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June 1, 2007

Quantization and Cohomology (Week 26)

Posted by John Baez

In this week’s seminar on Quantization and Cohomology, we finished sketching a proof that:

  • Principal GG-bundles over MM correspond to smooth anafunctors hol:Disc(M)Ghol: Disc(M) \to G, where Disc(M)Disc(M) is the smooth category with MM as the space of objects, and only identity morphisms.
  • Gauge transformations between principal GG-bundles over MM correspond to smooth ananatural transformations between such anafunctors.

We began by giving the definition of ‘smooth ananatural transformation’:

  • Week 26 (May 29) - The definition of smooth anafunctor (review), and the definition of smooth ananatural transformation (new). Why the first Cech cohomology of a smooth space MM with coefficients in a smooth group GG is isomorphic to the set of smooth anafunctors F:Disc(M)GF: Disc(M) \to G mod smooth ananatural isomorphisms. Generalization: the first Cech cohomology of a smooth category. Dreams of further generalizations.

Last week’s notes are here; next week’s note are here. Next week is the last week of this year-long seminar!

A couple of small technical points:

  1. When defining smooth ananatural transformations α:FG\alpha: F \Rightarrow G between smooth anafunctors F,G:CDF,G : C \to D, I mistakenly defined them only between anafunctors that were defined using the same open cover of Ob(C)Ob(C). It’s easy to generalize this definition to the case where FF and GG are defined using different covers: just take a common refinement. We need this to establish the connection to Cech cohomology.
  2. By “the first Cech cohomology of MM with coefficients in GG”, we really mean the first Cech cohomology of the sheaf of smooth GG-valued functions on MM. We can define first Cech cohomology with coefficients in a sheaf of groups even when those groups are nonabelian. More general nnth cohomology either requires abelian coefficients or, better, coefficients in a stack of nn-groups.
  3. We can afford to be sloppy about the difference between (ana)natural transformations and (ana)natural isomorphisms when our functors take values in a group, or groupoid — since then they’re the same thing! In the notes we’ve often been indulging in this sloppiness, without commenting on it.
Posted at June 1, 2007 7:30 PM UTC

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Read the post Quantization and Cohomology (Week 25)
Weblog: The n-Category Café
Excerpt: How describing bundles in terms of Cech cohomology secretly amounts to describing them in terms of smooth anafunctors.
Tracked: June 1, 2007 7:45 PM

Re: Quantization and Cohomology (Week 26)

After attending Stolz and Teichner’s recent talks in Edinburgh, I’ve been discussing with Urs in private emails the relationship of their idea of “smooth functors” with the philosophy presented, eg. in John’s Quantization and Cohohomology seminar above.

I think ultimately the two approaches are “almost” (up to some caveats) tautologically the same… though they’re wrapped slightly differently. I would say that Stolz and Teichner’s approach is a more openly stack-like approach, while the approach presented above by John is also “stacky”, though somewhat shy to come out of the closet and admit it :-) Just kidding guys.

I’ll compare these two approaches in the way these approaches think of a “principal GG-bundle”. Of course, everything works for GG-bundles with connection as well, but let’s stick to the case John has discussed explicitly in his lecture.

Both approaches are trying to formalize the notion of a “smooth assignment of a GG-torsor to every point xx of a manifold MM”.

John has demonstrated one way to do this; one thinks of a GG-bundle as a smooth anafunctor F:Disc(M)GF : Disc(M) \rightarrow G, where “smooth” means that Disc(M)Disc(M) is regarded as a groupoid internal to smooth spaces.

The other way to do it is the “proudly stacky” way. You think of the discrete path groupoid Disc(M)Disc(M) simply as a plain discrete groupoid MM… no smooth structure. Lurking in the background of course is the stack MM over smooth manifolds, whose value at XX is the discrete groupoid whose objects are elements of the hom-set Hom(M,X)Hom(M,X).

On the other hand, we also have the groupoid GTorG-Tor, whose objects are GG-torsors and morphisms are torsor morphisms. There is also a stacky version of this, GTorG-Tor , which associates to a manifold XX the category of principal GG-bundles over XX.

Ok. Here is my paraphrasing of the Stolz-Teichner approach. We are trying to decide when a functor

(1)F:MGTor F : M \rightarrow G-Tor

is smooth. Recall : MM is the discrete groupoid on MM.

Here’s the solution : we say that FF is smooth if it can be extended to a stack map

F:MGTorF : M \rightarrow G-Tor

such that the value of FF on the point *Man* \in Man equals FF, i.e.

FF (*)(*) =F= F.

Of course, that’s essentially the same as saying that our objects of interest are just stack maps from MM to GTorG-Tor . But it’s psychologically a little different… I would like to line it up with John’s approach, thus I would like to think of the primary geometric object as the assignment of a GG-torsor to every point xMx \in M :

(2)xP x. x \mapsto P_x.

The criterion that this extends to a stack map is just our formalization of the requirement that this assignment is “smooth”.

Anyhow, what I’m essentially saying is that the formalism of groupoids internal to smooth spaces, anafunctors, etc. can also be phrased in the language of stacks. John is well aware of this! I think it’s just been a matter of choosing the language which people will be most comfortable with in the long run.

Posted by: Bruce Bartlett on June 3, 2007 1:26 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

I think ultimately the two approaches are “almost” (up to some caveats) tautologically the same…

Not the same. Just equivalent. :-)

One approach is global, one is local. One is in terms of the descent data of the other.

A stack is a prestack such that global objects are exactly those obtained from gluing local objects.

A bunch of local objects together with their gluing is an object in a “descent category”. This is the cocycle data.

Here “is” means: “canonically equivalent”, as opposed to just equivalent. There are 1001 different ways to talk about the structures we are talking about here, and all of them are equivalent. But some are more equivalent than others.

An anafunctor is a neat way to re-encode an object in a category of descent data into a single functor:

the covering category involved in the definition of the anafunctor:

is generated

- from “local morphisms” (sitting in a cover)

- and from “jumps” between patches.

Modulo an obvious relation.

The value of the anafunctor on the first kind of generators is the “local object”. Its value on the second kind of generators is precisely the gluing data. Together, the one anafunctor defines one object in the descent category.

Its respect for the “obvious relation” is precisely the simplex-shaped thing in the definition of an object of a descent category.

When we have a functor from here to there, we have two ways of desciding whether or not it is smooth:

either there is a smooth structure globally on everything. (This is what Stolz and Teichner consider). Then: just check if your functor is smooth with respect to that.

or we don’t bother to put a global smooth structure on everything. Instead, we locally trivialize our functor and extract its descent data (this is what I do with Konrad).

Then one can say that the original thing was smooth if its descent data is.

This is useful, because we can entirely forget about the global thing in this case and just consider descent data only. This is what happens when one works entirely with anafunctors.

The advantage of this is that it is more “hands on”. Especially when we categorify, the gloabl smooth structures on the global things we need quickly becomes hard to come by. Already the total spaces of 2-bundles are hard to come by.

But in terms of cocycle data/descent data/anafunctors, it is very easy to say what a 2-bundle is: just a cocycle with values in a 2-group.

Whether or not we get get back to thet global object from there is a different issue, then.

Here are some links to things I mentioned:

the relation between anafunctor and descent data was discussed in Local Transition of Transport, Anafunctors and Descent of n-Functors. For the n=1n=1-case this is spelled out in great detail in my paper with Konrad. Some of the global vs. local issues are discussed in the comment section of Quantization and Cohomology (week 23) here.

I made some remarks on how this relates to Stolz-Teichner’s approach in a couple of comments here: Globally smooth parallel transport.

This pdf doesn’t work out the two main issues as cleanly as it should. There are really two of them:

there are two ways to put a global smooth structure on large categories like GTorG\mathrm{Tor} or Vect\mathrm{Vect}:

i) proceed the “standard” way as Stolz-Teichner do

ii) instead use the Chen-smooth space philosophy and instead of stacks of bundles consider sheaves of plots (which are such that there is a bundle with some properties).

In the very last section of these notes I talk about how both these approaches lead to something very much the same, up to some subtlety.

The other issue is which of these global smooth structures (either of kind i) or of kind ii) ) lead to which local data: to which kind of anafunctors.

Posted by: urs on June 3, 2007 5:12 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

One reason that I currently prefer to think in terms of stacks, rather than in terms of smooth categories and smooth functors, is because the “stacky way” works nicely for cobordisms, which I think is difficult to handle in the other approach. Perhaps you could persuade me otherwise!

Let’s say we’re thinking about the classic Segal-style problem of functors

(1)F:2Cob(M)Vect F : 2Cob(M) \rightarrow Vect

where 2Cob(M) is the category whose objects are closed 1d manifolds Σ\Sigma equipped with a smooth map into MM, i.e. ΣfM\Sigma \stackrel{f}{\rightarrow} M, and whose morphisms are abstract 2d cobordisms XX equipped with a smooth map XgMX \stackrel{g}{\rightarrow} M, together with boundary identifications of g\partial g with the incoming and outgoing boundaries… you know the deal.

We want to make sense of what it means to say that “FF is a smooth functor”.

The main point I want to stress here is that the class of objects and the class of morphisms of 2Cob(M)2Cob(M) are not sets! It seems difficult to think of 2Cob(M)2Cob(M) as a smooth category, which is supposed to be a category internal to smooth spaces… in other words, the objects and morphisms must form honest sets.

On the other hand, this problem is handled effortlessly in the stacky approach. As Stolz and Teichner stress, this is the viewpoint which says that :

A functor F:2Cob(M)VectF : 2Cob(M) \rightarrow Vect is smooth if it can be extended to smooth families .

In other words, we consider the stacks (I’m just thinking of a stack as a weak 2-functor Man opCatMan^op \rightarrow Cat satisfying gluing conditions) 2Cob s(M)2Cob_s (M) and Vect sVect_s. The value of 2Cob s(M)2Cob_s(M) on a manifold UManU \in Man is the category whose objects are “smooth families of 1d manifolds mapped into MM, parametrized by UU”.

In other words, the objects of 2Cob s(M)(U)2Cob_s(M)(U) are fiber bundles PUP \rightarrow U, whose fibers are closed 1d manifolds, equipped with a map f:PMf : P \rightarrow M.

And morphisms are “smooth families of cobordisms in MM parametrized by UU”. And so on.

Notice that the value of the stack 2Cob s(M)2Cob_s (M) on a point is just the category 2Cob(M)2Cob(M):

(2)2Cob s(M)(*)=2Cob(M). 2Cob_s(M) (*) = 2Cob(M).

Similarly Vect sVect_s is just the vector bundles stack : it assigns to UManU \in Man the category of vector bundles over UU. And the value of Vect sVect_s on a point is just VectVect:

(3)Vect s(*)=Vect. Vect_s (*) = Vect.

Finally, we say a functor

(4)F:2Cob(M)Vect F : 2Cob(M) \rightarrow Vect

is smooth if it can be extended to a map of stacks

(5)F s:2Cob s(M)Vect s F_s : 2Cob_s (M) \rightarrow Vect_s

such that the value of F sF_s on the point equals FF,

(6)F s(*)=F. F_s (*) = F.

One might argue that you could solve the “cobordisms don’t form a set” problem by dropping the requirement that smooth spaces actually be concrete sets equipped with extra structure.

But my reply to that would be : wouldn’t that just be another reluctant inch down the road that leads to fully embracing stacks? I’m concerned that smooth spaces are a kind of “God of the gaps”, if you know what I mean.

Posted by: Bruce Bartlett on June 3, 2007 8:26 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

This reminded me of KT Chen’s approach to differential forms and even connections in
the case of loop spaces and path spaces
in whihc smoothness is described in terms of
test spaces, i.e. ordinary smooth fin dim spaces mapped into the function space

but that was long before stacks

jim

Posted by: jim stasheff on June 4, 2007 12:56 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

Was it really before stacks?

I’m not so knowledgeable about the early history of stacks, but the first publication I’m aware of is Mumford’s “Picard groups of moduli problems” in the 1965 proceedings of a 1963 conference. From what I remember he didn’t use the word “stack”, but all the ideas are there. The concept was written formally in Deligne-Mumford, which appeared in 1969. Another key early publication would be Giraud’s thesis. I don’t know what it was finished, but his book appeared in 1971.

Anyway, Chen’s paper “On differentiable spaces” appeared in 1986, but maybe he had his theory going for a long time before that?

Posted by: James on June 4, 2007 8:06 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

Chen’s work on smooth spaces goes back at least to 1975:

  • K. T. Chen, Iterated integrals, fundamental groups and covering spaces, Trans. Amer. Math. Soc. 206, (1975), 83–98.

He modified his definition a bit over the years.

I don’t really know when stacks were first considered. But, they seem to go back at least to 1969, from what James writes. I said Grothendieck invented them. But maybe it was Deligne–Mumford?

Posted by: John Baez on June 4, 2007 8:06 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

I think it’s probably not accurate to say Deligne and Mumford invented stacks, but they probably did invent the word. I just checked their paper, and where they give the definition, they say they’re proposing it as the English translation of the champs of Giraud’s thesis.

From what I remember, Grothendieck often cites Giraud’s thesis in his unpublished writings on these things. On the other hand, I think he usually gives full credit to his students even when their theses were completely outlined by him. On the other other hand, according to Deligne-Mumford, Giraud’s thesis was submitted at U Columbia (presumably the one in New York), so maybe he wasn’t a student of Grothendieck’s after all. I’m confused. Probably everyone who went to the SGA seminars knows exactly what the situation is.

But I guess I’d still put my money on Grothendieck. If stacks weren’t in the set of his inventions, they’re certainly in its closure.

Posted by: James on June 5, 2007 4:39 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

OK. I looked a bit through Récoltes et Semailles (available here). Grothendieck leaves no doubt on the matter.

Footnote 573:

Je signale, parmi ces idées et outils que j’avais introduits, qui ont été enterrés et qui ont fini par s’imposer malgré le boycott instauré par Deligne et mes autres élèves cohomologistes : les catégories dérivées, les motifs (version étriquée, il est vrai) et le yoga des catégories de Galois-Poincaré-Grothendieck (rebaptisées “tannakiennes” pour les besoins de l’Enterrement), le formalisme de cohomologie non commutative autour des notions de champs, gerbes et liens (développé par Giraud d’après les idées de départ introduites par moi à partir de 1955).

And footnote 927:

Cette terminologie [champs,…] suggestive a été introduite par Giraud, à la place d’une terminologie provisoire (un peu à la va-comme-je-te-pousse) que j’utilisais à partir de 1955 (genre “catégories fibrées de nature locale” et autres noms mal venus, pour des notions dont la nature fondamentale exigeait des noms lapidaires et frappants).

(16 juin) A la première page de l’introduction à son livre, Saavedra parle du “formalisme pour l’algèbre homologique non commutative introduit par Giraud”. C’est un des nombreux endroits où j’ai pu sentir quelqu’un de plus futé que l’auteur de ce livre, qui lui a “tenu la main”… le même qui se plaît à ne parler de “catégories dérivées” que pour ajouter dans la foulée “introduites par Verdier” (alors qu’il sait parfaitement, dans l’un et l’autre cas, à quoi s’en tenir… ).

I’d say the case is closed.

Posted by: James on June 6, 2007 8:37 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

Thanks for straightening out the history.

Since I don’t read much French, I don’t understand the last passage. Is Grothendieck complaining about how Saavedra said Giraud introduced nonabelian cohomology, and how someone said Verdier introduced derived categories?

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

Re: Quantization and Cohomology (Week 26)

OK. Here is my attempt. This is a big step for me, attempting to translate in public…

I list here, among the ideas and tools which I introduced, which were buried and which wound up being essential in spite of the boycott organized by Deligne and my other cohomological students: derived categories, motives (a limited version, it is true) and the yoga of the categories of Galois-Poincaré-Grothendieck (renamed “tannakiennes” for the needs for the Burial), the formalism of noncommutative cohomology around the concepts of stacks, gerbes and liens (developed by Giraud following the initial ideas introduced by me beginning in 1955).

**

This suggestive terminology [stacks,…] was introduced by Giraud, in place of provisional terminology (not carefully thought out(?)) that I began using in 1955 (like “fibered categories of a local nature” and other ill-conceived names for concepts whose fundamental nature required names concise and striking).

(June 16) On the first page of the introduction to his book, Saavedra speaks about the “formalism for noncommutative homological algebra introduced by Giraud”. This is one of many places where I have been able to sense somebody smarter than author of the book, who “guided his hands”, the same person who likes to speak about “derived categories” only to add in the same breath “introduced by Verdier” (when he knows perfectly, in each case, what to believe).

Posted by: James on June 7, 2007 8:22 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

One reason that I currently prefer to think in terms of stacks, rather than in terms of smooth categories

Okay, there may be good reasons to pass from Chen-smooth spaces to stacks over manifolds.

Chen-smooth spaces are a very special sort of sheaves (hence even more special sort of stacks) over a special sort of manifolds (open subsets or convex subsets of n\mathbb{R}^n). The reason to restrict to such a special sub case of much more general sheaves or stacks on manifolds is that it is

a) sufficient for many applications

b) very useful for concrete computations.

I’d think the jury is still out on whether or not we can equip cobordisms with a Chen-smooth structure, but but quite possibly you are right, and this notion is too rigid for that case.

But what I would like to emphasize is how this issue of passing from Chen-smooth spaces to more general smooth spaces is independent of anything regarding anafunctors and the like.

You seemed to have started this discussion by pointing to the anafunctor way of doing things here as opposed to the stacky way of doing things there – and my point is that this is not a sensible distincion.

Here is a category Smth \mathrm{Smth} of smooth spaces. It may be either Chen-smooth spaces or stacks over manifolds or anything in between.

Here is the “global” way of talking about smooth functors f:CDf : C \to D, with respect to this notion of smoothness:

Realize both CC and DD as categories internal to Smth\mathrm{Smth}. Then let ff be a functor internal to Smth\mathrm{Smth}.

For instance, take CC to be a category of cobordisms and DD a category of vector spaces.

The anafunctor way of doing things would instead proceed like this:

we’d pick yet another category internal to Smth\mathrm{Smth}, called |F| |F| such that it comes with a surjection |F|C |F| \to C such that this is a surjective equivalence onece we forget the smooth structure.

Then consider not smooth functors on CC but smooth functors on |F||F|.

That would be a smooth anafunctor on CC, in this setup.

One reason for doing this would be that by passing to the cover |F||F| we would be able to pass to a “small local model” for DD, without loosing any information contained in the original functor ff.

If we have an inclusion GD G \hookrightarrow D with GG being a small category (still inside Smth\mathrm{Smth}, as everything else in sight), smooth functors F:|F|G F : |F| \to G may still well contain all the information there is in smooth functors f:CD. f : C \to D \,. (They will if ff admits a local trivialization with respect to the inclusion GDG \hookrightarrow D).

Working with the anafunctor may be useful or it may not be useful, depending on your application.

Posted by: urs on June 4, 2007 10:11 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

I feel that Urs, John and I all have different conceptions of the manner in which “stacks”, “anafunctors”, “Chen smooth spaces” and “smooth categories and smooth functors” are being compared with each other in this discussion.

Let me outline more clearly the actual way in which I am comparing these approaches. (I am trying to frame the debate my way, just as in the presidential debate last night!)

Firstly, I respond to Urs remarks.

The comparison is not between:

(1)smoothspacesvsstacksovermanifolds. smooth spaces \quad vs \quad stacks over manifolds.

Rather, it’s between

(2)"categoriesinternaltosmoothspaces" wayofthinkingvsstackywayofthinking. \text{"}categories internal to smooth spaces \text{" } way of thinking \quad vs \quad stacky way of thinking.

There’s a big difference!

Both of these approaches are trying to answer the following question:

What does it mean for a functor F:CDF : C \rightarrow D to be smooth?

Smooth spacey answer : Find a way to think of CC and DD as categories C^\hat{C} and D^\hat{D} internal to Smth, the category of smooth spaces. Then FF is smooth if it can be expressed as an internal anafunctor

(3)F^:C^D^. \hat{F} : \hat{C} \rightarrow \hat{D}.

Stacky answer : The stacky answer is :

Smooth is as smooth does.

Lol :-) Anyhow, you don’t put a smooth structure on anything, and you certainly don’t think in terms of internal categories at all!

Rather, you say :

F is smooth if it can be extended to smooth families.

This means you construct a stack C sC_s (and also a stack D sD_s), whose value on a manifold UManU \in Man is the category whose objects are smooth families of objects in CC parametrized by UU, and whose morphisms are smooth families of morphisms in CC parametrized by UU. I pointed out how to do this explicitly via fiber bundles in my previous comment.

The main point is that clearly, on a point *Man* \in Man, these stacks just give back the original categories:

(4)C s(*)=C,D s(*)=D. C_s (*) = C, \quad D_s (*) = D.

And then you say that our functor

(5)F:CD F : C \rightarrow D

is smooth (remember : CC and DD are just plain categories, not internal to anything, and FF is a plain functor!) if it can be extended to a stack map

(6)F s:C sD s F_s : C_s \rightarrow D_s

such that the component of F sF_s on the point *Man* \in Man is FF,

(7)F s(*)=F. F_s (*) = F.

Again, I want to stress that this is completely different to the picture Urs had in mind here :

Here is a category Smth of smooth spaces. It may be either Chen-smooth spaces or stacks over manifolds or anything in between.

….

Realize both CC and DD as categories internal to Smth. Then let FF be a functor internal to Smth.

Okay, now that I’ve stressed the difference between the two approaches, let me also sketch the ultimate similarity. Because… what is a stack map F s:C sD sF_s : C_s \rightarrow D_s anyway? Well, for each manifold UManU \in Man, we have a functor

(8)F s(U):C s(U)D s(U). F_s(U) : C_s (U) \rightarrow D_s (U).

So by definition we have something which assigns

(9)smoothfamiliesofobjectsinCparametrizedbyUsmoothfamiliesofobjectsinDparametrizedbyU. smooth families of objects in Cparametrized by U \mapsto smooth families of objects in D parametrized by U.

And the same for morphisms.

In other words, at the end of the day, a map of stacks is very similar to what you’d get out in the end if you started off thinking in terms of categories and functors internal to the category Smth of smooth spaces!! Let me say that once more, for effect:

(10)mapofstacksF s:C sD ssimilarinternalanafunctorF^:C^D^. map of stacks F_s : C_s \rightarrow D_s \quad \stackrel{similar}{\leftrightarrow} \quad internal anafunctor \hat{F} : \hat{C} \rightarrow \hat{D}.

But it’s a priori completely different, as I’ve explained. There are no explicit internal categories in the stacky “tao” at all!

I guess I’ve run out of space, so I’ll comment on the difference between John’s conception of the debate and my own, below.

But I can’t resist a final word : it’s clear from the way I’ve outlined the debate, that thinking of smooth functors in the “stacky” way is not a big generalization of the “smooth spacey” way. They really are just different ways of approaching the same problem - but I’m still of the opinion that the stacky way is a bit superior.

Posted by: Bruce Bartlett on June 4, 2007 11:38 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

This means you construct a stack C sC_s (and also a stack D sD_s), whose value on a manifold UManU \in \mathrm{Man} is the category whose objects are smooth families of objects in CC parametrized by UU, and whose morphisms are smooth families of morphisms in CC parametrized by UU

Yes. And this is the same as saying that we have a category internal to stacks on manifolds.

Think of a Chen-smooth structure explicitly as a sheaf, and you can give a description of a Chen-smooth category in exactly the words you use above!

Posted by: urs on June 4, 2007 11:50 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

Yes and no… but more “no” than “yes”, unless I am terribly confused!

You are forcing me to be completely explicit here. These issues are very subtle and certainly can be very confusing, even after one has written them down in black and white!

Let’s settle on some definitions, taking great pains over our notation. A smooth space X\mathcal{F}_X is a concrete sheaf. So it is a sheaf

(1) X:Man opSet \mathcal{F}_X : Man^op \rightarrow Set

whose value on a manifold UManU \in Man is actually a subset of the set-wise maps from UU to some fixed set XX:

(2) X(U)Hom Set(U,X). \mathcal{F}_X (U) \subset Hom_\text{Set} (U, X).

The collection of smooth spaces forms a category Smth which is a full subcategory of the category Sheaves.

A smooth category is a category CC internal to Smth. So it consists of objects Ob¯(C),Mor¯(C)Smth\overline{Ob}(C), \overline{Mor}(C) \in Smth together with the usual internal category data.

Also, we’ll say that a sheafy smooth category is a category internal to Sheaves. Thus in particular it consists of sheaves Ob¯(C),Mor¯(C)Sheaves\overline{Ob}(C), \overline{Mor}(C) \in Sheaves.

Finally, by a “stack” we’ll simply mean a contravariant weak 2-functor :Man opCat\mathcal{F} : Man^op \rightarrow Cat satisfying appropriate gluing conditions, as can be found in the nice expository article on smooth stacks by Metzler.

We think of the collection of stacks as forming a category Stacks, and thus we also have the concept of a category internal to stacks.

Finished with the definitions, now to respond to Urs’ comments.

——————————————

Urs wrote:

Yes. And this is the same as saying that we have a category internal to stacks on manifolds.

No. A category internal to Stacks will consist of stacks Ob¯(C),Mor¯(C)Stacks\overline{Ob}(C), \overline{Mor}(C) \in Stacks, together with the internal category data. That’s not what we have here.

Consider for the sake of explicitness the stack Vect sVect_s, which assigns to a category UU the category of vector bundles over UU. Notice that Vect sVect_s is not a category internal to stacks!

If it were, we’d have an object stack Ob¯(Vect s)\overline{Ob}(Vect_s) and a morphism stack Mor¯(Vect s)\overline{Mor}(Vect_s). What are these? What do they assign to manifolds UManU \in Man? Perhaps you might say that

(3)Ob¯(Vect s)(U)=categoryofvectorbundlesoverU. \overline{Ob}(Vect_s) (U) = category of vector bundles over U.

But then what about the morphisms stack? I think you’ll agree that this doesn’t make sense. In other words, there is no functor

(4)StackscategoriesinternaltoStacks. Stacks \rightarrow categories internal to Stacks.

That doesn’t make sense!

Next point. Urs wrote:

Think of a Chen-smooth structure explicitly as a sheaf, and you can give a description of a Chen-smooth category in exactly the words you use above!

Yes. But it doesn’t work the other way round. In other words, there is a functor

(5)A:CategoriesinternaltosmoothspacesStacksonmanifolds. A : Categories internal to smooth spaces \rightarrow Stacks on manifolds.

This functor works as follows. Suppose CC is a category internal to smooth spaces. Then A(C)A(C) is the stack whose value on a manifold UManU \in Man is the category with objects and arrows given by:

(6)Ob[A(C)(U)]=[Ob¯(C)](U),Mor[A(C)(U)]=[Mor¯(C)](U). Ob[A(C)(U)] = [\overline{Ob}(C)](U), \quad Mor[A(C)(U)] = [\overline{Mor}(C)](U).

The same formula would work for abstract sheaves; in other words we have a functor

(7)A:CategoriesinternaltosheavesStacksonmanifolds. A : Categories internal to sheaves \rightarrow Stacks on manifolds.

But there is no way to go back!

The point is that the stacks one obtains by applying AA to categories internal to sheaves are quite special : by definition (since a sheaf is a contravariant functor to Set), the categories they assign to manifolds UManU \in Man actually have a set of objects and a set of morphisms.

That can be quite restrictive, as the cobordisms example shows us.

In other words, the only obvious formula to construct a functor

(8)B:StacksCategoriesinternaltoSheaves B : Stacks \rightarrow Categories internal to Sheaves

would be to send a stack \mathcal{F} to the category internal to sheaves B(F)B(F), via

(9)[Ob¯(B())](U)=ObF(U) [\overline{Ob}(B(\mathcal{F}))](U) = Ob F(U)

and the same for morphisms. But that wouldn’t work, because (U)\mathcal{F}(U) was an ordinary category : it’s collection of objects and morphisms don’t form sets in general.

I suppose that there is a functor

(10)B:StackswhosefibersaresmallcategoriesCategoriesinternaltoSheaves B' : Stacks whose fibers are small categories \rightarrow Categories internal to Sheaves

but that’s a bit less interesting, since in most of the interesting examples of stacks, the categories (U)\mathcal{F}(U) assigned to open sets UU are not small. Like vector bundles. Or cobordisms.

There remains the possibility that I have some horrible misconception somewhere.

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

Re: Quantization and Cohomology (Week 26)

No. A category internal to Stacks will consist of stacks Obj¯(C)\bar \mathrm{Obj}(C), Mor¯(C)Stacks\bar \mathrm{Mor}(C) \in \mathrm{Stacks}, together with the internal category data. That’s not what we have here.

Okay, I see what you mean.

First, if we are being careful, the smooth category of cobordisms, say, does have

- a stack of morphisms, namely bundles of dd-dimensional (Riemannian) manifolds over each space SS

- a stack of objects, namely bundles of (d1)(d-1)-dimensional (Riemannian) manifolds over each space SS

Like on the bottom of p. 24 of SUSY Field Theories and Integral Modular Forms.

But then, on the very next page, they want to repackage that in the way you keep talking about, I think: the distinction between a category and its space of morphisms being a small one, they pass from thinking of assigning just SS-parameterized morphisms to thinking of assigning SS-parameterized categories.

But it’s not arbitrary categories they assign this way, but just those they call RB S d\mathrm{RB}^d_S, which are precisely such that they have SS-families of manifolds as objects and SS-families of cobordisms as morphisms.

But that’s exactly the information encoded in a category internal to stacks.

I think.

Just consider: a functor from Man op\mathrm{Man}^{\mathrm{op}} to Cat\mathrm{Cat} (with certain properties) is just a generalized space = 0-category. If we want to think of this as a category itself, it needs to be equipped with the extra structure that makes it a generalized space of morphisms. But that’s just another way of talking about a category internal to stacks.

Posted by: urs on June 4, 2007 2:01 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

To amplify this a little more: you may take comfort in the fact that I am not just disagreeing with you, Bruce, but also with def. 33 in that Stolz-Teichner preprint.

:-)

Def 33 says that a smooth category is just a stack on manifolds (with values in Cat instead of in Grpd, but never mind).

But a stack on manifolds is really just a generalized smooth space. It makes good sense to regard it as a category only when it comes with extra structure which makes it a generalized smooth space of morphisms.

In their concrete examples R dR^d and TVTV etc, they do provide this extra structure. And this amounts to, I think, actually defining a category internal to stacks – which consists of, among other things, a generalized smooth space of morphisms.

Posted by: urs on June 4, 2007 2:13 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

Bruce and I have further discussed this by private mail.

He finally got me to the point where I admit to have stubbornly ignored the premise he kept emphasizing all along, that any stack on manifolds itself can usefully be regarded as a smooth category.

The reason is that I kept thinking of the stack as a generalization of a manifold (a space) – like an orbifold, for instance.

But the latter of course “are” groupoids, just as well, so I was closing my eyes in front of the obvious.

On the other hand, thinking of stacks on manifolds as generalized smooth spaces this way, as usual, I’d think it still makes sense to consider categories internal to stacks over manifolds as general “smooth categories”.

Maybe that’s mislead. I need to think about it.

Anyway, Bruce has a very good point and I managed to not see if for a dozen of comments. Sorry.

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

Morphisms of Anafunctors

It’s easy to generalize [the definition of morphisms from anafunctor FF to anafunctor GG] to the case where FF and GG are defined using different covers: just take a common refinement.

I find morphisms of anafunctors over different covers a little tricky. It’s the same issue for all categories of descent data: when one allows different covers, things become much more involved. No?

It starts with making that passage to the common refinement precise. I found this took some thinking, back when I discussed this with Toby. Me, at least, I didn’t find this obvious from reading Toby’s thesis. It did help me to draw these diagrams, though.

But okay, after a while this becomes obvious. But then, it seems there are still some issues:

while passing to a common refinement makes it easy to define the composition of two given morphisms, it tends to break associativity and unit morphisms.

I tended to avoid these issues entirely by not considering morphisms over different covers at all, so maybe I am just missing some basic points.

But I do know that at least as soon as it comes to 2-anafunctors, this issue has caused quite some headaches.

A bundle gerbe is a kind of 2-anafunctor (with values in 1-dimensional vector spaces instead of in complex numbers). Danny in his thesis has a section on how to define morphisms of bundle gerbes over different covers using common refinements. It get’s quite intricate there, with the need of descending the line bundles back down to coarser covers.

Konrad thought about this a lot. I think he has some result on it in his More Morphisms between Bundle Gerbes.

Personally, I was happy when we learned one way to avoid this issue entirely: once we have a category of global objects equivalent to our category of anafunctors (i.e. the global stack of which the anafunctor category is the descent data category) the need to have morphisms between anafunctors with respect to different covers becomes somewhat less urgent.

Posted by: urs on June 3, 2007 5:55 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

Bruce wrote:

I would say that Stolz and Teichner’s approach is a more openly stack-like approach, while the approach presented above by John is also “stacky”, though somewhat shy to come out of the closet and admit it.

Perhaps a little personal history would help explain this.

I first started working on 2-bundles out of a feeling that lots of differential geometers and physicists are more comfortable with bundles than sheaves — so maybe these people would prefer 2-bundles to stacks!

The idea, of course, is that just as a bundle has a sheaf of sections, a 2-bundle should have a stack of sections. Not every sheaf is a sheaf of sections of some bundle, and similarly, not every stack will be the stack of sections of some 2-bundle. But, this case comes up often enough to make bundles worthwhile, and the same is true — I hoped, and still hope! — for 2-bundles.

I started thinking about this around 2001. I gave a course on it in 2002, and some talks, and wrote a paper. But, I decided to never publish this paper, for various reasons:

  1. I didn’t understand parallel transport for 2-connections. I just couldn’t believe the ‘fake flatness’ equation was required for well-defined parallel transport, even though some preliminary calculations seemed to show that.
  2. I didn’t understand what it means for a Lagrangian to be ‘gauge-invariant’, when the Lagrangian depends on a 2-connection instead of a connection.
  3. I didn’t understand how the total space of a 2-bundle is assembled from local pieces, in the case where the cocycle condition g ijg jk=g ikg_{ij} g_{jk} = g_{ik} holds only up to isomorphism.

Later, at some point Urs jumped in and helped me solve problem 1. I still don’t know the answer to problem 2 — and by now, I’m thinking about different things, so someone else may have to solve it.

As for problem 3, it was Toby who really solved this, in his thesis. The key was to not work in the 2-category I wanted to use:

[smoothcategories,smoothfunctors,smoothnaturaltransformations][smooth categories, smooth functors, smooth natural transformations]

Instead, Toby realized we have to work in

[smoothcategories,smoothanafunctors,smoothananaturaltransformations][smooth categories, smooth anafunctors, smooth ananatural transformations]

Someone else might have done this just to ‘get the theorem to work’ — to make the classification of 2-bundles match the known classification of stacks, in the case of a stack of sections of a 2-bundle. But Toby took a much more principled and ‘logical’ approach to the problem.

It was very lucky that Toby was deeply interested in intuitionistic logic, and an expert on matters concerning the axiom of choice. My 2-category

[smoothcategories,smoothfunctors,smoothnaturaltransformations][smooth categories, smooth functors, smooth natural transformations]

is, of course, a special case of ‘internal category theory’. For any category CC, we can define a 2-category

[categoriesinC,functorsinC,naturaltransformationsinC][categories in C, functors in C, natural transformations in C]

But, Toby noticed that this 2-category is ill-behaved when the axiom of choice fails in CC — that is, when not every epimorphism

p:EBp : E \to B

in CC has a section

s:BEs: B \to E

meaning a morphism with

ps=1 B.p s = 1_B.

For example, when the axiom of choice fails in CC, we can’t prove that a full, faithful and essentially surjective functor in CC is the same as an equivalence in CC.

Of course the axiom of choice does fail in the category of topological spaces, or smooth spaces. This is very much linked to how bundles work! So, Toby noticed that while the axiom of choice fails in these categories, it still holds ‘locally’ for certain ‘nice surjections’ p:EBp: E \to B, of the sort we study in bundle theory.

In other words, while bundles don’t usually have global sections, they always have local sections! So, in the category of smooth spaces, the axiom of choice only holds ‘locally’!

Starting with this insight, Toby decided to generalize Makkai’s idea of ‘anafunctors’ — which Makkai developed to do category theory without the axiom of choice. Toby defined ‘anafunctors in CC’ for any category CC that has well-behaved concepts of ‘locally’ and ‘nice surjection’. He then went on and set up a 2-category

[categoriesinC,anafunctorsinC,ananaturaltransformationsinC][categories in C, anafunctors in C, ananatural transformations in C]

In short: he invented a ‘local’ version of internal category theory.

After that, problem 3 was easily solved!

But, I’m not saying anything about this stuff in my lectures — there’s too much going on already, and too little time, to start talking about stuff like the axiom of choice.

So, I’m making Toby’s idea look like a ‘hack’ instead the wonderful insight it really is.

For example, I showed how you could cook up a concept of ‘smooth ananatural transformation’ simply by demanding that some theorem works: the one-to-one correspondence between 1st Cech cohomology of MM with coefficients in GG, and

{smoothanafunctorsF:MG}/{smoothananaturaltransformations}\{smooth anafunctors F: M \to G \}/\{smooth ananatural transformations\}

But, Toby thought about it all much more deeply.

Of course, after switching from smooth functors to smooth anafunctors, etcetera, 2-bundles start looking a lot more like stacks than I’d originally envisioned.

And here’s the curious thing. When we think about a category equipped with a general concept of ‘locally’, we naturally think about a ‘site’: that is, a category equipped with a Grothendieck topology. But, Grothendieck also invented stacks, and considered stacks on sites!

So, in a way, we’re all just catching up with ideas Grothendieck had almost half a century ago!

Posted by: John Baez on June 3, 2007 11:33 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

I believe I asked the following before. In any case, I don’t recall the answer:

Toby describes the construction of the total space of a 2-bundle from its transition data in the proof of proposition 22, starting on p. 67, by invoking a general theory of higher quotients and systematically “quotienting out” the identifications given by the 2-transitions.

By the power of this theory, it suffices for him to show that the 2-transition does define a 2-quotient. For, as it says before (203)

Therefore, I really do have an equivalence 2-relation, so some 2-quotient F×Uj˜EF \times U \stackrel{\tilde j}{\to} E must exist.

Here EE is the total space of the 2-bundle.

I find it hard to see where the notion of anafunctor enters this construction. Which step would fail would we work with plain internal functors here?

Posted by: urs on June 4, 2007 10:26 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 26)

Maybe I should amplify the point of my above question a little more:

from just reading Toby’s thesis, the impression I got, personally, was that the main and crucial insight was the theory of 2-quotients.

It is well known how to reconstruct a principal bundle from its transition data by quotienting out the action of the transition function. This action has an obvious categorification to a 2-transition, the way Toby describes it.

But what is rather non-obvious is that diving out by this 2-action in a suitable sense does yield a smooth total space. Somehow Toby achives this quite effortlessly once he has the theory of general 2-quotients.

This I find remarkable. But I am not sure I fully understand what makes this work (for instance, if it is the concept of anafunctor that makes this work, then it seems I not getting the point yet).

It is maybe remarkable that in the continuous as opposed smooth setup, Wirth and Stasheff worked hard, apparently, to get the analogous theorem, that dividing out the 2-transition does yield a good total space. (We talked about that in Wirth and Stasheff on Homotopy Transition Cocycles).

Somehow Toby manages to get an analogue of Wirth’s “mapping cylinder theorem” as a consequence of pure abstract nonsense on 2-quotients.

This I find quite remarkable. I would like to better understand it.

Posted by: urs on June 4, 2007 11:00 AM | Permalink | Reply to this

The case for stacks

Urs and I have been discussing above, and in private emails, the comparison between the “categories enriched in smooth spaces” approach and the “stacky” approach to the question,

What is a smooth category, and what are smooth functors between smooth categories?

I indicated in my third comment that this discussion is really at the meta-level of framing the debate itself .

John has made some helpful explanatory remarks about the history behind his choice to use categories internal to smooth spaces and smooth anafunctors - as opposed to an explicit use of stacks - to describe 2-bundles.

John wrote:

I first started working on 2-bundles out of a feeling that lots of differential geometers and physicists are more comfortable with bundles than sheaves — so maybe these people would prefer 2-bundles to stacks!

The idea, of course, is that just as a bundle has a sheaf of sections, a 2-bundle should have a stack of sections.

I have a feeling that I have a fundamentally different conception of how bundles and 2-bundles would be handled in the “stacky” paradigm, than the paradigm that John presents in this paragraph. In particular, I wouldn’t have thought of bundles and 2-bundles via their spaces of sections!

That might seem crazy - isn’t that the standard way of thinking? - but allow me to explain.

Let’s look at vector bundles, the simplest case. From the stacky perspective, what is a vector bundle on a manifold MM? (We’ll add connections later.) There are two answers:

1. “Space of sections” paradigm . A vector bundle is thought of via it’s sheaf of sections.

2. “Parallel transport” paradigm . A vector bundle is thought of as a smooth assignment of a vector space E xE_x to every point xMx \in M.

Before I explain Option 2, let me first say that I get the impression from John’s remarks that he decided to use the technology of (categories internal to smooth spaces and smooth anafunctors) because he felt geometers and physicists would be more comfortable with a “parallel transport” paradigm. That I would agree with; at least for me, that’s the paradigm I prefer.

What I would like to stress here is that the stacks paradigm presented in Option 2 already gives us the option of thinking in terms of parallel transport : we don’t need to use (categories internal to smooth spaces and smooth anafunctors)!

In other words, if John decided not to use stacks because his audience wouldn’t have been very comfortable with Option 1, then he threw out the baby with the bathwater! Stacks also have an Option 2.

Let me explain Option 2.

The stacky paradigm would proceed almost exactly the same way John proceeds, when he introduces smooth categories and smooth anafunctors… just subtly different.

Firstly, we set up a category Disc(M)Disc(M), which has objects the points of MM, and only identity morphisms. Take care now - this is where the stacks approach departs from John’s approach - we think of Disc(M)Disc(M) simply as an ordinary category , not as a “category internal to smooth spaces”. We don’t put any structure on it at all.

Then we say that a vector bundle on MM is a smooth functor

(1)E:Disc(M)Vect. E : Disc(M) \rightarrow Vect.

For this to make sense, I have to tell you what the words “smooth functor” mean, in the stacks paradigm. I’ve already explained this four times in this page : here, here, here and here.

But I’ll try once more. Every time I write it down, I learn a new aspect of it I didn’t see before.

Let me remind you first that I learnt this paradigm from Stolz and Teichner (see eg. this preprint). Metzler also has a nice expository article on smooth stacks. Here I’m using “stack” in the following sense : A stack is a weak 2-functor F:Man opCatF : Man^op \rightarrow Cat which satisfies some gluing conditions.

In the stacks paradigm, one uses the maxim

Smooth is as smooth does.

More precisely:

A functor is smooth if it can be made to work for smooth families.

In the stacks paradigm, a functor

(2)F:CD F : C \rightarrow D

between plain-old-ordinary categories is called smooth if there exist stacks C sC_s and D sD_s, such that the values they assign to the point *Man* \in Man are the original categories CC and DD,

(3)C s(*)=C,D s(*)=D, C_s (*) = C, \quad D_s (*) = D,

and there exists a stack map

(4)F s:C sD s F_s : C_s \rightarrow D_s

such that the component of F sF_s on the point equals the functor FF:

(5)F s(*)=F. F_s(*) = F.

Let’s apply this techonology to our vector bundles functor

(6)E:Disc(M)Vect. E : Disc(M) \rightarrow Vect.

Firstly, there is a stack Disc(M) sDisc(M)_s, which assigns to every manifold UManU \in Man the discrete category whose objects are smooth maps from UU into MM :

(7)Disc(M) s(U)=Maps(U,M). Disc(M)_s (U) = Maps(U, M).

There is also a vector bundles stack Vect sVect_s, which assigns to every manifold UManU \in Man the category of vector bundles over UU:

(8)Vect s(U)=categoryofvectorbundlesoverU. Vect_s (U) = category of vector bundles over U.

In other words, Disc(M) s(U)Disc(M)_s (U) is the category whose objects are “smooth families of objects in Disc(M)Disc(M)”. Similarly, Vect s(U)Vect_s(U) is the category whose objects are “smooth families of objects in VectVect”, and whose morphisms are “smooth families of morphisms in VectVect”.

Notice that, over *Man* \in Man, our stacks just return back our original categories :

(9)Disc(M) s(*)=Disc(M),Vect s(*)=Vect. Disc(M)_s (*) = Disc(M), \quad Vect_s (*) = Vect.

Finally, we thus say that our functor

(10)E:Disc(M)Vect E : Disc(M) \rightarrow Vect

is smooth if it works also for smooth families, i.e. if there exists a stack map

(11)E s:Disc(M) sVect s E_s : Disc(M)_s \rightarrow Vect_s

such that

(12)E s(*)=E. E_s (*) = E.

Okay. Having explained the stacks philosophy, let me say that at the end of the day, this is the same thing as you would have got out using categories internal to smooth spaces and anafunctors. But that’s not always true. I explained the difference between the two approaches at the end of this this post.

And finally, let me say that the whole thing works also for vector bundles with connection, and GG-bundles with connection, etc. In that game, our relevant source stack - which John would think of as the path groupoid P(M)P(M), a groupoid internal to smooth spaces - would be the stack P(M) sP(M)_s (excuse the pun!), which assigns to a manifold UU the groupoid P(M) s(U)P(M)_s (U), which has

(13)objects:smoothfamiliesofpointsinM,parametrizedbyU, objects : smooth families of points in M, parametrized by U,

and

(14)morphisms:smoothfamiliesofpathsinM,parametrizedbyU. morphisms : smooth families of paths in M, parametrized by U.

So, in the stacks approach, a GG-bundle with connection would simply be a smooth functor

(15)E:P(M)GTor, E : P(M) \rightarrow G-Tor,

(where I defined what smooth means above), which is exactly what all of John’s apparatus is trying to finally say - except we didn’t need the language of internal categories and internal functors to say it.

Okay, finally, finally, what would a “2-bundle” be in this approach? Crucially, contrary to what it appears John suggested, it would not be thought of as a stack (stacks are already reserved for ordinary bundles). Rather it would be a 2-stack! Thus : almost exactly the same philosophy as what Toby did, just subtly different. Admittedly though, 2-stacks are perhaps tough to work with (aren’t they? I don’t know anything about them.)

Posted by: Bruce Bartlett on June 6, 2007 4:34 AM | Permalink | Reply to this

Re: The case for stacks

I’m a little ashamed to have been expostulating so vociferously about this topic :-). In the spirit of John’s advice about the “Tao of mathematics”, let me add the following caveats to what I’ve been going on about.

I only learnt this part of the stacky philosophy very recently (i.e. what it means for a functor to be smooth) after attending some seminar talks in Ediburgh by Stolz and Teichner. And I must admit I don’t truly understand it yet… it’s weird, it’s all a big tautology , somehow. What’s nice about anafunctors is that the information is somehow much more concrete.

But the point is I’ve really just been knocking down a straw man when I say that I prefer the stacky paradigm of bundles with connection etc. than the anafunctor/smooth categories paradigm. Because John’s project is all about categorified gauge theory… 2-connections, fake curvature, and all that. And that’s where this stacky, “tautological”, paradigm will fail us - unless we go to 2-stacks, heaven help us :-)

On the other hand, John and Urs seem to have quite a sensible way of doing higher gauge theory via path 2-groupoids and so on. In this regard what I’ve been saying has been really “neither here nor there”!

Posted by: Bruce Bartlett on June 6, 2007 5:49 AM | Permalink | Reply to this

Re: The case for stacks

What’s nice about anafunctors is that the information is somehow much more concrete.

Bruce,

you did manage to make me agree that I missed one of your points. But this point I still won’t concede unless you try harder. :-)

We were talking about two different ways to define what a smooth category is.

That issue is rather independent of all things anafunctor. Once we fix one definition of smooth category, we can then talk about smooth anafunctors or smooth ordinary functors with respect to that notion.

But not in any way is using anafunctors or not an alternative or not to defining smooth categories the internal or the stacky way.

I think. Now disagree and we’ll sort this out.

(Have to run now, morning session starts in a few seconds. Won’t be able to respond until tonight.)

Posted by: urs on June 6, 2007 7:53 AM | Permalink | Reply to this
Read the post Quantization and Cohomology (Week 27)
Weblog: The n-Category Café
Excerpt: Review of what we've done in this course; prospectus of what's still to be done.
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