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Stream: learning: questions

Topic: Sheaves as the categorification of integrals?

John Onstead (Apr 01 2025 at 11:45):

As I've been learning about sheaves, I've noticed an interesting connection between them and integrals. First, they have similar properties: sheaves glue local data into global data, and integrals accumulate local quantities into global quantities. But it really seems to go deeper, with sheaves potentially even being a sort of categorification of integration. If we look at the most basic case of the real interval, a sheaf might assign to each open interval the set of functions defined on that interval. On the other hand, an integral of some particular function assigns to each interval the area under the curve of the function over that interval. In addition, just as a sheaf on a space can be defined in terms of its stalks which act to assign data to infinitesimally small neighborhoods, an integral is defined in terms of finding the area of infinitesimally tiny rectangles.

John Onstead (Apr 01 2025 at 11:46):

The reason why I argue for the use of "categorification" to describe this relationship is that integrals work with single functions and assign numbers to intervals while sheaves work with sets of functions and assign whole sets to intervals. In a way, sets are categorifications of numbers- though of course the glaring difference is that integrals work with real numbers rather than just natural numbers, and only natural numbers are categorified into sets. (There's also some interesting points in this nlab article, but there they view sheaves as categorifying functions rather than integrals). In any case, I was wondering- does this analogy actually make sense, and is there a real actual way to view sheaves as categorified integrals? Thanks for your help!

John Baez (Apr 01 2025 at 16:19):

It's an interesting idea I haven't seen. I would tend to think of sheaves as categorifying functions, and that idea is pretty useful. The pushforward of a sheaf along a map to the point is often seen as a kind of categorified integration.

David Corfield (Apr 01 2025 at 18:52):

John Baez said:

The pushforward of a sheaf along a map to the point is often seen as a kind of categorified integration.

And an indication that coends (as pushforwards) are nearer to integration.

John Onstead (Apr 01 2025 at 19:05):

John Baez said:

The pushforward of a sheaf along a map to the point is often seen as a kind of categorified integration.

David Corfield said:

And an indication that coends (as pushforwards) are nearer to integration.

Hmm... this sounds interesting! I'm just wondering what exactly it "means" to take the pushforward of a sheaf along a map to the point. Maybe an example would help- what is one instance of a specific sheaf where taking its pushforward is like integration, and why is this seen as like integration?

Matteo Capucci (he/him) (Apr 01 2025 at 19:13):

If anything I would compare measures (or distributions) and sheaves. Then Kock's paper on distribution monads provides a nice framework to compare the similarities!

Matteo Capucci (he/him) (Apr 01 2025 at 19:16):

The other POV is to see means (= integrals on probability spaces) as universal objects, i.e. the mean of a function $f:X \to \R$ is the number $m$ such that $\int_x f(x) = \int_x m$, just like the colimit of $f:X \to \bf Set$ can be substituted to $f$ for the purpose of mapping out of it. I discuss a bit (not as deeply as I'd like unfortunately) this idea here.

JR (Apr 01 2025 at 20:38):

John Baez said:

It's an interesting idea I haven't seen. I would tend to think of sheaves as categorifying functions, and that idea is pretty useful. The pushforward of a sheaf along a map to the point is often seen as a kind of categorified integration.

The pushforward of a function is a weighted sum. Seems like integration to me.

https://en.wikipedia.org/wiki/Transfer_operator#Definition

John Onstead (Apr 01 2025 at 21:15):

Matteo Capucci (he/him) said:

If anything I would compare measures (or distributions) and sheaves. Then Kock's paper on distribution monads provides a nice framework to compare the similarities!

That's interesting! I'll have to think about that more later; I've been planning to study a bit more measure theory anyways!

John Onstead (Apr 01 2025 at 21:24):

First, I want to make sure I even understand what a pushforward is. Let's start with a basic example, the unit interval $[0,1]$. We can define a sheaf $F: \mathrm{Open}([0,1]) \to \mathrm{Set}$ of smooth functions on this space. Since $\mathrm{Open}(*) = *$ the terminal category, there's also a unique functor $\mathrm{Open}([0,1]) \to *$, which gives the setup $* \leftarrow \mathrm{Open}([0,1]) \to\mathrm{Set}$, which is ripe for a left Kan extension. We can then define $\mathrm{Lan}_* F: * \to \mathrm{Set}$. I haven't worked out what this works out to yet. But is this what we mean by the "pushforward" of a sheaf? Or does it refer to something completely different?

John Baez (Apr 01 2025 at 21:48):

John Onstead said:

I'm just wondering what exactly it "means" to take the pushforward of a sheaf along a map to the point. Maybe an example would help - what is one instance of a specific sheaf where taking its pushforward is like integration, and why is this seen as like integration?

For any sheaf S over any space X, its pushforward to the point is the set of global sections, i.e. the set S(X). Or, if we have a vector space, it's the vector space of global sections, etc.

For example in geometric quantization we may take S to be the sheaf of holomorphic sections of a line bundle over X. Then the vector space S(X) is the "space of quantum states" of our system.

I think an example more like integration occurs when we have a span of spaces

$X \leftarrow Y \rightarrow Z$

and we pull back a sheaf from $X$ to $Y$ and then push it forward to $Z$. This is a categorified version of pulling back a function from $X$ to $Y$ and then "integrating over the fibers" of the map $Y \to Z$ to get a function on $Z$.

With some extra twists this idea gives a categorification of the Fourier transform called the Fourier-Mukai transform.

John Baez (Apr 01 2025 at 22:03):

John Onstead said:

But is this what we mean by the "pushforward" of a sheaf? Or does it refer to something completely different?

I don't understand the stuff you wrote, but the pushforward of a sheaf along a map is also called its direct image, and its definition is quite simple.

Grothendieck's yoga of six operations with sheaves includes pushforwards and pullbacks.

Mike Shulman (Apr 01 2025 at 23:10):

In sheaf theory and algebraic geometry the word "pushforward" is generally used for the right adjoint of pullback. For sheaves on discrete spaces, this is a product rather than a sum. More generally, for toposes of presheaves this is a right Kan extension, computed by an end rather than a coend.

John Baez (Apr 01 2025 at 23:33):

Yes, everything I said was consistent with what Mike just said, though if someone had asked me if pushforward was the right or left adjoint of pullback I would have gotten it wrong with probability 49%.

E.g. the "set of global sections" $S(X)$ of a sheaf $S$ on a discrete space $X$, is a product rather a coproduct.

So maybe it's bad to think of it as resembling an ordinary integral - maybe it's some sort of "product integral". (When I was a kid I developed the theory of product integrals, which are like ordinary integrals but with multiplication replacing addition.)

Mike Shulman (Apr 01 2025 at 23:37):

While I'm familiar with this terminology, I still haven't really made my peace with it. My intuition says that if you have a bunch of things stacked up in piles and you push them forwards, they end up stacked up in bigger piles, i.e. you took a sum.

John Baez (Apr 01 2025 at 23:47):

Yes, that intuition is nicely generalized by the idea of pushing forward a measure along a measurable map between measure spaces. This allows the "things" to be continuous rather than discrete (e.g. mud rather than rocks), and also allows for a continuous space of "piles" rather than a discrete set of piles. But the intuition is supposed to be what you said.

Kevin Carlson (Apr 01 2025 at 23:52):

I guess part of the reason for this suspicious terminology is that the left adjoint pushforward just doesn't usually exist for sheaves, right?

Mike Shulman (Apr 01 2025 at 23:54):

Presumably.

John Onstead (Apr 02 2025 at 00:51):

John Baez said:

So maybe it's bad to think of it as resembling an ordinary integral - maybe it's some sort of "product integral". (When I was a kid I developed the theory of product integrals, which are like ordinary integrals but with multiplication replacing addition.)

Mike Shulman said:

While I'm familiar with this terminology, I still haven't really made my peace with it. My intuition says that if you have a bunch of things stacked up in piles and you push them forwards, they end up stacked up in bigger piles, i.e. you took a sum.

This is all very interesting yet confusing in certain ways!

John Onstead (Apr 02 2025 at 00:55):

John Baez said:

For any sheaf S over any space X, its pushforward to the point is the set of global sections, i.e. the set S(X). Or, if we have a vector space, it's the vector space of global sections, etc.

For example in geometric quantization we may take S to be the sheaf of holomorphic sections of a line bundle over X. Then the vector space S(X) is the "space of quantum states" of our system.

Good to know, I think I was going off in a completely different direction. Originally I kind of hoped that there's be some natural transformation between a sheaf of smooth real functions on a space and the constant sheaf on $\mathbb{R}$ that somehow acted like an integral (IE, sends a section to the area under it on that space), but of course this is silly since some integrals are infinite or even worse undefined!

John Onstead (Apr 02 2025 at 11:46):

Another way to relate integrals and sheaves I have seen is through cohomology. One can do a certain cohomology with sheaves known as sheaf cohomology, and a special case of this is "de Rham cohomology". In this cohomology a central result is Stokes' theorem on manifolds, which is the greatest generalization of the fundamental theorem of calculus, and which deals with integrals. My question is: what does the fact that both sheaves (via sheaf cohomology) and integrals (via Stokes' theorem) connect to de Rham cohomology say about the relation between sheaves and integrals more directly?

John Baez (Apr 02 2025 at 14:37):

My initial impression is: almost nothing.

Jonas Frey (Apr 02 2025 at 16:11):

John Baez said:

It's an interesting idea I haven't seen. I would tend to think of sheaves as categorifying functions, and that idea is pretty useful. The pushforward of a sheaf along a map to the point is often seen as a kind of categorified integration.

I just came across this intuition of pushforward as integration on page 2 of Shapira's commentary on Recolltes et Semailles in the context of 6 functor formalisms! But probably it's a quite wide spread intuition.

John Baez (Apr 02 2025 at 16:47):

Thanks! It feels widespread - e.g. it's the reason people talk about the Fourier-Mukai transform as being a categorified Fourier transform - but I look forward to seeing someone actually discuss it out loud.

Fernando Yamauti (Apr 02 2025 at 16:47):

Jonas Frey said:

I just came across this intuition of pushforward as integration on page 2 of Shapira's commentary on Recolltes et Semailles in the context of 6 functor formalisms! But probably it's a quite wide spread intuition.

I think there he's referring to the fonctions-faisceux correspondence. That indeed gives a functorial assignement from sheaves with an endomorphisms to functions by defining those pointwise as the trace of th endo acting on the stalk. On the side of functions, the pushforward (with comp supp) becomes integration along the fibers.

That's the content in the end of SGAV. A summary also in Laumon's paper.

Also, in a sufficiently good 6 functor formalism (where we have a Thom isomorphism), the (derived) pushforward (with compact support) can be given that interpretation of integration along the fibers. But all that seems very far from the initial purpose of this question.

John Onstead (Apr 02 2025 at 19:19):

John Baez said:

My initial impression is: almost nothing.

That wasn't quite what I was expecting! But maybe this is a good opportunity for me to learn more about sheaf cohomology, since it seems like an interesting subject. I'll certainly be circling back to that in the future.

John Baez (Apr 02 2025 at 19:43):

Well, Fernando may offer a way to get what you want:

Also, in a sufficiently good 6 functor formalism (where we have a Thom isomorphism), the (derived) pushforward (with compact support) can be given the interpretation of integration along the fibers.

The problem that this approach confronts is that you want some sort of 'clever pushforward' of an $n$-form $\omega$ on an $n$-manifold $M$ to a point which produces a $0$-form on the point: the integral $\int_M \omega$. (This is the simple idea lurking behind the Thom isomorphism, which does this trick 'fiberwise'.)

Fernando Yamauti (Apr 03 2025 at 19:01):

Another point of view that doesn't depend on linear stuff (the integration along the fibers is very linear as it shift degrees via duality) which talks about integration is the theory of Lawvere distributions. That seems more in accordance with the point of view that sheaves are set-valued functions.

John Onstead (Apr 03 2025 at 19:43):

Fernando Yamauti said:

Another point of view that doesn't depend on linear stuff (the integration along the fibers is very linear as it shift degrees via duality) which talks about integration is the theory of Lawvere distributions. That seems more in accordance with the point of view that sheaves are set-valued functions.

That's interesting. It seems if you have a sheaf on a space, then you have a process that can take in a point on the space and return a set, the stalk of the sheaf for that point. This is kind of like a categorification of a natural numbers valued function, since every set can be decategorified into a natural number representing its cardinality. Too bad there's no obvious way to do something like this for real numbers! Though my main confusion is, couldn't the same logic work for presheaves? I'm pretty sure you can define stalks for presheaves, so why is it only sheaves that are seen as categorifications of functions?

John Baez (Apr 03 2025 at 20:33):

If you're willing to get sophisticated and Grothendieckian, presheaves on any category can always be seen as sheaves on that category with a suitable Grothendieck topology, called the trivial topology. So if you prove results for sheaves on an arbitrary site, those results apply to presheaves.

Kevin Carlson (Apr 03 2025 at 20:49):

The non-obvious way to do this for real numbers is via groupoid and even $\infty$-groupoid cardinality.

John Onstead (Apr 03 2025 at 21:21):

John Baez said:

If you're willing to get sophisticated and Grothendieckian, presheaves on any category can always be seen as sheaves on that category with a suitable Grothendieck topology, called the trivial topology. So if you prove results for sheaves on an arbitrary site, those results apply to presheaves.

Ah, that makes sense, presheaves are in some sense trivially sheaves.

Kevin Carlson said:

The non-obvious way to do this for real numbers is via groupoid and even $\infty$-groupoid cardinality.

I see, it's one more good reason to generalize to infinity category theory! Though I wonder if any real number is represented by some groupoid in this way, in the same way as for every natural number there is a set with that cardinality. The cardinality of a groupoid makes use of a formula which makes me think there might be some real numbers (for instance uncomputable numbers) that don't have groupoids with that cardinality. (but maybe such numbers are irrelevant anyways, since they won't ever show up as the value of some integral, at least with computable bounds!)

John Baez (Apr 03 2025 at 23:10):

Any nonnegative real number is the cardinality of a groupoid, because any nonnegative real number is a limit of finite sums of reciprocals $1/n$ with $n \in \mathbb{N}^+$.

John Baez (Apr 03 2025 at 23:10):

Negative numbers are far more mysterious.