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Stream: theory: algebraic topology

Topic: complexes of bundles

Tim Hosgood (Oct 18 2020 at 12:27):

There is a well known equivalence between $\mathrm{GL}$-principal bundles and locally free sheaves. Lots of people are very interested in (co)chain complexes of the latter, but google shows zero results for "complex of (G-)principal bundles". Do people not study complexes of principal bundles?

Todd Trimble (Oct 18 2020 at 14:47):

I can believe that there is an equivalence between the groupoid of locally free sheaves and the groupoid of $GL$-principal bundles, but not between the corresponding categories. Or do I misunderstand?

Reid Barton (Oct 18 2020 at 14:48):

Yes, I think this is right.

Reid Barton (Oct 18 2020 at 14:50):

Also, I think the original statement is lacking an $n$ in a couple places, but maybe there's a version of it which doesn't need one (involving a principal bundle for a groupoid?)

John Baez (Oct 18 2020 at 17:14):

What would a complex of principal bundles even be? A chain complex is a bunch of objects with maps $d_i : C_i \to C_{i+1}$ s.t. that $d_{i+1} d_i = 0$. For this "0" to make sense we at least need the hom-sets to be pointed sets, though typically go much further and demand that our category be enriched over abelian groups. The category of $G$-principal bundles for fixed $G$ doesn't have these properties, so $d_{i+1} d_i = 0$ doesn't make sense for these.

John Baez (Oct 18 2020 at 17:16):

As Todd+Reid pointed out, the "equivalence between locally free sheaves and GL-principal bundles" is really an equivalence between the groupoid of locally free sheaves of vector spaces of rank n and the groupoid of GL(n)-principal bundles.

John Baez (Oct 18 2020 at 17:19):

In neither of these groupoids do chain complexes make sense.

John Baez (Oct 18 2020 at 17:19):

Of course we can formally get rid of the dependence on n by taking the "union" (coproduct) of all these groupoids, but that doesn't help make sense of chain complexes.

John Baez (Oct 18 2020 at 17:20):

We can define chain complexes in the category of locally free sheaves of vector spaces.

John Baez (Oct 18 2020 at 17:42):

In much less fancy term: chain complexes live in the world of linear algebra, but principal bundles don't.

Tim Hosgood (Oct 18 2020 at 17:57):

hm, I must admit, I've only ever thought of (locally free sheaves <-> principal bundles) as a naïve algebraic geometer, and just how to turn one object into the other — I'd never stopped to think whether or not this association gave something on the level of morphisms

Tim Hosgood (Oct 18 2020 at 17:58):

so is there no functor from the category of locally free sheaves (of some fixed rank $n$) to the category of $\mathrm{GL}_n$-principal bundles such that it acts on objects by sending a locally free sheaf to the associated principal bundle?

Tim Hosgood (Oct 18 2020 at 17:58):

(or vice versa, I guess)

John Baez (Oct 18 2020 at 17:59):

What "category of $\mathrm{GL}_n$-principal bundles" are you talking about? There really aren't interesting morphisms between principal G-bundles other than isomorphisms... in general.

John Baez (Oct 18 2020 at 18:01):

There's a functor from the groupoid of principal $\mathrm{GL}_n$-bundles to the groupoid of rank-n locally free sheaves, and this is an equivalence.

John Baez (Oct 18 2020 at 18:02):

Then there's an inclusion of the groupoid of rank-n locally free sheaves into the category of rank-n locally free sheaves.

John Baez (Oct 18 2020 at 18:05):

So you can create a category of principal $\mathrm{GL}_n$-bundles by taking the groupoid of these and throwing in extra morphisms that are secretly morphisms between their associated rank-n locally free sheaves. You get a category equivalent to the category of rank-n locally free sheaves.

But this is a rather roundabout way of saying "forget principal $\mathrm{GL}_n$-bundles: I'd rather work with rank-n locally free sheaves".

Tim Hosgood (Oct 18 2020 at 20:12):

John Baez said:

What "category of $\mathrm{GL}_n$-principal bundles" are you talking about? There really aren't interesting morphisms between principal G-bundles other than isomorphisms... in general.

this explains why I've never read anything about morphisms between principal bundles then! you've piqued my interest with your "... in general" though — what sort of secrets are you hiding there? :wink:

John Baez (Oct 18 2020 at 20:14):

Well, I later explained a way you can "cook up" a concept of morphism between principal $\mathrm{GL}_n$ bundles.

Tim Hosgood (Oct 18 2020 at 20:17):

OK, yes, and this seems quite nice to me, but I get that you might as well just work with locally free sheaves at this point... but I'm still interested in this category anyway! is it studied somewhere at all?

John Baez (Oct 18 2020 at 20:20):

Btw, the automorphisms of a principal G-bundle can still be interesting. For example consider the trivial $\mathrm{U}(1)$-bundle on the circle, or the trivial $\mathbb{C}^\ast$ bundle if you prefer. I think this has some automorphisms that aren't homotopic to the identity...

John Baez (Oct 18 2020 at 20:22):

Tim Hosgood said:

OK, yes, and this seems quite nice to me, but I get that you might as well just work with locally free sheaves at this point... but I'm still interested in this category anyway! is it studied somewhere at all?

The category of locally free sheaves has been studied a lot. I doubt anyone has bothered to study this other equivalent category - I just made it up to please you. But all "non-evil" results about the category of locally free sheaves results will automatically transfer to this equivalent category.

Simon Pepin Lehalleur (Dec 27 2020 at 20:13):

There is a Tannakian interpretation of principal bundles (and of connections and Higgs fields on them) as strict exact faithful tensor functors from Rep(G) to Vect(X) due to Nori and Simpson, which is kind of related to this discussion. I am on my phone so I can't really go into details, and I will just refer to section 9 of Simpson's "Moduli of representations of the fundamental group of a smooth projective variety II" (http://www.numdam.org/article/PMIHES_1994__80__5_0.pdf)

Simon Pepin Lehalleur (Dec 27 2020 at 21:43):

It's an old discussion so maybe you won't see this? I will tag just in case: @Tim Hosgood @John Baez

John Baez (Dec 27 2020 at 21:47):

I saw it, thanks.

Tim Hosgood (Dec 27 2020 at 21:47):

oh, that sounds interesting, I’ll check it out thanks