuss the notion of functor between measurable groupoids, but not of natural transformation. Further, I expect there could be a good notion of anafunctor between Haar groupoids. But the measure-theoretic side of it is really not my cup of tea, so I await their reply with some anticipation.

Posted by: David Roberts on January 21, 2011 1:23 AM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

Whoops!

Groupoidification is a form of categorification, introduced by John Baez and James Dolan.

I parsed that sentence incorrectly. I took it to say that John and Jim introduced categorification, rather than groupoidification, which now I see is the intended meaning. Well of course Yetter was not involved in that (was he?) :)

Posted by: David Roberts on January 21, 2011 6:08 AM | Permalink | Reply to this

Re: Higher-Dimensional Algebra VII: Groupoidification

For us, the really interesting notion is that of Morita equivalence, which is fundamental in the study of measurable and topological groupoids, and their use in non-commutative geometry, operator algebras, etc. We are still working on figuring out the precise way to relate Morita equivalence to the groupoids we call Haar groupoids in our paper.

We plan to address the issue of Morita equivalence, along with other other categorical issues (like natural transformations) in a separate paper about the category HG, which we currently started working on.

Posted by: Aviv Censor on January 21, 2011 8:36 AM | Permalink | Reply to this

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August 23, 2009