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I have a very strange question. When people tell me that stacks are abstract nonsense that only algebraic geometers care about (or something similar) I try to point out that stacks occur in most branches of math even if you don’t think of them as stacks (most prominently the stack of vector bundles for differential geometers).
A joke I often make is that there is probably even a stack of measures on a (sufficiently nice) topological space that might interest measure theorists. This question is for people who know some measure theory. Today I was talking about this with an analyst, but the language barrier made it hard to actually verify.
Suppose $X$ is nice enough so that nothing fishy goes on with Borel measures. Then take $Op(X)$ to be the standard site of open sets. Form the category of pairs $(U, \mu)$ where $U$ an open and $\mu$ a Borel measure on $U$. We’ll do something kind of dumb for the arrows and say $(U,\mu)\to (V, \nu)$ is an inclusion $V\subset U$ together with an actual equality of measures $\mu|_V=\nu$.
Just take the forgetful functor and since everything is so rigid this forms a category fibered in sets over $Op(X)$ and is actually seems to be a stack. Now it probably isn’t interesting at all considering you can’t ever have a non-trivial automorphism of $(U,\mu)$. I was wondering if anyone has ever thought of this, or checked this, or come up with something more interesting in a similar vein that I can start using as an example. In theory you could try to use cohomology or something to study measures in this way or talk about “deformations of measures” or something.
The thing that the analyst did say is that measure theorists might care if you could somehow do this up to “bi-Lipschitz equivalence”. We thought about that. The arrows $(U,\mu)\to (V,\nu)$ would then be a bi-Lipshcitz homeomorphism $f:U\to U$ so that $(f_* \mu)|_V=\nu$. It also seems to form a stack (in sets) and moreover you could get lots of automorphisms (for instance take the disk in $\mathbb{R}^2$ and area measure, then any rigid automorphism gives the same measure back) so the stackiness is actually useful. Can anyone else quickly see if this is obviously true or not true for some reason?
So you observe that there is a sheaf of measures, right? And every sheaf is a special kind of stack (which is a special kind of 2-stack…which is a special kind of infinity-stack, if you wish).
There is for instance also a sheaf of metrics on smooth manifolds, and so forth, along the lines of your example.
Sure, sheaves, stacks, etc. : they are everywhere . The topic is called topos theory and higher topos theory and it is everywhere. Algebraic geometry is but a tiny application of this grand topic.
The idea that stacks and other general abstract theory that Grothendieck introduced into algebraic geometry are somehow endemic to algebraic geometry is as wide-spread as it is weird. I hear amazing things being attributed to algebraic geoemtry. Recently I heard a member of an important committee claim to an applicant that the little disk operad is a concept in algebraic geometry.
I realize that I am not replying to your question at the end, but let me just expand on the part at the beginning a bit more:
a discussion of the general notion of sheaves, stacks and $\infty$-stacks of “geometric structure”, such as measures, but also Riemannian metric, complex structures;, symplectic structures etc. is in
He discusses a geometric refinement of the Galatius-Tilman-Madsen-Weiss theorem on the homotopy type of the cobordism category, where now cobordisms are equipped with a morphism into such an $\infty$-stack of geometric structure.
Some stacks in mainstream differential geometry
There is an inherent reason while advanced points of view like stacks appeared first and brewed for long solely in algebraic geometry (apart from the obvious: Grothendieck). Namely in differential geometry you can do so much by direct intutive means: tubular neighborhood, partition of unity, local deformation, implicit function theorem, codimension counting etc. You have lots of functions there. Now in algebraic geometry, deform a polynomial, the curve deforms globally, one can not do things so much locally in algebraic geometry. You want to have implicit function therem you need to go itnto henselian rings and alike. You need infinitesimals, you need formal functions, and get out of category of schemes. So in differential geometry you may have a hack fix for everything, becuase of abundance of flexibility. In algebraic geometry, you have so muh rigidity that some notions like homotopy looked hopeless for a long while. Unless you have a correct deep abstract point of view you are doomed in algebraic geometry. So in algebraic geometry the solution exists and one can eventually get to the same flexibility once one gets to the botton of the things (the main modern idea being crossing with topology to get derived geometry). So derived things like stacks had to be invented early on to move on. Unlike in differential geometry where the classical stuff could resist the need for a long while, but eventually one has more ambition and getting full way through there as well.
There is an inherent reason while advanced points of view like stacks appeared first and brewed for long solely in algebraic geometry (apart from the obvious: Grothendieck). Namely in differential geometry you can do so much by direct intutive means:
Yes! This is exactly what I use to say: in differential geometry one can fake it for too long and avoid seeing the truth.
This is a nice explanation, Zoran.
I should point out that I realized this from a private “lecture” by the algebraic geometer Tomasz Maszczyk who opened my eyes there. I am glad that Urs sees it independently the same way and that the explanation makes sense to Todd. I like Urs’s way to say it so much!
in differential geometry one can fake it for too long and avoid seeing the truth
Edit: I also hope (am convinced) that yet another layer of the story, yet not explicit, is hidden in the difficulties of (nonderived) noncommutative algebraic geometry which will once force us to complete picture of the geometry further at places where it was easier to overlook in commutative situation.
Another way to highlight the issue is this:
over CRing$^{op}$ very few objects are affine.
over SmoothRing$^{op}$ every smooth manifold and much more is affine.
Right, this is one very important aspect.
Thanks. I myself am pretty convinced of the viewpoint as well. That’s why I was explaining to people that although the definitions look scary at first, practically everything satisfies it. These references are really good, but still very much focus on the differential geometry side (it isn’t that hard to make a differential geometer at least see that there might be some use since they are already using stacks secretly). It is kind of pointless to keep pressing this issue, but is there an example of someone studying say the stack of measures using the some general yoga about stacks and then translating back to get a nice proof of something that I could bring up for further removed crowds?
For one thing, merely noting that they form a stack should be kind of impressive since you’ve in some since made a moduli space parametrizing all measures on the space. So if a measure theorist cares about classifying the measures, then this should be an interesting object to study.
Domenico Fiorenza and Elena Martinengo have an article, currently submitted to the $n$Journal, whose abstract sets out to address people as you seem to be facing:
A short note on infinity-groupoids and the period map for projective manifolds (arXiv:0911.3845)
Abstract:
A common criticism of infinity-categories in algebraic geometry is that they are an extremely technical subject, so abstract to be useless in everyday mathematics. The aim of this note is to show in a classical example that quite the converse is true: even a naive intuition of what an infinity-groupoid should be clarifies several aspects of the infinitesimal behaviour of the periods map of a projective manifold. In particular, the notion of Cartan homotopy turns out to be completely natural from this perspective, and so classical results such as Griffiths’ expression for the differential of the periods map, the Kodaira principle on obstructions to deformations of projective manifolds, the Bogomolov-Tian-Todorov theorem, and Goldman-Millson quasi-abelianity theorem are easily recovered.
This is still not about measures, but about complex geometry. Do you strictly need something that involves measure theory?
I definitely don’t care if measures come up. The key property I’m looking for is something that a “pure” analyst might find interesting. I still feel like the results there an analyst could write off as just more geometry (complex, algebraic, what’s the difference?) which is why I tried measures. Maybe there is a stack of PDE’s or harmonic functions or maybe a stack on $L^2[0,1]$ that makes some proof transparent…
I could almost see something along the following lines working. You define a stack of measures on $\mathbb{R}$ in some way and there is a notion of deformations of measures. Unraveling the definition one way gives that a deformation means there is some function, $f$, for which $\nu(A)=\int_A fd\mu$ and unraveling another way gives that the two measures are absolutely continuous, so this gives some stack theoretic proof of the Radon-Nikodym Theorem.
Maybe I’m just being silly though.
Ah, sorry, only now do I understand: it’s not supposed to be about any geometry. Hm, need to think..
Urs #8
a better way might be to say:
among sheaves/generalised spaces over CRing, very few are representable
among sheaves/generalised spaces over SmoothRing, many are representable
It’s just now that people need ’smooth spaces’ e.g. inf dim manifolds more and more that we find that there are actually lots of non-representable things over SmoothRing. People have used the hack of passing to Frechet manifolds etc so far. But that requires a lot of care to make things rigorous, and not in the toolbox of ordinary differential geometers.
Passing to stacks, and especially by using some presentation like anafunctors, we find that everything becomes representable, namely by an internal groupoid (assuming we’re looking at stacks of groupoids).
[Aside – A recent proto-theorem of mine: every 2-category of geometric stacks on a site is equiv to a slice 2-category of some bicategory of anafunctors internal to that site. You heard it here first :-) ]
’Representability’ is the same as ’affine’, but brings a whole lot of other connotations and technology with it.
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