@Slereah Dunno, but Immanuel Kant claimed that it's a priori knowledge that space has 3 dimensions. Sorry, I can't give a good ref for that, I guess it's somewhere in Critique of Pure Reason (1781), but here's a recent (paywalled) article on the topic: jstor.org/stable/24704497

Around that time, there was some progress being made towards non-Euclidean geometry, but Kant published 5 decades before Lobachevsky & Bolyai published their works on hyperbolic geometry.

I wonder what impact it would've had on Kant's philosophy of space if he knew about hyperbolic geometry...

Is there a categorical idea for how one goes from a local description to a global description

Like going from a vector field to a diffeomorphism, or a PDE to a solution of that PDE, or a distribution to a foliation

How does one generally integrate objects

@Slereah isn't that just stating that the functor from the category of the global description (e.g. Lie groups) to the category of the local description (e.g. Lie algebras) has certain properties?

@ACuriousMind I mean it's category theory, so most probably it's about a functor

But that doesn't narrow it down much

like, look at how nlab phrases Lie's theorems

An "obvious" functor

I'm trying to find of such a description to go from the connection form to the exponential map, basically

The exponential map is basically integrating the geodesic spray, I guess

Which depends on the connection form

from what connection form to what exponential map?

Any affine connection, in this case

sure, but what has any exponential map to do with the connection

Aren't they related?

nlab claims that one of the property of the exponential map is basically $\nabla \exp_p = \exp_p$

I mean the exponential map traces the geodesics of the connections

but I suspect you're talking about a different exponential map than the one I'm thinking about, that's why I'm asking you what exponential map you mean

the one of the normal neighbourhood

I want to say "not the Lie group one" but then again they're mildly related

Also from what I can tell in a related sector, there is the Wario's equivalent evil twin of differential geometry called integral geometry

Which doesn't get discussed as much

@Slereah Okay, so what's wrong with saying that $\mathrm{exp}$ is defined by $\nabla \mathrm{exp} = \mathrm{exp}$ and $\mathrm{exp}(0) = 1$?

Well the whole going from local objects to global objects happens a lot in differential geometry

I am wondering if there are more general ideas on the topic

I'm not so sure this is a case of going from local to global - you get an $\mathrm{exp}_p$ at every point, you're just turning the global data of the connection form into equally global data in form of a collection of maps $\mathrm{exp}_p$

this isn't like the Lie group case where the exponential map at one point essentially determines the structure of the whole group

The only global-local passage here is that for any single $\mathrm{exp}_p$, it turns local data at $p$ into paths in its neighbourhood, where I'm not sure whether "paths in a neighbourhood" should be seen as local or global

it is a tricky one

The reason it bothered me is that the log map seems even less local, since for two separate points, you get a vector at one of them

The real local/global dichotomy for a connection is going between the description in trivializing patches and the description on the bundle as a whole

@Slereah but couldn't you equally well define that one to give you vectors at both points :P

I vaguely recall that sometimes people use three levels of things

The level of the whole manifold, a neighbourhood or a single point

@ACuriousMind Well sure, you just get the parallelly propagated vector going the other way

but then using the parallel propagator is pretty non-local

It's literally integrated!

I'd really say that a connection is always global data, but a large part of the theory of connections can be done with the local data in a single coordinate patch

I'm not convinced trying to distinguish between "more local" or "more global" data for the various ways in which you can present the connection data in such a patch is a good way to approach this

The only thing I can think of that's kind of what I'm trying to grasp at is the De Rham theorem I guess

the integration of forms over simplexes

Maybe that can be related here, idk

Curves are just 1-simplex after all

From the general vibe I am getting maybe Cech cohomologies are involved

Cech cohomology with values in $G$ is essentially counting how many different $G$ bundles you can build - the expression of a bundle in local patches essentially defines Cech cocycles, and two local expressions define isomorphic bundles when they are cohomologous in any refinement

sometimes science is more art than science, unfortunately

And I must rely on gut feeling for a while

I just feel the need to drop in and say that I just had chocolate and pistachio gelato and that no other gelato I've ever had comes close

ew, pistachios :P

could be worse

@Mithrandir24601 pistachio is in my mind essentially the ideal ice cream flavour. Pity you can't easily (if at all) find a decent one is non-gelato countries =)

@glS I tasted it in the shop (cafe) and was completely like 'ohhh wow, this is amazing!', asked for a scoop of that and chocolate, then when I got home and had them both together, was just totally and utterly blown away. Just sublime

I've had gelato in NI fairly recently and it's nice yes, but it's not the same as what the Italian cafe in Bristol does

definitely not the same, I can imagine

@Mithrandir24601 if you want to try the real thing, there's going to be a quantum info conference in Palermo in September, is all I'm saying =)

well "levers are wrong" is certainly a new one for me

^^ please mods close and delete this nonsense which has nothing to do with the meta site.