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Stream: learning: questions

Topic: What is "local to global" in the CT POV?

John Onstead (May 07 2024 at 14:33):

The recent discussion on Baez's topos theory blog posts is exploring sheaves, and has recently connected sheaves with the concept of a bundle and section of a bundle. Both concepts are certain abstractions of what is known as a "local to global" correspondence, which is usually when local data can be "glued together" in a certain way to extend it globally. While sheaves are seen as a powerful way to do this, the concept holds for a wide variety of cases. For instance, some cohomology theories deal with the failure in your ability to extend local to global. There's also descent theory, which I don't understand but apparently has something to do with local-global and gluing conditions. In analysis/calculus, a large number of the machinery depends on local to global relations, since calculus deals with the infinitesimal (local) and its relation to the finite or even infinite (global). Another example is in an atlas, which "glues together" a whole bunch of charts and local homeomorphisms and helps to define the global concept of a manifold. But the examples do not stop there: these correspondences hold even outside topology, as this stack exchange post seems to demonstrate.
Whenever some repeating pattern happens across math, one can usually try to reinterpret it as a singular concept in the category theory point of view (CT POV). For instance, across math, there are many ways to "multiply" mathematical objects together. While they seem unrelated at first, if we zoom out to the CT POV, we see they are really just different manifestations of the singular concept of the categorical product, being internalized to all the different categories where these diverse mathematical objects live. What I want to know is, what is the CT POV version of local to global? For which singular categorical concept (an "internal local-global correspondence" if you will) do the local-global properties of sheaves, atlases, bundles and sections, infinitesimal analysis, Hasse Principle, etc. all arise as different manifestations of, just internalized in different ways into different categories?

Tim Hosgood (May 07 2024 at 15:26):

I can't speak about the Hasse principle (maybe it's true there but idk), but to me (and I think you'll find this to be a pretty opinion-based/taste-based question) the underlying thing is "descent", where by this I mean "something commuting with a homotopy limit"

David Egolf (May 07 2024 at 17:00):

I don't yet know what a "homotopy limit" is, but I recently learned a possibly related fact, which I found interesting. Roughly, "a sheaf is a functor that commutes with limits coming from open covers". The phrase I just quoted is from page 20 of "Sheaves, Cosheaves, and Applications". A very nice discussion and precise elaboration is then provided over the next few pages!

Tim Hosgood (May 07 2024 at 17:06):

yeah this is very related! a brief sketch of the story I have in mind is the following: you can write a space $X$ as the (homotopy) colimit of the Čech nerve $\check{C}\mathcal{U}=U_\alpha\leftarrow U_{\alpha\beta}\leftarrow\ldots$ of some nice cover $\mathcal{U}$; descent, or the sheaf condition, or whatever you would like to call it then essentially says "a contravariant functor $F$ is a sheaf iff $F(X)\simeq\lim(F(U_\alpha)\to F(U_{\alpha\beta})\to\ldots)$", which is the same as saying "$F$ commutes with the (co)limit".

Tim Hosgood (May 07 2024 at 17:07):

in very rough notation, you could say that the sheaf condition is $F(\operatorname{colim}\check{C}\mathcal{U}) \simeq \lim F(\check{C}\mathcal{U})$ (and me saying homotopy limit is just because I often care about sheaves of spaces, or other higher-categorical replacements for sets)

Tim Hosgood (May 07 2024 at 17:10):

but the relevance to some of the other things that John mentioned is that you can write lots of things as homotopy limits, e.g. the space ("higher category") of bundles can be written as a homotopy limit of some diagram involving the Čech nerve

John Baez (May 07 2024 at 17:34):

Descent is a very general idea, which makes it hard to define in full generality without being very abstract, but very roughly the idea is that you have a map $p: E \to B$ which is like a covering space or bundle, and you're asking when a thing on $E$ 'descends' to define a similar thing on $B$.

The very simplest case would be if you have a map $f: E \to X$ and you want to know if it 'descends' to a map $g: B \to X$. This means: can you find a function $g: B \to X$ such that $f = g \circ p$.

John Baez (May 07 2024 at 17:39):

Notice that any function $g: B \to X$ automatically 'lifts' to a function $f : E \to X$, defined by $f = g \circ p$. That's easy. But in descent we are deliberately asking the harder reverse question: given $f$ we are trying to find $g$ with $f = g \circ p$. That's why descent is interesting.

John Baez (May 07 2024 at 17:44):

The more typical examples of descent are categorifications of this simple example. For example $E$ could be an open cover of some topological space $B$ (remember, we can think of an open cover as giving a map $p: E \to B$ where $E$ is the disjoint union of the sets in the open cover). We could have a bundle on $E$, and then ask whether it 'descends' to a bundle on $B$.

Again we are trying to do the reverse of some automatic process of 'lifting', also called 'pulling back', a bundle from $B$ to $E$.

But now this is not a yes-or-no question anymore: we're not asking about a mere property anymore, but rather a structure! That is, we are interested in what extra structure we must equip our bundle on $E$ with, to get it to descend to a bundle on $B$. This extra structure is called 'descent data'. There might be many choices that work, or one, or none.

In this particular example, involving bundles, the descent data is called 'transition functions', and these functions need to satisfy a famous 'cocycle condition' that looks like

$g_{ij} g_{jk} = g_{ik}$

where $i,j,k$ index sets in our open cover of $B$.

John Baez (May 07 2024 at 17:48):

As we go to more and more categorified versions of descent, the descent data becomes more complex, and the above equation becomes a more complicated set of equations.

John Baez (May 07 2024 at 17:53):

What I just explained is a watered-down version of what Tim Hosgood said, but I tried to strip off some technical terms like 'Čech nerve', 'homotopy colimit', etc. because while these concepts are important, I think it's hard for beginners to understand them until they get the basic gist of what 'descent' is all about - for example, why it's called 'descent'.

John Onstead (May 07 2024 at 20:17):

Thanks @Tim Hosgood @David Egolf @John Baez for the help! I'll look more into descent. I'm especially intrigued by how the descent data is given by transition functions, since these do appear often in many local-global correspondences. For instance, in manifolds a transition function compares between charts within the atlas in the areas where they may overlap. I figured that limits and colimits would show up, since in many contexts a colimit can be seen as a "gluing" operation and gluing is how to get from local to global.
Before asking this question I had also tried to work out my own interpretation of what "local-global" could be in a generic category, and I came up with a hypothesis involving this very fact about colimits, but I'm not sure how good it is. Maybe I can refine it a bit and present it tomorrow. Perhaps there's a relation between it and descent, but I'll have to learn more about descent to be sure!

Tim Hosgood (May 07 2024 at 20:28):

I'd love to hear if anybody else has any good stories about why transition functions show up when talking about descent! I know one story, which has a lot of technical words, but which is really quite nice when you get past them all (the holim can be computed by the totalisation of a cosimplicial simplicial set, and this is a very explicit thing you can write down and draw pictures of, and then you find that something happens to be 2-coskeletal so the cocycle condition falls out from the fact that $(0<1)\circ(1<2) = 0<2$)

Tim Hosgood (May 07 2024 at 20:31):

John Baez said:

What I just explained is a watered-down version of what Tim Hosgood said, but I tried to strip off some technical terms like 'Čech nerve', 'homotopy colimit', etc. because while these concepts are important, I think it's hard for beginners to understand them until they get the basic gist of what 'descent' is all about - for example, why it's called 'descent'.

this is definitely a good historical point of view too! descent via descent data is very much how this appeared in Grothendieck's FGA

Tim Hosgood (May 07 2024 at 20:33):

if I wanted to have a really slick motto for all this "local to global"/descent/holim stuff, I would maybe say "we want the Čech nerve of a cover to be a good substitute for the space itself; everything else follows from trying to make this be true"

Tim Hosgood (May 07 2024 at 20:34):

so this is another way of trying to approach this: start by looking at what the Čech nerve is all about, what nerves are about in general, and build things up from here :)

Tim Hosgood (May 08 2024 at 12:25):

not to derail this thread, but there are lots of parts of descent (in the more traditional sense) that I still don't really understand. could anybody explain why we would say that a morphism $f\colon x\to y$ in a category $\mathcal{C}$ (with pullbacks) should be said to have "effective descent" if the change of base $f^*\colon \mathcal{C}/y\to\mathcal{C}/x$ is monadic?

John Onstead (May 08 2024 at 14:04):

As mentioned I've come up with a "hypothesis" for this question. What I noticed was that every local-global correspondence has the same features: local data (which can possess "local properties"), global data (with global properties), transition maps (that compare between parts in how they "fit together"), gluing (that takes you from local to global), and "zooming" (that allows you to get back to the local from global). My "hypothesis" is that a "local-global correspondence internal to C" is a set of guidelines to express when and how a certain object in that category can be expressed as a (weighted) colimit of (some class of) its subobjects.

I came up with this because, oftentimes, when we "glue" mathematical objects together (like two sets into their disjoint union), we use some sort of colimit to do it, so it can be seen as a sort of abstract gluing operation. In this hypothesis, the "local" is given by subobjects, the "global" is the object itself, the gluing that gets us from local to global is the colimit, and the local-global correspondence tells us which diagram involving subobjects to take this colimit of. Usually, the morphisms included within this diagram are, or are related to, some notion of "transition map" that describes in some way how two parts of an object compare.

A good illustrative example is in atlases and topological manifolds. There's a result, which is like a "local-global correspondence for topological manifolds", that states how a topological manifold is a colimit of its atlas. So if we go into Top and find a certain diagram encoding the atlas (in this case, the class of subobjects are the open sets of the space), taking that diagram's colimit gives us the manifold, a good expression of my hypothesis. However, I'm still trying to figure out how this could work for sheaves- does the descent condition of colimits and limits commuting imply something like my hypothesis or is in any way related to my hypothesis? And does my hypothsis have anything to do with Čech nerves in the sense mentioned by Tim above?

Kevin Carlson (May 08 2024 at 17:07):

Tim Hosgood said:

not to derail this thread, but there are lots of parts of descent (in the more traditional sense) that I still don't really understand. could anybody explain why we would say that a morphism $f\colon x\to y$ in a category $\mathcal{C}$ (with pullbacks) should be said to have "effective descent" if the change of base $f^*\colon \mathcal{C}/y\to\mathcal{C}/x$ is monadic?

Tim, basically you can turn the category of descent data along this map into the category of algebras for the monad. nLab has the start of a pretty useful tree of related links.

Tim Hosgood (May 08 2024 at 17:24):

John Onstead said:

A good illustrative example is in atlases and topological manifolds. There's a result, which is like a "local-global correspondence for topological manifolds", that states how a topological manifold is a colimit of its atlas. So if we go into Top and find a certain diagram encoding the atlas (in this case, the class of subobjects are the open sets of the space), taking that diagram's colimit gives us the manifold, a good expression of my hypothesis. However, I'm still trying to figure out how this could work for sheaves- does the descent condition of colimits and limits commuting imply something like my hypothesis or is in any way related to my hypothesis? And does my hypothsis have anything to do with Čech nerves in the sense mentioned by Tim above?

The Čech nerve of a cover of a manifold is exactly a specific diagram built out of the open sets such that if you take its colimit then you recover the manifold!

Tim Hosgood (May 08 2024 at 17:29):

and you can think of the sheaf condition for a presheaf on a manifold a saying "it should commute with exactly this colimit diagram", a sheaf on $X$ is a functor $F\colon\mathsf{Open}(X)^\mathrm{op}\to\mathsf{Set}$ (where $\mathsf{Open}(X)$ is the poset of open subsets of $X$) such that, if $X\simeq\operatorname{colim}N$, then $F(\operatorname{colim}N)\simeq\lim F(N)$ (and we already know that this left-hand side is $F(X)$, by the hypothesis on $N$)

Tim Hosgood (May 08 2024 at 17:30):

there are some technical details missing in this brief version, so don't take it as literally completely true in all possible scenarios, but that's one way of understanding what's going on

John Onstead (May 08 2024 at 22:09):

Tim Hosgood said:

The Čech nerve of a cover of a manifold is exactly a specific diagram built out of the open sets such that if you take its colimit then you recover the manifold!

That sounds really cool! I have so many questions about this, I really wish I could ask them all...
I guess if I had one question for now it'd be how to relate this sense of a Cech nerve, where it is some sort of diagram, with the notion given on the nlab, where it is instead given as a simplicial object related to some morphism and doing some pullback on itself over and over again.
Edit: I didn't see there's a whole separate thread on descent! I'll see if my questions may be answered there before asking any more here...

Tim Hosgood (May 08 2024 at 22:23):

a "simplicial object" is another way of saying "a diagram of a certain specific shape", and the nlab uses "pullback" instead of saying "intersection". if you unwrap the big definition, you'll find that the Čech nerve is a diagram that looks like $N_0\leftleftarrows N_1 \xleftarrow[\leftarrow]{\leftarrow} N_2\ldots$, and each of the $N_i$ are a disjoint union of some intersections of your open sets

Tim Hosgood (May 08 2024 at 22:24):

but as you can see in the thread on descent, there are a lot of ways of understanding these things — i've been thinking about and working with Čech nerves for quite some time, and i still have lots to learn from other viewpionts!