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Stream: deprecated: algebraic geometry

Topic: Descent

Joe Moeller (May 12 2020 at 02:39):

What else do we have to know before we can move on to Galois descent?

John Baez (May 12 2020 at 16:43):

Nothing! So, I'll make this a new topic.... though eventually it will loop back to Galois cohomology.

John Baez (May 12 2020 at 16:45):

So, descent is fundamentally a question like this: you have a functor

$F: C \to D$

and you take an object of $D$ and you ask "did it come from an object of $C$"?

John Baez (May 12 2020 at 16:46):

Of course it might have come from more than one object of $C$, so instead of asking about the property of whether it came from an object of $C$ it's better to think of this as a structure!

John Baez (May 12 2020 at 16:47):

So, the question of descent becomes "how can we nicely describe the structure on objects of $D$ that says how they came from a object of $C$?"

John Baez (May 12 2020 at 16:47):

Get it?

Joe Moeller (May 12 2020 at 16:53):

It's pretty clear how fibrations connect to descent now.

Joe Moeller (May 12 2020 at 16:54):

John Baez said:

So, the question of descent becomes "how can we nicely describe the structure on objects of $C$ that says how they came from a object of $D$?"

is this backwards?

John Baez (May 12 2020 at 17:01):

It's backwards, let me fix it in my original comment.

John Baez (May 12 2020 at 17:03):

Yes, fibrations are connected to descent.

John Baez (May 12 2020 at 17:03):

And here's another way they're connected to descent - maybe it's a different way than what you were thinking of, but maybe it's connected.

John Baez (May 12 2020 at 17:06):

Suppose you have a map of spaces $p: E \to B$. Say you've got a category of bundles or sheaves up on $E$ and a category of bundles or sheaves down on $B$. Can you say how descent, as I've described it, shows up here?

John Baez (May 12 2020 at 17:07):

(Just pick bundles or sheaves of any kind you like; it doesn't matter much.)

John Baez (May 12 2020 at 17:18):

(Btw, for people joining, I like to teach math by asking lots of questions, which are usually supposed to be fairly easy for my intended audience - in this case @Joe Moeller.)

sarahzrf (May 12 2020 at 17:24):

well, there's an adjunction between the categories of sheaves on E and B—are you talking about descent from sheaves on E to sheaves on B, or vice versa? :thinking:

sarahzrf (May 12 2020 at 17:24):

or is that the puzzle? :)

John Baez (May 12 2020 at 17:25):

That's half the puzzle. You can push forwards and pull back sheaves, so you could in theory study descent going either way.

Joe Moeller (May 12 2020 at 17:26):

I think you want the functor from bundles on E to bundles on B given by composing with p. Then the question of descent for that functor is when does a bundle on B factor through p? Right?

John Baez (May 12 2020 at 17:27):

Right!

sarahzrf (May 12 2020 at 17:27):

ooooh right, that's exactly what pullback of bundles is

John Baez (May 12 2020 at 17:27):

Wait, wrong!

sarahzrf (May 12 2020 at 17:27):

and then that's exactly what lifting problems are

sarahzrf (May 12 2020 at 17:27):

oh oops lmao

John Baez (May 12 2020 at 17:27):

Joe said something sorta wrong.

John Baez (May 12 2020 at 17:28):

It sounded so much like the right answer that at first I assumed he'd gotten it right.

John Baez (May 12 2020 at 17:29):

I'll let you two clear it up.

sarahzrf (May 12 2020 at 17:29):

oops

John Baez (May 12 2020 at 17:30):

I'll just say this: when Sarah said "that's exactly what a pullback of bundles is", it didn't match what Joe had just described. Why not? Sort it out!

sarahzrf (May 12 2020 at 17:30):

yeah i kinda started to notice something wrong when i went back to see what you were complaining about...

Joe Moeller (May 12 2020 at 17:30):

What I described is not pulling back.

John Baez (May 12 2020 at 17:30):

You were "pushing forward" a bundle.

sarahzrf (May 12 2020 at 17:31):

i always super super hate how it's contravariant to go from p to p⁻¹ and then it's contravariant again to go from p⁻¹ to certain topos functors

John Baez (May 12 2020 at 17:31):

If your "bundle" means just a map of spaces, you can push it forward. But most people want bundles to be slightly nice, like "locally trivial".

John Baez (May 12 2020 at 17:32):

Then you can't push them forward, in general. You can only pull them back.

John Baez (May 12 2020 at 17:32):

For example, suppose I have a Mobius strip bundle over a circle - that's a certain locally trivial bundle whose fibers are lines.

sarahzrf (May 12 2020 at 17:32):

the double contravariance trips me up every goddamn time when i forget one of the two flips

Joe Moeller (May 12 2020 at 17:33):

So if we're talking about pulling back bundles from B to E, then the descent question is when is a bundle on E a pullback of a bundle on B.

John Baez (May 12 2020 at 17:33):

Say I map the circle to a point. What happens if I try to push forward this Mobius strip bundle? What do I get?

John Baez (May 12 2020 at 17:33):

You're right, Joe - and that's why it's called DESCENT and not ASCENT.

John Baez (May 12 2020 at 17:34):

You're trying to see if a bundle "up on $E$" descends to one on the "base" $B$. This is the same as seeing if it's a "lift" of a bundle on $B$.

John Baez (May 12 2020 at 17:34):

But please answer my latest question, just to make sure we know what's going on....

sarahzrf (May 12 2020 at 17:35):

ooh, a whole bundle, not just a section of a given bundle?

sarahzrf (May 12 2020 at 17:35):

:eyes:

Oscar Cunningham (May 12 2020 at 17:36):

Surely there's only one bundle on a point with a given total space, so it will just be that

John Baez (May 12 2020 at 17:37):

So what's that?

Oscar Cunningham (May 12 2020 at 17:37):

The map from the Mobius strip to the point

Joe Moeller (May 12 2020 at 17:38):

The bundle should be the line over the point.

John Baez (May 12 2020 at 17:39):

Oh, goody! The great thing about having more than one student is that they give contradictory answers and learn by fighting it out.

John Baez (May 12 2020 at 17:39):

So we've got two answers to what happens if you push forward the Mobius strip bundle over the circle along the map from the circle to the point.

John Baez (May 12 2020 at 17:40):

Joe gave a clear recipe for how this pushing forward should work, so one can just follow that recipe and see what it gives.

Oscar Cunningham (May 12 2020 at 17:40):

The line over the point is what you would get if you pulled back the bundle along an inclusion of the point into the circle

sarahzrf (May 12 2020 at 17:41):

meanwhile im sitting here unsure what it means to push a bundle forward :sob:

John Baez (May 12 2020 at 17:42):

What it "means" in a deep philosophical way, or how to do it?

sarahzrf (May 12 2020 at 17:42):

the latter!!

John Baez (May 12 2020 at 17:42):

The most general concept of a "bundle" over a space $X$ is just a map $p: Y \to X$.

sarahzrf (May 12 2020 at 17:42):

well, i know that

sarahzrf (May 12 2020 at 17:42):

im pretty happy with pulling bundles back and what that means

John Baez (May 12 2020 at 17:43):

Can you guess how to push it forward along a map $f: X \to X'$?

sarahzrf (May 12 2020 at 17:43):

.........is it a pushout :face_palm:

sarahzrf (May 12 2020 at 17:43):

noooo wait that doesnt even

John Baez (May 12 2020 at 17:43):

You can't draw a pushout with the data I gave.

sarahzrf (May 12 2020 at 17:43):

the arrows go the wrong way what am i saying

John Baez (May 12 2020 at 17:43):

Right - wrong!

sarahzrf (May 12 2020 at 17:44):

yeah im not sure

sarahzrf (May 12 2020 at 17:44):

other than to wildly guess it's what direct image of sheaves does

John Baez (May 12 2020 at 17:44):

So what do you do? You gotta do what you can do with the stuff you've got.

Oscar Cunningham (May 12 2020 at 17:44):

Draw the diagram: $Y\to X\to X'$

John Baez (May 12 2020 at 17:44):

Come on, you're "overthinking" it Sarah. Usually people say that when they're underthinking it, but here you are really overthinking it.

sarahzrf (May 12 2020 at 17:44):

\>.>

sarahzrf (May 12 2020 at 17:45):

omg oscar

sarahzrf (May 12 2020 at 17:45):

literally just compose?

John Baez (May 12 2020 at 17:45):

Yeah!

sarahzrf (May 12 2020 at 17:45):

for fucks sake

John Baez (May 12 2020 at 17:45):

You have reinvented "composition of morphisms" :tada:

John Baez (May 12 2020 at 17:45):

I figure I can be a bit sarcastic without permanently damaging you...

sarahzrf (May 12 2020 at 17:45):

:O cant believe i managed to do what took eilenberg and mac lane previously .....

Joe Moeller (May 12 2020 at 17:46):

That's what he meant when he said that I gave a recipe for it, I said you get a functor by composing with p.

John Baez (May 12 2020 at 17:46):

Right!

sarahzrf (May 12 2020 at 17:46):

well, i wasnt even wrong about direct image of sheaves then was i?

John Baez (May 12 2020 at 17:46):

So this very general concept of "bundle" easily pushes forward.

Joe Moeller (May 12 2020 at 17:46):

My answer came from saying "vector bundle" in my head. So I was trying to think what should the vector space over the point be when I compose with the map from the circle to the point. I think this is an error. The answer should be the mobius strip over the point.

John Baez (May 12 2020 at 17:46):

Right!

John Baez (May 12 2020 at 17:47):

So this way of pushing forward bundles is pretty bad if you like bundles with a fixed fiber, like "line bundles".

John Baez (May 12 2020 at 17:47):

Or even more generally "vector bundles".

John Baez (May 12 2020 at 17:47):

We took a vector bundle over the circle, pushed it forward to the point, and got something that wasn't a vector bundle.

John Baez (May 12 2020 at 17:48):

So geometers generally consider pushing forward bundles to be a horrible thing to do.

John Baez (May 12 2020 at 17:48):

It's not to be discussed in polite society.

John Baez (May 12 2020 at 17:49):

Henceforth we will only pull back bundles.

John Baez (May 12 2020 at 17:49):

And it sounds like y'all know how that works. It's literally just a pullback in the categorical sense.

John Baez (May 12 2020 at 17:50):

You've got your bundle $p: E \to B$ and a map $f: B' \to B$ and you do the pullback you can do with that data.

John Baez (May 12 2020 at 17:51):

This is nice because line bundles pull back to give line bundles, vector bundles pull back to give vector bundles, principal $G$-bundles pull back to give principal $G$-bundles, blah blah blah.

John Baez (May 12 2020 at 17:51):

The reason is that the fibers of your new bundle over points in your new base $B'$ are just copies of fibers over the the points of your old base $B$.

John Baez (May 12 2020 at 17:52):

So the fun starts when you ask: given a bundle over my new base $B'$, could it be the pullback of a bundle over the old base $B$?

John Baez (May 12 2020 at 17:53):

I see my letters have changed by now.

John Baez (May 12 2020 at 17:53):

But this is the "descent" question.

John Baez (May 12 2020 at 17:53):

Let me see if you get it. Say $B = B'$ is a circle but $f$ is the 2-1 cover of the circle by itself.

John Baez (May 12 2020 at 17:54):

Say $p: E \to B'$ is the Mobius strip bundle over $B'$. Does it descend to $B$?

Oscar Cunningham (May 12 2020 at 17:55):

One condition is that points of $B'$ that go to the same place in $B$ should have the same fiber

Joe Moeller (May 12 2020 at 17:56):

I'll say no. I think the double twist would descend to the Mobius strip though.

sarahzrf (May 12 2020 at 17:56):

wait wait i wanna think about this

sarahzrf (May 12 2020 at 17:56):

i havent visualized it yet

John Baez (May 12 2020 at 17:56):

Yes, all of you should visualize it. A couple of comments: 1) "the same" is a very tricky concept.

John Baez (May 12 2020 at 17:56):

2) "the double twist" line bundle over the circle is a funny thing to talk about.

sarahzrf (May 12 2020 at 17:58):

so this is sort of an extension problem, huh

sarahzrf (May 12 2020 at 17:58):

if we assume that there's a classifying space for the kind of bundle we're talking about

sarahzrf (May 12 2020 at 17:58):

right?

Oscar Cunningham (May 12 2020 at 17:58):

The fibers of the Mobius bundle are all homeomorphic. But I think Joe's right, there's some 'global' obstruction.

John Baez (May 12 2020 at 17:58):

People would usually call this a "lifting" problem not an extension problem.

sarahzrf (May 12 2020 at 17:58):

i wanted to say it was a lifting problem but i think the arrows go the wrong way around??

Reid Barton (May 12 2020 at 17:59):

they call it ... a "descent" problem

John Baez (May 12 2020 at 17:59):

Classifying spaces are too fancy for what we're doing right now, btw. This is just pictures of circles and stuff.

sarahzrf (May 12 2020 at 17:59):

classifying spaces are great for pictures of circles >:(

Joe Moeller (May 12 2020 at 18:00):

How I'm visualizing this is that the question is whether there's a space over the circle such that if you un-doublecover the circle, and you bring along the space for the ride, it turns into the Mobius strip.

sarahzrf (May 12 2020 at 18:00):

yeah

John Baez (May 12 2020 at 18:00):

Yeah, my comment about a lifting problem was backwards. I guess it is an extension problem, abstractly - I got confused because we're "extending" along a 2-1 map, not an inclusion!

Reid Barton (May 12 2020 at 18:00):

if you want to think of it in terms of classifying spaces, though, you have a map from $B'$ to some fancy classifying object and you want to "descend" the domain of the map to $B$.

John Baez (May 12 2020 at 18:00):

I really don't want to talk about classifying spaces now.

John Baez (May 12 2020 at 18:01):

But yes, you are right.

sarahzrf (May 12 2020 at 18:01):

why, does it make the answer too easy >:)

John Baez (May 12 2020 at 18:01):

It's too fucking abstract!

sarahzrf (May 12 2020 at 18:01):

:cry:

John Baez (May 12 2020 at 18:01):

We're doing stuff that could be done in kindergarten!

Joe Moeller (May 12 2020 at 18:01):

I think un-doublecovering any line bundle over the circle should double the number of twists.

John Baez (May 12 2020 at 18:01):

Joe is saying the right kind of stuff... let me actually focus on it.

sarahzrf (May 12 2020 at 18:02):

no yeah im trying to think about that sort of concurrently

John Baez (May 12 2020 at 18:02):

What the hell is "un-doublecovering"? :upside_down:

Joe Moeller (May 12 2020 at 18:02):

pulling back over the double cover

John Baez (May 12 2020 at 18:02):

Okay.

Joe Moeller (May 12 2020 at 18:02):

but it's what I made up to convey my visualization

John Baez (May 12 2020 at 18:03):

So you're saying: if I have a line bundle over the circle with n twists, and I pull it back along the 2-1 cover of the circle by itself, I get a line bundle with 2n twists.

Joe Moeller (May 12 2020 at 18:03):

yes

John Baez (May 12 2020 at 18:03):

What do the rest of y'all think about that?

sarahzrf (May 12 2020 at 18:04):

hmmmmmmmmmm

John Baez (May 12 2020 at 18:04):

That's very true, Sarah.

John Baez (May 12 2020 at 18:05):

Joe: what's funny about the line bundle over the circle with 2 twists? I asked this before...

sarahzrf (May 12 2020 at 18:06):

i suspect it may be topologically indistinguishable from the trivial line bundle

John Baez (May 12 2020 at 18:06):

You can make one out of paper...

John Baez (May 12 2020 at 18:06):

If your line is sufficiently short...

sarahzrf (May 12 2020 at 18:06):

sure, but that's an embedding into R³

sarahzrf (May 12 2020 at 18:07):

the horned sphere is a thing too

Joe Moeller (May 12 2020 at 18:07):

They are not homeomorphic.

sarahzrf (May 12 2020 at 18:07):

as an abstract space, u get the möbius strip from a square by just gluing one side to the opposite side w/ a reversing map

John Baez (May 12 2020 at 18:08):

Oh, good, a fight! Are they homeomorphic or not?

sarahzrf (May 12 2020 at 18:08):

the reversing map is the only thing that distinguishes from the untwisted bundle

sarahzrf (May 12 2020 at 18:08):

but a double-twist strip doesnt have a reversing map

John Baez (May 12 2020 at 18:08):

Notice there are two questions: are the total spaces homeomorphic, and are they isomorphic as bundles.

sarahzrf (May 12 2020 at 18:08):

wait

sarahzrf (May 12 2020 at 18:08):

god dammit

Joe Moeller (May 12 2020 at 18:09):

I don't know what a reversing map is.

sarahzrf (May 12 2020 at 18:09):

well, whats yr argument joe? im kinda charging in w/o full thought :upside_down:

sarahzrf (May 12 2020 at 18:09):

reversing map—sorry, i mean you quotient one copy of [0, 1] to another by 1 - x

sarahzrf (May 12 2020 at 18:10):

where these copies of [0, 1] are the edges

John Baez (May 12 2020 at 18:12):

So, we've got two questions on the burner now:

1) Is Joe's idea true: if we have a line bundle over the circle with n twists, and we pull it back along the 2-1 cover of the circle by itself, we get a line bundle with 2n twists.

2) Is the bundle with 2 twists isomorphic to the bundle with 0 twists?

Joe Moeller (May 12 2020 at 18:12):

I take it back, I think the double twist is homeomorphic to the cylinder.

John Baez (May 12 2020 at 18:13):

So have you changed your mind too, Sarah, so you can keep arguing with Joe but on opposite sides?

John Baez (May 12 2020 at 18:13):

It's always fun when that happens.

Joe Moeller (May 12 2020 at 18:14):

I was thinking too much about the embedding before.

sarahzrf (May 12 2020 at 18:14):

on a möbius strip there is only one side

John Baez (May 12 2020 at 18:14):

I think I can imagine a homeomorphism between the double twist bundle and the no-twist one.

John Baez (May 12 2020 at 18:15):

It's easiest to describe this way. I can make both these bundles out of a rectangle of paper.

John Baez (May 12 2020 at 18:15):

The obvious "identity map" between rectangles of paper gives a homeomorphism between these bundles.

sarahzrf (May 12 2020 at 18:16):

(...how are you defining the double twist bundle?)

John Baez (May 12 2020 at 18:16):

I'm taking a long thin rectangle of paper, twisting it twice, and attaching the two ends in the sorta obvious way.

Joe Moeller (May 12 2020 at 18:16):

yes, but critically, you can evaluate the map on two chunks, away from the gluing area, and around the gluing area.

sarahzrf (May 12 2020 at 18:17):

@John Baez a dangerous thing to rely on when making the leap to formal claims :flushed:

John Baez (May 12 2020 at 18:17):

In topology if you can't prove things with twisted pieces of paper and pictures you're on really shaky ground.

sarahzrf (May 12 2020 at 18:17):

this is also true :thinking:

sarahzrf (May 12 2020 at 18:18):

i await your picture proof that locally compact spaces are exponentiable :)

John Baez (May 12 2020 at 18:18):

Once people start scribbling weird letters they can get in all sorts of trouble. :upside_down:

John Baez (May 12 2020 at 18:19):

Anyway, I'm claiming the total space of the the double twist bundle is "obviously" $S^1 \times [0,1]$, where now I'm making the line compact just for fun...

sarahzrf (May 12 2020 at 18:19):

end compactification :eyes:

John Baez (May 12 2020 at 18:19):

I make it by taking a long thin rectangle of paper, twisting it twice, but then identifying points using the identity map on [0,1].

Joe Moeller (May 12 2020 at 18:19):

use the open interval instead.

John Baez (May 12 2020 at 18:20):

True, cheap paper leaves out the edges.

sarahzrf (May 12 2020 at 18:20):

having a compact total space seems nice :>

John Baez (May 12 2020 at 18:21):

Anyway, so please classify "line bundles with n twists over the circle", for all n.

Joe Moeller (May 12 2020 at 18:21):

cylinder, Mobius.

John Baez (May 12 2020 at 18:21):

Right: Z gets mapped to Z/2.

John Baez (May 12 2020 at 18:22):

Okay, so you've classified real line bundles over $S^1$.

John Baez (May 12 2020 at 18:22):

You got Z/2.

John Baez (May 12 2020 at 18:22):

For fans of classifying spaces, this is because $\pi_1$ of the relevant classifying space is Z/2.

John Baez (May 12 2020 at 18:23):

Or better: we've proved that $\pi_1$ of the relevant classifying space is Z/2.

sarahzrf (May 12 2020 at 18:23):

is it rp2

John Baez (May 12 2020 at 18:23):

It's $\mathbb{RP}^\infty$, actually.

sarahzrf (May 12 2020 at 18:23):

O:

John Baez (May 12 2020 at 18:24):

It's the Eilenberg-Mac Lane space of Z/2.

John Baez (May 12 2020 at 18:24):

Classifying spaces tend to be ginormous.

sarahzrf (May 12 2020 at 18:24):

right that sounds familiar

John Baez (May 12 2020 at 18:25):

Okay. Now that you understand all line bundles over the circle, describe the process of lifting a line bundle on the circle to one on its double cover!

John Baez (May 12 2020 at 18:26):

This is how to completely crush this example of "descent": completely understanding lifting.

Joe Moeller (May 12 2020 at 18:27):

Lifting anything gives the cylinder.

sarahzrf (May 12 2020 at 18:28):

thats what id guess as long as you really do double the number of twists

John Baez (May 12 2020 at 18:28):

Can you say it nice in terms of Z/2?

sarahzrf (May 12 2020 at 18:28):

:eyes:

sarahzrf (May 12 2020 at 18:28):

Z/2 is annihilated by uh

sarahzrf (May 12 2020 at 18:28):

2Z

Joe Moeller (May 12 2020 at 18:28):

multiply by 0

John Baez (May 12 2020 at 18:28):

There's actually a group of line bundles on the circle, and it's Z/2.

John Baez (May 12 2020 at 18:29):

The reason is that you can "tensor" line bundles - the tensor product of two 1d vector spaces is a 1d vector space.

John Baez (May 12 2020 at 18:29):

So there's always a group of line bundles over any space.

John Baez (May 12 2020 at 18:29):

And we've seen this group is Z/2 for the circle.

Joe Moeller (May 12 2020 at 18:29):

Yes, I'm imagining doubling the twists, but now that we established you can cancel two twists, I'm simplifying.

John Baez (May 12 2020 at 18:30):

I don't like how you're simplifying.

John Baez (May 12 2020 at 18:30):

It's okay, but I have something else in mind.

John Baez (May 12 2020 at 18:30):

We have a map Z/2 -> Z/2.

Joe Moeller (May 12 2020 at 18:31):

The zero map.

John Baez (May 12 2020 at 18:31):

You can call it "multiplying by 0", but I'd rather not.

Oscar Cunningham (May 12 2020 at 18:31):

Do you want us to say the word 'homomorphism'?

John Baez (May 12 2020 at 18:31):

Well, we've been lifting along a double cover.

Oscar Cunningham (May 12 2020 at 18:31):

'Doubling'?

John Baez (May 12 2020 at 18:31):

What if we lifted along the triple cover of the circle by itself?

John Baez (May 12 2020 at 18:31):

We'd get some map Z/2 -> Z/2 again...

sarahzrf (May 12 2020 at 18:32):

coprime

sarahzrf (May 12 2020 at 18:32):

doesnt annihilate

sarahzrf (May 12 2020 at 18:32):

i think

John Baez (May 12 2020 at 18:32):

All these words are good, but I like what Joe said.

sarahzrf (May 12 2020 at 18:32):

~_~

John Baez (May 12 2020 at 18:33):

Lifting along an n-fold cover gives a map Z/2 -> Z/2, and it's just multiplication by n.

John Baez (May 12 2020 at 18:33):

So I wanted Joe to say "multiplying by 2", but he said "multiplying by 0".

John Baez (May 12 2020 at 18:33):

You really do get a double twist bundle when you lift the single twist bundle along the double cover, but it just happens that 2 = 0.

John Baez (May 12 2020 at 18:34):

I'm getting a bit worn out, but I'll wrap up.

John Baez (May 12 2020 at 18:35):

We've seen that descent quickly gets us into some algebra... some algebraic topology for example.

John Baez (May 12 2020 at 18:36):

Bundles over the circle are particularly simple because they're completely described (up to iso) by a "twist", which is an element of some group.

John Baez (May 12 2020 at 18:36):

So descent in this case boils down to stuff about group homomorphisms - what's their image, and how many group elements map to one (so what's their kernel).

John Baez (May 12 2020 at 18:38):

We saw that 2: Z/2 -> Z/2 was two-to-one and half-onto, so only half the line bundles "descend" along the double cover of the circle by itself, and the one that does *descends in two different ways".

John Baez (May 12 2020 at 18:39):

As soon as we leave the circle and work with more fancy spaces, we'll need to think about "transition functions" also known as "2-cocycles".

John Baez (May 12 2020 at 18:39):

So this gets us pulled into sheaves and sheaf cohomology pretty quick...

John Baez (May 12 2020 at 18:41):

But where I really want to go is algebraic geometry, where we think of fields as fields of functions on certain spaces called "schemes"; then the fundamental group we were just playing around with - the fundamental group of the circle was hiding in what we did - will be called a "Galois group".

sarahzrf (May 12 2020 at 18:42):

isnt the only kind of space whose ring of functions is a field, a point

John Baez (May 12 2020 at 18:42):

The question "does an algebraic gadget defined on a big field K come from extending an algebraic gadget defined over some subfield k" turns out to be another case of descent.

John Baez (May 12 2020 at 18:43):

Yeah, it's funny: fields are like points, but they still have holes in them, leaving a fundamental group.

John Baez (May 12 2020 at 18:43):

This is one of the enduring mysteries of algebraic geometry.

sarahzrf (May 12 2020 at 18:43):

hmmmmmmmmmm

John Baez (May 12 2020 at 18:44):

James Dolan spoke of "voodoo mathematics" - the study of spaces that have been poked so full of holes that every point is a hole.

John Baez (May 12 2020 at 18:45):

The "etale fundamental group" of a field is its absolute Galois group - this is what Artin and Mazur showed, I guess.

sarahzrf (May 12 2020 at 18:46):

/me opens the nlab page on galois theory again

John Baez (May 12 2020 at 18:46):

Anyway, this is just a taste of stuff... I basically prefer to move sort of slowly, so we all have time to really get all the concepts clearly, and so I have time to study enough to continue seeming smart.

sarahzrf (May 12 2020 at 18:46):

OH NO THE NLAB IS DOWN

sarahzrf (May 12 2020 at 18:46):

me but with nlab https://xkcd.com/903/

John Baez (May 12 2020 at 18:50):

Ugh. It's the 0Lab now, eh?

John Baez (May 12 2020 at 18:51):

Btw, @Joe Moeller - one fun part is how all the stuff I just toured at the very end is connected to Brauer groups. Brauer groups are really connected to descent and "bundles".

Joe Moeller (May 12 2020 at 18:56):

Great! How?

John Baez (May 12 2020 at 19:04):

That's for later...

John Baez (May 13 2020 at 23:06):

Okay, last time we discussed a nice example of descent. We looked at pulling back real line bundles from the circle to its double cover (which happens to also be a circle). We studied the "descent" question: when is a line bundle on the double cover the pullback of a line bundle on the circle?

We kinda roughly saw that there was an abelian group of line bundles on any space, which for the space $S^1$ is just $\mathbb{Z}/2$. We saw that pulling back line bundles gives a homomorphism between such groups.

John Baez (May 13 2020 at 23:07):

Thus, the descent problem in this case is just about figuring out the image and kernel of a group homomorphism! The image tells us which line bundles are pullbacks. The kernel tells us how many different line bundles pull back to give a given bundle.

John Baez (May 13 2020 at 23:08):

In our example the homomorphism was

$\mathrm{multiplication \; by \;2}: \mathbb{Z}/2 \to \mathbb{Z}/2$

John Baez (May 13 2020 at 23:08):

which Joe called

$0 : \mathbb{Z}/2 \to \mathbb{Z}/2$

John Baez (May 13 2020 at 23:10):

I called it multiplication by 2 because pulling back a line bundle to the double cover of the circle doubles the number of twists it has: you can see this very clearly.

But a real line bundle over the circle has either 0 or 1 twists, and we're working mod 2, so doubling the number of twists is the same as getting rid of all the twists!

John Baez (May 13 2020 at 23:10):

The image of our homomorphism is $\{0\}$ so the only line bundle that "descends" - the only one that's a pullback - is the untwisted bundle.

John Baez (May 13 2020 at 23:11):

The kernel of our homomorphism is $\mathbb{Z}/2$ so if a line bundle descends, it does so in two ways.

John Baez (May 13 2020 at 23:12):

Now, how should we go further? We could look at real line bundles on all possible spaces, not just the circle. This is still pretty manageable. It would get us into a little bit of sheaf cohomology.

John Baez (May 13 2020 at 23:15):

We could go further and look at all vector bundles over all spaces.

John Baez (May 13 2020 at 23:15):

We could go further and look at all principal bundles over all spaces.

John Baez (May 13 2020 at 23:15):

We could go further and look at all sheaves over all spaces.

John Baez (May 13 2020 at 23:16):

We could go further and look at all sheaves over all sites.

John Baez (May 13 2020 at 23:17):

We could then specialize and look at sheaves over schemes with their etale topology.

John Baez (May 13 2020 at 23:17):

We could then specialize even more and look at schemes that come from fields. Then we'd be doing "Galois descent".

John Baez (May 13 2020 at 23:18):

Or, we could head straight for Galois descent - this is a subject people study without necessarily knowing all the more general cases I've just listed!

John Baez (May 13 2020 at 23:22):

I guess someone should say what they want to do.

sarahzrf (May 14 2020 at 00:05):

i would say something except i have some loose ends ive been procrastinating on tying up that ive finally gotten around to trying to look at so i cant pay attention right now

John Baez (May 14 2020 at 00:17):

Okay, no problem! I've got some other customers...

Morgan Rogers (he/him) (May 14 2020 at 10:13):

It might be a longer journey, but I would love to understand how the descent that's been discussed so far relates to the theory of descent in topos theory (perhaps this is the "all sheaves over all sites" option). There seems to be a comparable motivation of wanting to identify how data descends from the domain of a geometric morphism, but (in the Elephant, at least) there is an extra step: we construct a category of "things which behave as if they are in the image" and call a geometric morphism "descent" if the codomain is equivalent to that category.
However, the choice of data here is still quite mysterious to me, and is not arrived at very organically in that text.
If that option is too much of a tangent for this topic, I'm happy to move it to #theory: topos theory.

John Baez (May 14 2020 at 18:34):

Morgan Rogers said:

It might be a longer journey, but I would love to understand how the descent that's been discussed so far relates to the theory of descent in topos theory (perhaps this is the "all sheaves over all sites" option).

That's a fun journey. We'd been studying descent for real line bundles: i.e. given a map $p: E \to B$, when is a line bundle over $E$ the pullback of one over $B$? In fact we just did it when $p$ is the double covering of a circle by itself.

But line bundles over space are a special case of vector bundles over spaces, so we can ask the same question for those.

And vector bundles over spaces are a special case of sheaves on spaces, so we can ask the same question for those.

And sheaves over spaces are a special case of sheaves on sites, so we can ask the same question for those.

And sheaves on sites are a special case of objects in elementary topoi, so we can probably ask them question for those, though here my understanding gets very fuzzy.

Category theorists might be tempted to quickly jump to the highest level of generality and then climb on down. Normal people might prefer climbing up the ladder of generality one rung at a time. But either way, each level has its own special features that are worth understanding!

sarahzrf (May 14 2020 at 18:35):

category theorists VS normal people :thinking:

John Baez (May 14 2020 at 18:35):

I think I'm more comfortable near the bottom of the ladder, since I've spend lots of time thinking about line bundles and vector bundles, and less about sheaves on spaces, and so on.

John Baez (May 14 2020 at 18:36):

Normal people: "what is an example of this?" Category theorists: "what is this an example of?" They are dual.

John Baez (May 14 2020 at 18:40):

Anyway, I'm happy to explain anything I actually understand.... and happy to try to understand anything anyone else explains, in the subject of "descent".

Maybe one place to start is this very brief snippet on descent for vector bundles:

• Wikipedia, Descent for vector bundles.

Here $p: Y \to X$ is an open cover of a topological space $X$ by a bunch of open sets $U_i$, so $Y = \coprod_i U_i$.

John Baez (May 14 2020 at 18:41):

Does everyone get the idea here?

Morgan Rogers (he/him) (May 15 2020 at 11:35):

Ooh expressions in terms of pullbacks are starting to appear, and the description of "being able to glue the bundles together into a bundle over $X$" feels a lot like an extension of the sheaf condition

Jens Hemelaer (May 15 2020 at 11:53):

Morgan Rogers said:

Ooh expressions in terms of pullbacks are starting to appear, and the description of "being able to glue the bundles together into a bundle over $X$" feels a lot like an extension of the sheaf condition

This is why the category of bundles over a space is a stack (or 2-sheaf). The descent condition is roughly the same as the sheaf condition, after replacing $=$ by $\cong$.

Jens Hemelaer (May 15 2020 at 12:03):

Funny story about this:
I was in Bonn for 1 year as a MSc student, and one of the algebraic geometry exam questions was something like:
"Let F be a sheaf on an irreducible topological space X, such that F is locally the constant sheaf. Show that F is a constant sheaf itself."

I already spent too much time on the other questions, and I was incredibly confused by this. If it is locally equal to the constant sheaf, then of course it is also globally equal? Of course, what was meant is "F is locally isomorphic to the constant sheaf" (because equality is an "evil notion").

My problem was that the professor was working in the 2-sheaf of sheaves on X, while I was working in the sheaf of sheaves on X. :smiley:

sarahzrf (May 15 2020 at 18:54):

i have made the exact observation before that uh

sarahzrf (May 15 2020 at 18:54):

the reason why locally constant sheaves on a connected space can be nontrivial even tho locally constant functions on a connected space are always constant

sarahzrf (May 15 2020 at 18:55):

is precisely b/c you generally assume contractibility of your notion of "equality" for the codomain of a "function on a space"

sarahzrf (May 15 2020 at 18:55):

whereas stalks of sheaves can be isomorphic to one another in different ways

sarahzrf (May 15 2020 at 18:56):

so u get monodromy.

John Baez (May 15 2020 at 19:32):

All this is nice. Maybe I'll ask a lowbrow concrete question or two just to keep us from floating into the stratosphere and dying of oxygen deprivation.

Puzzle. Say we have an cover of a topological space $X$ by open sets $U_i$ and a trivial $\mathbb{R}^n$-bundle on each $U_i$. What data do we need to choose to glue these together and get a vector bundle on $X$?

Puzzle. If I choose some data of this sort and so do you, when do they give isomorphic vector bundles on $X$?

John Baez (May 15 2020 at 19:34):

So, the first question asks for the descent data in this situation. The second asks when two choices of descent data are isomorphic. When we answer these we'll be well on the road to having a groupoid of descent data - though knowing when two things are isomorphic is not as good as knowing what's an isomorphism.

sarahzrf (May 15 2020 at 19:36):

hmm :thinking:

sarahzrf (May 15 2020 at 19:37):

oooooooooh! i see how this relates to higher sheaf stuff now 🤯

sarahzrf (May 15 2020 at 19:37):

but uh thinking thru what the concrete situation actually looks like...

sarahzrf (May 15 2020 at 19:39):

we definitely at least need $f_{ij} : E_i|_{U_{ij}} \cong E_j|_{U_{ij}}$, where $E_i|_V$ means the restriction of our trivial bundle over $U_i$ to the subset $V \subseteq U_i$, and $U_{ij}$ means $U_i \cap U_j$

John Baez (May 15 2020 at 19:40):

Right!

sarahzrf (May 15 2020 at 19:40):

and i think we want a clique condition

sarahzrf (May 15 2020 at 19:40):

like $f_{ij} ; f_{jk} = f_{ik}$

sarahzrf (May 15 2020 at 19:40):

no wait

sarahzrf (May 15 2020 at 19:40):

no wait do we

sarahzrf (May 15 2020 at 19:40):

yeah i think we do

John Baez (May 15 2020 at 19:40):

Yes, we do!

sarahzrf (May 15 2020 at 19:41):

i was thinking that would make the bundle trivial, but

sarahzrf (May 15 2020 at 19:41):

right, this only applies to like... shrinking down an intersection to include intersecting more pieces

sarahzrf (May 15 2020 at 19:41):

not zipping over to some distant place

John Baez (May 15 2020 at 19:41):

Can you tell me how to build the Mobius strip bundle this way?

sarahzrf (May 15 2020 at 19:41):

no im hungry and cant think at full capacity im gonna go get a snack

John Baez (May 15 2020 at 19:42):

We can cover the circle with just two contractible pieces of we want.

sarahzrf (May 15 2020 at 19:42):

:sob:

John Baez (May 15 2020 at 19:42):

Okay. Either you or someone else should see how what you just said works in our good old example.

John Baez (May 15 2020 at 19:42):

(People who know all this stuff too well should stay quiet.)

John Baez (May 15 2020 at 19:43):

We can wait for Sarah to eat.

Kenji Maillard (May 15 2020 at 19:47):

For the mobius strip, don't you just repeat what you would do with the paper strip ? Like if we cover the circle by two contractible covers, we take 2 strips of papers (our trivials bundles) and glue them "normally" on one side and with a twist on the other side

John Baez (May 15 2020 at 19:49):

Right!

John Baez (May 15 2020 at 19:50):

We discussed it earlier in this thread; now we're trying it again with a bit more math.

John Baez (May 15 2020 at 19:51):

I think I want to get Sarah to work through the details.

John Baez (May 15 2020 at 19:53):

But it'd be great to have you join this "class".

Kenji Maillard (May 15 2020 at 19:54):

Sorry to barge in suddenly, I have been silently following this thread with a lot of attention, it really helps making these things clearer :)

John Baez (May 15 2020 at 19:55):

Great! I'm glad you barged in! Since I have no idea how many people are paying attention, I tend to feel almost nobody is, which makes me unhappy.

Nathanael Arkor (May 15 2020 at 19:59):

With so many users in the server, there are certainly many more silent watchers for any given topic than you would imagine.

sarahzrf (May 15 2020 at 20:41):

hello i got distracted after coming back

sarahzrf (May 15 2020 at 20:41):

mobius...

John Baez (May 15 2020 at 20:42):

Did you ever watch Mobius strip?

sarahzrf (May 15 2020 at 20:42):

so your base space is S¹. say it's covered by U₁ and U₂, defined by intersecting with half planes

sarahzrf (May 15 2020 at 20:42):

is mobius strip the name of a show or something

John Baez (May 15 2020 at 20:43):

No, he was actually a strip-tease artist in his spare time.

sarahzrf (May 15 2020 at 20:43):

:weary:

John Baez (May 15 2020 at 20:43):

Anyway, yes, you're doing great! Two open subsets!

sarahzrf (May 15 2020 at 20:45):

i feel like maybe thats not an appropriate joke for this community

sarahzrf (May 15 2020 at 20:45):

<.<

sarahzrf (May 15 2020 at 20:45):

but, uh

John Baez (May 15 2020 at 20:46):

Okay, I'll try to be more serious in the future.

sarahzrf (May 15 2020 at 20:46):

(idk about failure to be serious being the problem, more the risque-ness)

sarahzrf (May 15 2020 at 20:47):

we need f₁₂ and f₂₁, and the clique condition (i shouldve included a clause about f_{ii} = id) says they're inverse, so that's just f₁₂

sarahzrf (May 15 2020 at 20:47):

so a vector bundle automorphism of $(U_1 \cap U_2) \times \mathbb R$

sarahzrf (May 15 2020 at 20:49):

$U_1 \cap U_2$ has two connected components, so we can just define this piecewise—on one, it's the identity; on the other, it's negation on the $\mathbb R$ coordinate

sarahzrf (May 15 2020 at 20:51):

hmm, giving the data like this kinda reminds me of the tiny tiny bit ive read about čech methods...

John Baez (May 15 2020 at 20:52):

Great! By the way, this "clique condition" is exactly right now, and it's exactly what people call the 2-cocycle condition.

sarahzrf (May 15 2020 at 20:52):

i always forget the 0-ary clause in definitions like that >_<

sarahzrf (May 15 2020 at 20:52):

i need to remember to use unbiased versiosn more often

John Baez (May 15 2020 at 20:52):

Right! But you seem to know the 0-ary thing has to be in there.

sarahzrf (May 15 2020 at 20:53):

i mean

sarahzrf (May 15 2020 at 20:53):

it always is

sarahzrf (May 15 2020 at 20:53):

in things like this

John Baez (May 15 2020 at 20:53):

Good! So, what you've just given is called a 2-cocycle in Cech cohomology.

sarahzrf (May 15 2020 at 20:54):

HA i was right about the čech thing

sarahzrf (May 15 2020 at 20:54):

🇨🇿

John Baez (May 15 2020 at 20:54):

Yes, you can čech that off your bucket list now.

John Baez (May 15 2020 at 20:55):

Was that a suitable joke for this forum?

sarahzrf (May 15 2020 at 20:55):

🪣

John Baez (May 15 2020 at 20:56):

That symbol doesn't display for me, which may be all for the best.

John Baez (May 15 2020 at 20:56):

Let's mine this example a bit more. You said $f_{ij}$ was 1 or -1. What's the deal with these numbers?

sarahzrf (May 15 2020 at 20:57):

i said no such thing

John Baez (May 15 2020 at 20:57):

You said "on one, it's the identity; on the other, it's negation on the R coordinate."

John Baez (May 15 2020 at 20:57):

That's a long-winded way of saying 1 or -1.

sarahzrf (May 15 2020 at 20:57):

that was all a single $f_{ij}$ :)

sarahzrf (May 15 2020 at 20:58):

but i'm being obnoxious, i know what you're getting at

sarahzrf (May 15 2020 at 20:58):

we can refine the cover until the intersections are connected, right?

John Baez (May 15 2020 at 20:58):

Okay, fine. But my question was: what in general could your $f_{ij}$ take values in?

John Baez (May 15 2020 at 20:59):

Yes, on a tolerably nice space you can always use a "good cover", where all the the opens and all their intersections are contractible (or empty).

John Baez (May 15 2020 at 20:59):

https://en.wikipedia.org/wiki/Good_cover_(algebraic_topology)

sarahzrf (May 15 2020 at 21:00):

well, we're giving automorphisms of trivial line bundles, so that's a continuous function to GL(1) right

John Baez (May 15 2020 at 21:00):

Great!

John Baez (May 15 2020 at 21:01):

So yes, you gave me a "Cech 2-cocycle valued in the sheaf of continuous GL(1)-valued functions"... I think that's what the experts call it.

sarahzrf (May 15 2020 at 21:01):

hmm..... something something do we only need a homotopy class to determine the bundle up to isomorphism :thinking: isn't that a condition that appears in the theory of classifying spaces

John Baez (May 15 2020 at 21:02):

Right, and that sort of thing shows up in this Cech story too.

John Baez (May 15 2020 at 21:02):

Now, you'll notice that while your cocycle took values in GL(1), it actually took values in something smaller!

John Baez (May 15 2020 at 21:02):

What's that?

sarahzrf (May 15 2020 at 21:03):

S⁰

sarahzrf (May 15 2020 at 21:04):

cuz we can homotope anything in GL(1) to something in S⁰

sarahzrf (May 15 2020 at 21:07):

what is GL(2) shaped like

John Baez (May 15 2020 at 21:09):

Great! In this game $S^0$ is usually called $O(1)$.

sarahzrf (May 15 2020 at 21:10):

dammit

sarahzrf (May 15 2020 at 21:10):

and here i was feeling good about noticing it was S⁰ and not just {-1, +1} :D

John Baez (May 15 2020 at 21:10):

It's okay; I just want to be able to generalize to higher dimensions!

sarahzrf (May 15 2020 at 21:11):

mostly being tongue in cheek :)

John Baez (May 15 2020 at 21:11):

$O(n)$ is the group of rotations/reflections of $\mathbb{R}^n$.

John Baez (May 15 2020 at 21:11):

One great thing is that it's a maximal compact subgroup of $GL(n)$.

sarahzrf (May 15 2020 at 21:11):

oh cool

John Baez (May 15 2020 at 21:12):

An amazing fact about Lie groups is that they're all homotopy equivalent to any maximal compact subgroup!

sarahzrf (May 15 2020 at 21:12):

:o

sarahzrf (May 15 2020 at 21:12):

i think i remember SO(3) being the same as RP³... is there something similar for O(n)

John Baez (May 15 2020 at 21:12):

Even better, any Lie group is, as a manifold, a product of "the" (any) maximal compact subgroup and a vector space.

sarahzrf (May 15 2020 at 21:12):

or at least SO(n)

sarahzrf (May 15 2020 at 21:13):

wow i should learn some lie theory

sarahzrf (May 15 2020 at 21:13):

that shit sounds crazy

John Baez (May 15 2020 at 21:13):

Yes, SO(3) is diffeomorphic to $RP^3$.

John Baez (May 15 2020 at 21:14):

There's a nice way to see it, by saying any rotation is a clockwise rotation around some axis... so rotations are described by points in a ball in 3d...

John Baez (May 15 2020 at 21:14):

but a rotation by pi around one axis is the same as a rotation by pi around the axis pointing the opposite direction!

John Baez (May 15 2020 at 21:15):

So we take the ball and identify opposite points on its boundary!

sarahzrf (May 15 2020 at 21:15):

yeah thats how i remember it :)

John Baez (May 15 2020 at 21:15):

Yes, so that gives $SO(3) \cong RP^3$.

John Baez (May 15 2020 at 21:16):

So that means $GL(3) \cong RP^3 \times V$ for some vector space $V$, where the isomorphism is only of manifolds, not of groups.

John Baez (May 15 2020 at 21:16):

And you can figure out how big $V$ needs to be just be figuring out the dimension of $GL(3)$.

John Baez (May 15 2020 at 21:16):

Oh, whoops, I lied.

John Baez (May 15 2020 at 21:17):

$GL(3) \cong O(3) \times V$

John Baez (May 15 2020 at 21:17):

SO(3) is just the rotations. O(3) is the rotations and reflections.

John Baez (May 15 2020 at 21:17):

In fact O(3) $\cong$ SO(3) $\times$ Z/2, as groups and as manifolds.

John Baez (May 15 2020 at 21:18):

Anyway, I could ramble on about this stuff all day, since this is the bread and butter of mathematical physics.

John Baez (May 15 2020 at 21:19):

I should probably get to work on a paper... but I think it's no surprise to you now when I wrap up and say "n-dimensional real vector bundles over a space can be described using Cech 2-cocycles on that space valued in the sheaf of continuous GL(n)-valued functions".

John Baez (May 15 2020 at 21:20):

But we didn't get around (yet) to figuring out when two 2-cocycles give isomorphic bundles! That's the next step. I gotta quit now, but maybe you can think about that, if you want.

sarahzrf (May 15 2020 at 21:21):

i was thinking earlier... the thign i said about seeing the connection to higher sheaves

sarahzrf (May 15 2020 at 21:21):

so in an ordinary sheaf you have a set of sections for each open. in a higher sheaf you have a category of sections for each open

sarahzrf (May 15 2020 at 21:22):

so im seeing how here you have a category of vector bundles over each open

John Baez (May 15 2020 at 21:22):

Yes! That sort of "higher sheaf" is called a "stack".

sarahzrf (May 15 2020 at 21:22):

so ive heard o:

sarahzrf (May 15 2020 at 21:23):

and restriction functors would be given by pullback?

John Baez (May 15 2020 at 21:23):

Yeah! One of the nice examples of a stack that's not a mere sheaf is the stack of vector bundles (say with some chosen fiber).

sarahzrf (May 15 2020 at 21:23):

but so id heard that "descent" is what you have for stacks instead of the typical gluing condition and it gets hairier

John Baez (May 15 2020 at 21:23):

Well, there's "descent" at all levels of hairiness.

sarahzrf (May 15 2020 at 21:23):

and i think i get it—the issue is that you can no longer just require that sections be equal on the overlaps, you have to require that they be isomorphic on the overlaps, but

sarahzrf (May 15 2020 at 21:24):

then you also need to require that the isomorphisms are coherent

John Baez (May 15 2020 at 21:24):

Right!

sarahzrf (May 15 2020 at 21:24):

:triumph:

sarahzrf (May 15 2020 at 21:24):

but there was one other thing that occurred to me earlier—

sarahzrf (May 15 2020 at 21:24):

if u have an ordinary sheaf given by continuous functions into some codomain, say

John Baez (May 15 2020 at 21:24):

A lot of this stuff can be understood to some extent using Cech cohomology.

sarahzrf (May 15 2020 at 21:25):

u can already upgrade it to some sort of stack, right, by just considering the set of functions as a category by the fact that they naturally have higher structure bc of homotopy

sarahzrf (May 15 2020 at 21:25):

and what youre talkling about in this thread is like—establishign a correspondence between that and the directly-described stack of vector bundles?

John Baez (May 15 2020 at 21:26):

I don't know if that's what I'm doing. I don't think it quite is.

John Baez (May 15 2020 at 21:26):

But it might turn out to be, in the end.

John Baez (May 15 2020 at 21:27):

I think it'll be good if you just figure out a really simple condition for when two choices of $f_{ij}$ (with the same cover) give isomorphic bundles.

sarahzrf (May 15 2020 at 21:27):

hmmmmmmm maybe later >.>

sarahzrf (May 15 2020 at 21:27):

adhd engage

John Baez (May 15 2020 at 21:27):

Yeah, later.

John Baez (May 15 2020 at 21:28):

Then you'll be able to cook up a groupoid with $f_{ij}$ 's as objects and some other things as isomorphisms.

John Baez (May 15 2020 at 21:28):

And people usually just take isomorphism classes and call that "the 2nd Cech cohomology".

John Baez (May 15 2020 at 21:29):

But yeah, we may be getting our hands on a stack here, secretly!

John Baez (May 15 2020 at 21:30):

Okay, seeya! Gotta keep editing Kenny Courser's thesis.

sarahzrf (May 15 2020 at 21:30):

thanks for the lecture :)

sarahzrf (May 20 2020 at 08:13):

hmm, sorry i havent been around

sarahzrf (May 20 2020 at 08:13):

been off my adderall... :upside_down:

sarahzrf (May 20 2020 at 08:13):

and will be again.......

sarahzrf (May 20 2020 at 08:15):

can you describe a stack as a pseudofunctor O(X) → Cat w/ a condition like... some kind of "turning certain diagrams into lax limits", maybe :eyes:

Todd Trimble (Nov 07 2020 at 12:14):

If $\text{Nat}(F, G)$ denotes the category of pseudonatural transformations and modifications between pseudofunctors, then the condition for $G$ to be a stack is that $G(X) \simeq \text{Nat}(\hom(-, X), G) \to \text{Nat}(F, G)$ is an equivalence for each covering sieve $F \hookrightarrow \hom(-, X)$. Analogous to a sheaf, but one categorical dimension up.

ADITTYA CHAUDHURI (Nov 07 2020 at 14:03):

What are some good examples of stacks over a site $C$ which are are not gerbes over $C$?

John Baez (Nov 07 2020 at 15:41):

I think the stack of vector bundles and vector bundle morphisms over a topological space is an example.

Ulrik Buchholtz (Nov 07 2020 at 15:42):

Or simply the stack of (small) sheaves and either isomorphisms or all morphisms.

ADITTYA CHAUDHURI (Nov 07 2020 at 20:25):

@John Baez @Ulrik Buchholtz Thank you Sir.

ADITTYA CHAUDHURI (Nov 07 2020 at 20:38):

@John Baez I understand your point , I think you mean that "Existence of a global section does not ensure Triviality in vector bundles but ensures triviality in $G$-torsors. Hence , local connectedness property of the gerbe holds for stack of $G$-torsors over a space but fails in the case of stack of Vectors bundles over a space. Am I correct ?

John Baez (Nov 07 2020 at 20:45):

I'm not an expert on this stuff, so I don't even remember what the "local connectedness property of a gerbe" is. But it's certainly true that existence of a global section does not ensure triviality in vector bundles but ensures triviality in a $G$-torsor.

ADITTYA CHAUDHURI (Nov 07 2020 at 21:07):

@John Baez Let $F:D \rightarrow C$ be a stack over a site $C$. Then we say that $F$ is locally connected if for each object $U$ in $C$ and for any $a,b \in F(U)$, there exist a cover $\lbrace U_i \rightarrow U \rbrace$ of $U$ such that $a|_{i}$ is isomorphic to $b|_{i}$ in $F(U_i)$ for all $i$. While proving this property for the stack of $G$ torsors over a space $X$ I used the fact that a $G$-torsor is trivial if there exist a global section. So, I thought that the same procedure will not work for the stack of vector bundles over $X$. Though I agree it does not ensure that the stack of vector bundles over $X$ is not a gerbe

Reid Barton (Nov 07 2020 at 21:44):

If we consider two vector bundles over $U$ of different ranks, then we won't be able to make them isomorphic by pulling back to a cover.

Reid Barton (Nov 07 2020 at 21:47):

But if we consider the stack of vector bundles of a fixed rank $n$ (and bundle isomorphisms between them) then I think it will be a gerbe. (We could allow $n$ to be a locally constant nonnegative-integer-valued function on $X$.)

ADITTYA CHAUDHURI (Nov 07 2020 at 21:48):

@Reid Barton But the category of vector bundles over a space is not a groupoid . Since Gerbes are always defined to groupoid valued stacks hence it will not be a gerbe I think.

Reid Barton (Nov 07 2020 at 21:49):

That's right, that's why I specified that we consider just vector bundle isomorphisms and not all vector bundle maps.

ADITTYA CHAUDHURI (Nov 07 2020 at 21:50):

@Reid Barton Ohh!! Sorry! I got it.. Thanks.

John Baez (Nov 07 2020 at 22:24):

In my original remark I specified the stack of vector bundles and all vector bundle morphisms, and I was hoping all those non-invertible morphisms would prevent this stack from being a gerbe. I am a simple man given to simple counterexamples.

But okay, I guess even the stack of vector bundles and vector bundle isomorphisms is not a gerbe.

John Baez (Nov 07 2020 at 22:25):

The stack of rank-n vector bundles is equivalent to the stack of GL(n)-torsors, so yes, that should be a gerbe.

sarahzrf (Nov 07 2020 at 22:53):

Todd Trimble said:

If $\text{Nat}(F, G)$ denotes the category of pseudonatural transformations and modifications between pseudofunctors, then the condition for $G$ to be a stack is that $G(X) \simeq \text{Nat}(\hom(-, X), G) \to \text{Nat}(F, G)$ is an equivalence for each covering sieve $F \hookrightarrow \hom(-, X)$. Analogous to a sheaf, but one categorical dimension up.

this is actually the only precise definition of the descent condition i remember lol

sarahzrf (Nov 07 2020 at 22:54):

but i think it also moves some amount of difficulty into bicategory yoneda...

John Baez (Nov 07 2020 at 23:07):

Yeah, for lowly folks like me who have mainly thought about stacks on topological spaces, it's probably some super-efficient way of saying you get an object of $G(X)$ when you've got objects on each open set $U_i$ (or whatever) covering $X$, and isomomorphisms $g_{ij}$ between their restrictions to intersections $U_i \cap U_j$, such that $g_{ik} = g_{ij} g_{jk}$ on intersections $U_i \cap U_j \cap U_k$. And that last thing, that "coherence" law, looks like something you see whenever you replace equations by isomorphisms.

Todd Trimble (Nov 08 2020 at 01:05):

sarahzrf said:

but i think it also moves some amount of difficulty into bicategory yoneda...

The way I look at it, that's the kind of thing you can calculate to your heart's content, if you're motivated enough to pursue it. Bicategorical Yoneda is not after all difficult!

ADITTYA CHAUDHURI (Nov 08 2020 at 06:25):

@John Baez Sir, I undersrtand your point. I should have directly said that the Stack of vector bundles over a space X is not a gerbe from the fact that it is not groupoid valued stack. But I was so much into understanding "what local connectedness property actually says about the gerbe" I forgot to check the most elementary one. For example in the case of stack of $G$-torsors over a space, the local connectedness property tells that "Existence of a local sections ensure local triviality in $G$-torsors".

sarahzrf (Nov 08 2020 at 06:33):

John Baez said:

Yeah, for lowly folks like me who have mainly thought about stacks on topological spaces, it's probably some super-efficient way of saying you get an object of $G(X)$ when you've got objects on each open set $U_i$ (or whatever) covering $X$, and isomomorhpisms $g_{ij}$ between their restrictions to intersections $U_i \cap U_j$, such that $g_{ik} = g_{ij} g_{jk}$ on intersections $U_i \cap U_j \cap U_k$. And that last thing, that "coherence" law, looks like something you see whenever you replace equations by isomorphisms.

this isnt the full condition, is the thing, cuz u need to know about how the object you get is related to the family

John Baez (Nov 08 2020 at 17:55):

You're right.

John Baez (Nov 08 2020 at 17:57):

I was just trying to figure out where the cocycle conditions $g_{ik} = g_{ij} g_{jk}$ comes from, if you start with Todd's definition in terms of $\mathrm{Nat}(F,G)$.

John Baez (Nov 08 2020 at 17:59):

They remind me both of "weak preservation of composition" in the definition of pseudofunctor, and the prism diagram in the definition of pseudonatural transformation, so I'm a bit confused, but also too lazy to straighten it out in my mind.

sarahzrf (Nov 08 2020 at 22:13):

it's gonna be the latter, because a descent datum is a pseudonatural transformation

John Baez (Nov 08 2020 at 22:15):

Okay, thanks. I used to know this stuff better, I think. I can't even remember what I forgot.

David Michael Roberts (Nov 15 2020 at 09:31):

I would phrase local connectivity as say that locally, everything looks the same, and that local non-emptiness is what tells you things are locally trivial. Vector bundles are locally trivial, after all, but as discussed don't locally look the same.

Tim Hosgood (Nov 15 2020 at 15:33):

just in case it's of any help to anybody, I'm now onto translating the parts of FGA that are about descent (which is really the vast majority of it...), and I think it's a pretty nice (albeit "Grothendieck-y") reference

Tim Hosgood (Nov 15 2020 at 15:33):

(I'm about a third of the way through FGA 3.I : https://latexonline.cc/compile?git=https%3A%2F%2Fgithub.com%2Fthosgood%2Ftranslations&target=_in-progress%2Ffga%2Ffga-3-I.tex&trackId=1605454299350)