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3:00 AM
soooooo, groupoids which have the same pi_0?
 
Transitive groupoids are those such that all objects are iso.
Or, their pi_0 = 0
 
oh i see what you're saying
 
or 1, depending on how you count
Howbeit in the internal case we say that a certain map has local sections rather than is just onto (which in a geometric setting makes no sense - awful maps can be onto)
 
so, the kernel
 
Not sure what you mean
there
 
3:01 AM
nvm
got it
 
Sorry, a better way to say it is this:
A groupoid in sets is transitive iff the map X_1 -> X_0 \times X_0 is onto
In the geometric setting we say this map has local sections
 
So locally it's a split epimorphism?
 
okay that makes sense
at least, i get the surjective statement
 
Yes, but locally may be in some Grothendieck topology (e.g. etale)
 
Right
 
3:03 AM
OK, so now we have a groupoid X with a functor to X_0/X_1 =: pi_0(X)
 
really, the point is, $\pi_0=0$
 
No, only for the fibres of the projection map
 
What exactly do you mean by X_0 / X_1?
 
okay.... sorry. sigh, i think i'm mixed up
oh i get what you're saying
just parsing your language wrong
 
X_0/X_1 is the coequaliser of the source and target maps X_1 -> X_0, equivalently the coarse moduli space, equivalently the space of iso classes of the groupoid etc
 
3:05 AM
yes, absolutely, the fibers of that functor have trivial $pi_0$
 
Ah... Does that necessarily exist?
Or are you assuming it exists?
 
In spaces, yes, for schemes, let's take the source and target maps to be smooth, or possibly etale
 
i was thinking you were taking the functor $F:Groupoids\to Sets$ or something
but i get it now
 
And perhaps we want (s,t) : X_1 -> X_0 \times X_0 separated. But that is icing on the cake
 
Icing is delicious.
 
3:07 AM
Alright, now we change perspective slightly, and work in spaces over X_0/X_1
 
alright
 
So our groupoid has maps X_1 -> X_0/X_1 and X_0 -> X_0/X_1 compatible with all the groupoid structure
(commutes with source/target/identities/composition)
 
alright, yeah, that makes sense
so they're spaces over X_0/X_1
 
Yes.
Let's let M := X_0/X_1
 
3:09 AM
OK. THEN, we want to consider the map X_1 -> X_0 \times_M X_0
one moment
 
the one that factors thru just regular X_0\times X_0
 
The other way around.
 
ahah right
 
This is the 'relative to M' version of X_1 -> X_0^2
and NOW this is surjective, or rather, it has local sections
 
so, yeah, just, coproduct in $M$-spaces
 
3:11 AM
product
 
jeez sorry right
 
Alright. That's the set up
 
lol ok
 
We say that X -> M is an A-bundle gerbe if X_1 -> X_0 \times_M X_0 is a principal A-bundle
 
aha okay
 
3:12 AM
using the Grothendieck topology we have lying around
 
Or A-torsor or whathaveyou
 
yeah, i had a feeling they were kind of like principle G-bundles in some complicated way
i mean, they are
just, over specific things
 
Another way to see this is that the groupoid X has an action by the 2-group A => 1
with quotient M
One usually starts with a space/scheme M and then takes X such that X_0/X_1 = M, though
 
alright
cool!
 
3:14 AM
Note that A here must be abelian
 
For the same reason that pi_2 is
 
because you can't deloop nonabelian groups twice
 
alright. right on. thanks for the crash course
 
3:16 AM
So there are some nice ways to generalise this just a little bit, without even going to nonabelian things, but that's probably enough for now
 
i guess, intuitively, how do you think of such things?
 
so what about Eric's question?
 
what do they do?
 
The trick is, as usual, what it means for two of these to be equivalent
 
for me the fact that you can't deloop GLn/U(n) more than once for n > 1 is a total show stopper
 
3:17 AM
What you need to do is take more general 2-groups, rather than just trying to deloop a nonabelian group more times.
 
sure
but what 2-group has the right to replace a twice delooped U(n)?
 
Giraud got stuck at this point, because he defined nonabelian gerbes using the crossed module (same thing as a 2-group) Ad: G -> Aut(G)
And nothing was functorial in homomorphisms G-> H
Ah, so here's the key example from higher gauge theory
 
omg, i have to leave and now you start talking about this
 
transcripts, bro
 
Consider the group Omega G of smooth loops in a compact simple simply-connected Lie group G
Email me, @Eric
Omega G has a canonical extension hat(Omega G) by U(1)
It turns out this is 2-connected (just to keep the homotopy theory up)
In fact the 2-connected cover of Omega G.
I've never seen this fact mentioned in print before
Then recall we have the space of connections on the trivial G-bundle on a circle A_G
This can be identified with a certain space of smooth paths [0,1] -> G (periodic derivatives and something like that), which send 0 to e \in G
 
3:22 AM
Sorry, catching up, is A here the same as $\pi_1(X)$?
 
Sorry, getting ahead of myself
Not quite - changed gears completely.
 
I'm confusing letters.
 
I'm describing a 2-group that we can consider gerbes/2-bundles for
 
I'm talking about way back.
 
Ah, that was X
Yes, A = pi_1(X) = Aut(x) for any x \in X_0
For the ordinary bundle gerbe, all the automorphism groups are abstractly isomorphic, but the groupoid X is not transitive
I'll push on, but do stop me if need be
Now Omega G sits inside A_G, as periodic paths certainly have periodic derivatives
 
3:26 AM
Is anybody else still here?
 
yes
 
Just checking.
 
Glad I'm not talking to myself (and let's not start that again)
We then consider the homomorphism hat(Omega G) -> Omega G -> A_G
There is a conjugation action of A_G on Omega_G
And this lifts to a 'conjugation' action on hat(Omega G)
And this makes hat(Omega G) -> A_G a very interesting crossed module.
 
I'm not too familiar with crossed modules
 
As it represents the next step in the Whitehead tower of G
 
3:29 AM
but with this you've constructed a 2-group, right?
 
yes
We take the groupoid of hat(Omega G) acting on A_G via the given map
Then the action of the latter on the former means we have a monoidal groupoid
which is a 2-group
I'm afraid I don't know any interesting examples of 2-group schemes
Aside from the obvious ones
 
so wait
what is the Whitehead tower of G?
 
Ah, going back a step, if we assume G = Spin(n)
 
then this is String?
 
Then we get Spin(n) -> SO(n) -> O(n)
Yes
And String_G for any old 1-connected G
 
3:32 AM
I heard a talk by Christoph Wockel once that sounded a lot like this.
Is this related?
 
Yes, he's all over this
 
It certainly is. It's the Baez-Crans-Schreiber-Stevenson model
 
I have a related silly question
is there a sensible way of doing this modulo 8 for O(n)?
 
3:35 AM
lol
 
Hmm, not sure what that would mean
Do you mean like Clifford algebras
Or over Z/8?
 
well
I know doing a Whitehead tower relies on you having killed off lower groups before proceding higher
 
Or Bott periodicity
Yes
 
but is there a way of killing off groups mod 8 here
yes because of Bott periodicity
 
Ah. Hmm
Not that could even imagine
So you want to kill off pi_i(O(n)) (n >>0) for i = 3mod 8 or similar
 
3:38 AM
that's the idea
although fixing n there is going to cause troubles
 
Man, that's pretty crazt
crazy
 
I said it was silly
 
So you want O, rather than O(n)
 
yes
 
3:41 AM
I don't think that link works for people other than yourself
 
hmm, it should
Never mind
Let's just say I don't feel optimistic
 
that worked
 
:-)
So a question I would be interested in is what sort of interesting 2-groups one can get in schemes
Hi Jon
 
I'll pretend to be competent and say: won't String be such an example?
 
3:47 AM
oh hai
 
Spin is an algebraic group, seems like String should be a 2-group scheme
or is that nonsense?
 
did Mathematica not give you the polynomials you wanted?
 
Perhaps we can take formal loops in G, and get an Ind-scheme
 
no it did. they're just big
plus i've been really silly about typesetting all of them really nicely
 
3:49 AM
I've seen worse
 
I guess it's at this point that my lack of understanding of cohesion betrays me
 
yeah. it won't take too long
sorry for interrupting, continue
 
good performance
 
@Will - is that to Jon or myself
(re: lack of understanding)
 
3:51 AM
to you
 
Ah, about String
 
I understand his polynomials pretty well
 
ok, some history there....
 
yes, of cyberbullying
he told me not to do it again
 
Ah, that was just before I came in
So can we replace Omega G by the formal loops ind-scheme, when G is a 1-connected (reductive, if that makes sense) group scheme?
 
3:53 AM
cyberbullies
 
so you mean Hom(Spec(k((t))),G) ?
sorry I don't know what formal loops are
 
That's something like it
Perhaps I'm thinking along the lines of polynomial loops
Just looking it up
OK, something like R |-> G( R((t)) ), but that's probably what you wrote
(source: arxiv.org/abs/math/0607130, Pappas-Rapoport)
 
ok
well that certainly seems like something you can do but I have no idea if it leads anywhere
 
Then we'd need something like the transgression of the canonical 3-cocycle (assume everything nice, so this exists) on G to this loop group
Yeah, I'm just talking off the top of my head. I'm not an algebraic geometer, after all
Other than that, perhaps something like the automorphism 2-group of a stack, although that's not going to be a scheme except in special cases
sorry, 2-group in schemes
 
well that's where cohesive structures come in right, but I don't know how you do it really
 
4:04 AM
Well, I suppose it's probably a group stack
Cohesion is probably overkill just to get one simple example :-)
OK, so here's an idea: what's the automorphism 2-group of the stack of elliptic curves?
 
do we expect to be able to name it?
 
No, compute it
It would be nice if the action lifted to the sheaf of E_oo ring spectra which is elliptic cohomology
Perhaps people already know bits and pieces of this already
 
dang ol mathematica won't solve my equations
 
Shalosh B. Ekhad would be able to.
 
i don't know what that means
 
4:12 AM
so you mean you don't know who to call?
 
gerst bersters
 
I enjoy that some of the papers have no coauthors
 
+1 Will
 
Another Hanukkah Miracle: The Gaps Between Consecutive Christmas-in-Hanukkah Years is ALWAYS a Fibonacci Number!
 
4:16 AM
Anyway, you were mentioning, @JonBeardsley about the groupoid of fgls
 
yes, use mathematica to compute an automorphism 2-group Jon
 
dear mathematica, what is the homotopy of the 893-sphere? localized at 17
@DavidRoberts yeah, that's the only one i've ever significantly bumped into
well, i'm sort of interested in descent groupoids, but i don't really know what that means yet
i gotta get this book by janelidze from the library
 
Ah, the one on Galois theory?
 
oh god 2-vector spaces
 
4:20 AM
Why?
 
so re:Eric's question, what is a 2-vector bundle?
 
Know anything bad?
Well, let's just wait and see if I get my research grant
And then I'll be able to work on them for 3 years
Seriously, though...
 
i want to understand Kontsevich's stuff about polyvector fields and Hochschild homology
 
so what is the 2-group corresponding rank n 2-vector bundles?
 
yeah the one on galois theory
sorry, just saw that
 
4:21 AM
Well, that's a slightly tricky question...
 
because i'm interested in descent groupoids associated to galois and hopf-galois extensions of ring spectra
 
yeah that's why I haven't been convinced to like 2-vector spaces
 
but i don't know that stuff too well w/r/t dang ol regler rings
 
Just a moment... brb
The main problem, to my mind, is that we have all these flavours of 2-vector spaces flying around, and then a massive general version due to Urs Schreiber
 
good old good old
 
4:27 AM
And it's not entirely clear which applications they are good for.
 
so do I need to just accept there is no one such thing as "2-vector bundle" and that different notions of 2-vector space give different notions of 2-vector bundles?
 
For instance, Kapranov-Voevodsky ones are used for 2-K-theory by Baas-Dundas-Rognes - but only at the beginning
then it's all spectra calculations
Yes - that is right
 
hrmpf
 
Well, there is not one sort of 1-vector space, when you think of it
 
?
 
4:29 AM
Fin. dim, separable Hilbert, Banach, Frechet, etc
And then we have all sorts of different fields
Things just bifuricate
 
ok sure
 
OK - the REAL problem: what is a 2-field?
 
but if you settle on finite dimensional, and settle on a field
are there still bifurcations in the 2-theory?
 
Well, what's a 2-field?
 
4:31 AM
yeah fair enough, I see
 
It seems like R and C, which are the easy ones, are very special anyway
KV 2-vector spaces take Vect as the base ring, and right there you have lost the topology on the objects
A given vector space doesn't admit any deformations, say
Man, I've been hogging the airwaves.
 
I've enjoyed it
 
Don't get to talk about this stuff much at my department
Thanks @Will !
 
no worries
glad there's some math going on around here
 
And it's all even vaguely relevant :-)
I might even ask a question on MO about Aut(Ell)
imagine that!
 
4:35 AM
could turn out quite interestingly
 
good!
quote this chatroom!
haha
 
Ok, let me thrash out a question here
What is Eq(Ell)?
 
What is Eq?
 
The stack of elliptic curves is a very concrete, nice stack [Q on which site?]
OK, that's why I wanted to do it here
Aut(Ell)
What is Aut(Ell)?
The stack of elliptic curves is a very concrete, nice stack [on X site]
and as an object of the 2-category of stacks, its endofunctors which are equivalences are the objects of a 2-group. That is, a monoidal groupoid with weakly invertible monoidal product.
[I think I need a link to the description of Ell.]
 
you might as well do it over the complex numbers
and draw a punctured sphere with two cone points as an explanation :P
 
4:41 AM
alright, have fun guys, these stupid polynomials have made me want to put my eyes out with an icepick. i gotta go to bed
 
careful with the icepick
 
ok, so assuming 6 is invertible, what's that ring again?
Yes, g'night @JonBeardsley
 
i always sleep with an ice pick. later.
 
which ring are you asking about, sorry?
 
Ah, there's a very simple ring from which one can get a lot of information about M_{1,1}.
I'm trying to find it
OK, so there's a Hopf algebra (=groupoid in Aff) which presents the moduli stack of elliptic curves
 
4:45 AM
There are some lecture notes by Mike Hopkins that describe this nicely.
Section 20
Page 70
 
ah so you mean the formula for an elliptic curve in Weierstrass form with j-invariant j, j not 0 or 1728?
 
That's the one, @EvanJenkins
And yes, @Will
ok, so I now have my affine groupoid scheme
(cont.) ... So as the stack is so concrete, we can try to calculate this automorphism 2-group.
My first question is, has this been done?
At least, away from the primes 2 and 3 there shouldn't be massive problems.
Then, if we have this 2-group, one can ask whether the action lifts to objects related to Ell.
For instance, we have the universal elliptic curve over Ell, which we can view as the total space of a stack over Ell
[I think? Does this sound reasonable? Perhaps it's not a stack over Ell, but over the base site, but with a map to Ell...]
Then we also have the sheaf of ring spectra from which we get elliptic cohomology theories: is this an equivariant sheaf? (in the appropriately weak way)
 
Lurie does something like this in his survey of elliptic cohomology.
 
using Aut(Ell)? I know there's some equivariant stuff in there, but it's not in front of me
In fact I'd be mildly surprised if Aut(Ell) acted on the sheaf of spectra, but there's no harm asking.
 
Ah, no, probably not.
 

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