Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
It would good to have some general discussion here, but I don't know enough to get it started. I mean especially the relationships between G-structures, integrability conditions (including those of higher degree), transitive pseudogroups, and Cartan geometry.
Maybe you should open a corresponding stub nLab entry with some keywords and some references and the like, so that I for instance have a chance of knowing what it might be I could contribute.
Ok, so started Cartan geometry, Klein geometry, and G-structure.
This is a topic which has been systematized several decades ago; is there some fundamental new vision recently there ?
No, nothing especially new, unless there was something worth developing about torsoroids. I was just looking around a bit after John's last TWF, and came across some things, which I still don't know how to tie together.
Surely this stuff needs categorifying. I mean looking at a frame bundle, what is a 2-frame bundle? What is a G-2-structure for G a 2-group? I stop there because you all are capable of giving the continuation...! Some has been categorified I think, at least tentatively and in the blog but was there a conclusion. Reduction of structure group also leads to obstructions and cohomological interpretations ....? and what are the analogues of pseudogroups etc.
Yes, Tim, that thought occurred to me. Then I had the sneaking suspicion that Urs had started to do this anyway with his fivebrane(n)- and string(n)- structures as these aren’t Lie groups, yet can be realised as Lie 6-group/Lie 2-groups. But from the discussion at string 2-group, I see that this introduces a distinction irrelevant to all the examples at $G$-structure. Do we have a case of topological 2-groups $G$ about which we might want to say that a space has a $G$-structure?
Also Urs has phrased things in terms of classifying spaces, e.g., $X \to \mathcal{B} Spin(n)$. Could one not categorify that quite easily?
Wait a minute. Kobayashi defines G-structures in Transformation groups in differential geometry for $G$ a Lie subgroup of $GL(n, R)$. That raises two questions.
1) From the Wikipedia article it seems $G$ is allowed to be more generally a Lie group with a map to $GL(n, R)$, so $G = spin(n)$ is fine. Is this generally accepted?
2) Should Urs have written that $spin(n)$ as a topological group gives rise to a spin-structure?
I do not claim to understand this area well, but it seems to me that traditionally one had either a subgroup or an extension so there were obstructions either to getting the transition functions to be in the subgroup or to lift to the `overgroup', but turning the subgroup inclusion into a fibration made the two similar if not the same problems. That however still leaves the integrability condition to handle and there I get a bit lost.
I have a nice example where things aren't smooth so forms etc do not enter but there is a sense in which the same thing seems to be happening, namely Pl-structures. The Hauptvermutung problem was solved by showing first that the fibre of the `obvious' fibration between the classifying spaces (done simplicially) was homotopically a K(C_2,1) and then finding how to construct a manifold where the lifting did not exist using that information. In other words it uses simplicial groups instead of topological or Lie ones.
David,
there are differen concepts called "G-structure". Spin-structure etc. refers to lifts of the structure group of the tangent bundle of a manifold. In other contexts "G-structure" means that the holonomy group of a metric manifold is G, hence that the structure group can be reduced .
We should have an nLab entry explaining this...
2-frame bundle?
presumably given some 2-vector bundle, we could try to take at each point on the base some groupoid that is meant to be something like a basis, and see what the symmetries are. I think that in the ’groupoidification’ programme there was some way of going from a groupoid to a 2-vector space, how about going the other way?
Going down a dimension for a moment, a fin. dim. vector space gives at least a number - its dimension, which is the decategorification of any set of basis elements. This arises in part from the free-forget adjuction between Set and Vect. Is there a way to get a similar adjunction between 2Vect and Gpd? Is the 2-vector space with basis a groupoid $\Gamma$ the functor category $Vect^\Gamma$, in the same way that the vector space with basis $S$ is the function space $k^S$?
Given this, can we talk about a groupoid which is some sense ’linearly independent’? Can we do this with polynomial functors? By checking that the only solution to
$\bigoplus_g V_g \otimes A_g \simeq ?$is trivial (is the trivial rep)? Some of these questions are probably nonsensical :)
Is the 2-vector space with basis a groupoid $\Gamma$ the functor category $Vect^\Gamma$, in the same way that the vector space with basis $S$ is the function space $k^S$?
Yes, that’s how I like to think of ($Bimod =$) $Vect Mod \subset 2 Vect$ as being the sub-2category of 2-vector spaces that admit a basis. (Noticing that $Vect^\Gamma$ is the same as $k[\Gamma]-Mod$ for $k[\Gamma]$ the category algebra of $\Gamma$.)
In this sense the “dimension” of a 2-vector space would be a Morita equivalence class of algebras (algebroids).
And that idea of speaking of a 2-frame bundle of a 2-vector space seems to be a good one. We are this very moment having some discussion about this in Utrecht…
I should maybe clarify: when I write $Vect Mod$ here I mean it in the sense of the bicategory of modules and bimodules in the sense of Vect-enrriched category theory: so objects are Vect-enriched categories = algebroids, which we think of as placeholders for their categories of Vect-valued presheaves = modules over them. Morphisms are bimodules/profunctors between these Vect-enriched categories, that we think of as placeholders for colimit-preserving linear functors between the corresponding presheaf categories.
Urs, I wonder how close your initial guess at a 2-frame bundle will turn out to be.
Morita equivalence class of algebras (algebroids)
or better: some sort of stack of algebras?
Some new comments on an old discussion:
re #4
is there some fundamental new vision recently there ?
Yes: its all about twisted c-structures in $Smooth\infty Grpd$.
re #6
Surely this stuff needs categorifying.
Done. :-)
I mean looking at a frame bundle, what is a 2-frame bundle?
The frame bundle construction is just a way to get the principal bundle to which the tangent bundle is associated. It helps here to forget vector bundles entirely for the time being and just look at everything in terms of the underlying principal bundles.
Then for instance an example of a 2-frame bundle is the $String$-2-bundle that lifts the ordinary frame bundle over a manifold with String-structure.
What is a G-2-structure for G a 2-group?
For $G \to K$ any morphism of smooth $\infty$-groups, where we might consider $K = GL(n)$ for a give $n$-dimensional manofold $X$, a $G$-structure on $T X$ is an object in the 2-groupoid given as the homotopy pullback
$\mathbf{H}(X, \mathbf{B}G) \times_{\mathbf{H}(X, \mathbf{B}K)} \{T X\} \,.$More is at twisted differential c-structure and the links given there.
1 to 16 of 16