Bruce wrote:
I would say that Stolz and Teichner’s approach is a more openly stacklike 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 2bundles out of a feeling that lots of differential geometers and physicists are more comfortable with bundles than sheaves — so maybe these people would prefer 2bundles to stacks!
The idea, of course, is that just as a bundle has a sheaf of sections, a 2bundle 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 2bundle. But, this case comes up often enough to make bundles worthwhile, and the same is true — I hoped, and still hope! — for 2bundles.
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:

I didn’t understand parallel transport for 2connections. I just couldn’t believe the ‘fake flatness’ equation was required for welldefined parallel transport, even though some preliminary calculations seemed to show that.

I didn’t understand what it means for a Lagrangian to be ‘gaugeinvariant’, when the Lagrangian depends on a 2connection instead of a connection.

I didn’t understand how the total space of a 2bundle is assembled from local pieces, in the case where the cocycle condition $g_{ij} g_{jk} = g_{ik}$ 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 2category I wanted to use:
$[smooth categories, smooth functors, smooth natural transformations]$
Instead, Toby realized we have to work in
$[smooth categories, smooth anafunctors, smooth ananatural transformations]$
Someone else might have done this just to ‘get the theorem to work’ — to make the classification of 2bundles match the known classification of stacks, in the case of a stack of sections of a 2bundle. 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 2category
$[smooth categories, smooth functors, smooth natural transformations]$
is, of course, a special case of ‘internal category theory’. For any category $C$, we can define a 2category
$[categories in C, functors in C, natural transformations in C]$
But, Toby noticed that this 2category is illbehaved when the axiom of choice fails in $C$ — that is, when not every epimorphism
$p : E \to B$
in $C$ has a section
$s: B \to E$
meaning a morphism with
$p s = 1_B.$
For example, when the axiom of choice fails in $C$, we can’t prove that a full, faithful and essentially surjective functor in $C$ is the same as an equivalence in $C$.
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’ $p: E \to B$, 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 $C$’ for any category $C$ that has wellbehaved concepts of ‘locally’ and ‘nice surjection’. He then went on and set up a 2category
$[categories in C, anafunctors in C, ananatural transformations in C]$
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 onetoone correspondence between 1st Cech cohomology
of $M$ with coefficients in $G$, and
$\{smooth anafunctors F: M \to G \}/\{smooth ananatural transformations\}$
But, Toby thought about it all much more deeply.
Of course, after switching from smooth functors to smooth anafunctors, etcetera, 2bundles 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!
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 stacklike 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 $G$bundle”. Of course, everything works for $G$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 $G$torsor to every point $x$ of a manifold $M$”.
John has demonstrated one way to do this; one thinks of a $G$bundle as a smooth anafunctor $F : Disc(M) \rightarrow G$, where “smooth” means that $Disc(M)$ 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 $Disc(M)$ simply as a plain discrete groupoid $M$… no smooth structure. Lurking in the background of course is the stack $M$ over smooth manifolds, whose value at $X$ is the discrete groupoid whose objects are elements of the homset $Hom(M,X)$.
On the other hand, we also have the groupoid $GTor$, whose objects are $G$torsors and morphisms are torsor morphisms. There is also a stacky version of this, $GTor$ , which associates to a manifold $X$ the category of principal $G$bundles over $X$.
Ok. Here is my paraphrasing of the StolzTeichner approach. We are trying to decide when a functor
is smooth. Recall : $M$ is the discrete groupoid on $M$.
Here’s the solution : we say that $F$ is smooth if it can be extended to a stack map
$F : M \rightarrow GTor$
such that the value of $F$ on the point $* \in Man$ equals $F$, i.e.
$F$ $(*)$ $= F$.
Of course, that’s essentially the same as saying that our objects of interest are just stack maps from $M$ to $GTor$ . 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 $G$torsor to every point $x \in M$ :
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.