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added to Cartan connection
definition
a standard reference
a standard example
Elie, of course.
I didn’t yet. I always mix them up. (Not the concepts, but whose name goes with which).
It is about time to add some abstract axiomatization of Cartan connections in cohesive homotopy theory. (Thanks to David Corfield for pushing me.)
I have now added to Cartan connection a Definition – In terms of smooth moduli stacks of the traditional concept, formulated in terms of lifts of modulating maps $X \to \mathbf{B}G_{conn}$.
This is closely related to the discussion at orbit method and I have added cross-links. (I’d dare say that this relation is something one would never have seen without the stacky formulation in both these entries).
The fully general definition for $\infty$-groups in any cohesive $\infty$-topos is now rather straightforward. I’ll add it once I have thouhgt about it a bit more.
Does this help with Gabriel’s question about whether Cartan geometry fits with the simple idea of gluing together model spaces? I see in Def 2.3 the gluing, but it’s followed by conditions.
Would it be necessary to modify what I had from your claim from here on those slides in Paris?
A Cartan connection, being a connection on a space and not just a space, is
a) first of all a space $X$ that is locally glued from $G/H$;
b) second a $G$-principal connection on $X$ equipped with a reduction to $H$ such that this reduction matches the local $G/H$-geometry.
My claim that you link to refers to a), and I think there is nothing that needs correction. In that discussion we were talking about gluing spaces, and a Cartan connection in particular involves gluing spaces form cosets $G/H$. But Cartan geometry is a bit more, namely also compatible differential data on such a space.
I should tweak the diagram in the entry a bit more to make clearer how the universal lift really arises, since the diagram currently makes it look like we are using that the connection vanishes on the cover, while of course we are using only that the underlying bundle is trivial on the cover.
I have started to make some quick informal notes on what I am really after here, related to globalizing higher super $p$-brane WZW models over sugra spacetimes via higher Cartan connections. This is mostly just to clear up my own thouhgts, here is a pdf
That stronger version of Cartan connection where one requires the manifold to have a cover by $G/H$s is something that one may consider also without connection data around. This must have a standard name in the literature, but I am not sure right now which one:
namely given a $G$-principal bundle on a manifold $X$ equpped with reduction along $H \to G$, then one may ask for a trivializing cover $\{U_i \to X\}$ such that the canonically induced maps $U_i \to G/H$ are equivalences.
What, if any, would be a traditional name for this concept?
I have thought a bit more about the synthetic formalization of Cartan connections. My previous formalization was not accurate enough to the traditional definition. Now I have improved on it. I’d dare say now I got it right, but let’s see.
First I have added this remark on the role of first-order infinitesimal disks in the formulation.
Then I adapted the synthetic definition accordingly.
I have expanded the entry a bit more:
added a decent (I hope) Idea section;
started a “History”-section (which certainly could be further expanded);
added after the central definition a remark that says what this comes down to in the Cech cocycle incarnation and in the Ehresmann incarnation of principal connections and added pointers to the literature for these;
added some more references.
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