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That's true, yes.
fine. so I guess that stackification [,,,]
Yes, that's rigtht.
what I like of this path groupoids point of view is that one talks of connections without differenial forms.
Yes!
What is curvature?
But we said this before: curvature for a transport is some transport
But I'm still unable to see this directly. And, most of it, I'm still unable to see a priori that curvature is a globally defined 2-form, despite a connection is so only locally.
Okay, so what you need is the full story of how one derives this.
Let's do the simpler case first, where the coefficient object is a group object. Recall the abstract definition of "abstract de Rham cohomology" not in terms of differential forms, but just in terms of flat oo-bundles whose underlying oo-bundle is trivial. (See the diagrams there.)
Then it is a theorem that we have a fibration sequence
.
This allows us to define non-flat differential cohomology with coefficients in A to be the homotopy fiber of over the given curvature class. This curvature class is by definition of globally defined. Have a look at the link for the case of groupal coefficients.
and, more in general to a functor \Pi(X)\to \mathbf{B} INN(G): it is a flat \mathbf{B} INN(G)-connection!
Yes, that's the idea, exactly.
There are very close connections with reduction of structure group. In fact in non_abelian cohomology where the construction INN(G) sort of originates is is usual (following Dedecker) to use a 2-group as coefficients or rather the corresponding crossed module in that setting. I have been trying to look at Turaev's homotopy quantum field theory from this point of view, but I find it quite hard going. In algebraic geometry some useful ideas have been emerging with work by Aldrovandi and Noohi on Butterflies (which in n-lab speak are ......? I leave someone else to say because I always choose the wrong term!) Their stuff would seem to be connected with what you are talking about.
Butterflies (which in n-lab speak are ......?
a butterfly is a butterfly is a butterfly... :-)
Butterflies are just a way to present morphisms in between one-object 2-groupoids.
Pradines certainly used the term but I am not sure it is in exactly the same context.
I did not know that there was an entry on butterflies!
There is a second part of Butterflies: ArXiv 0910.1818. I have added a link to the entry.
<blockquote><bockquote >I was only looking at 2-connected covers, originally. </blockquote><br /><br />this is interesting, since it perfectly agrees with the geometric picture I was trying to describe in the posts above: functors <latex >\mathcal{P}_1(X)\to \mathbf{B}G</latex> lift to functors <latex >\mathcal{P}_2(X)\to Something</latex>, and this something one has to expect to be the 2-connected cover of <latex >\mathbf{B}G</latex>. <br />it is precisely what I was trying to mean by saying I didn't like the appearance of <latex >INN(G)</latex>, and would have liked a more geometric description on the first spot, which then one could prove to be explicitly given by the INN construction.<br /><br />thanks a lot :-)</bockquote>
There is a second part of Butterflies: ArXiv 0910.1818. I have added a link to the entry.
Thanks. I added a remark on terminology to this at butterfly and then added the reference also to principal 2-bundle.
By the way: somebody wrote at principal 2-bundle that while that concept is more general than that of gerbe, that of gerbe could "easily be generalized to match".
I am not so sure about this. If you start generalizing the simple statement "locally non-empty and connected stack" such that it describes sections of general 2-bundles, you have to say things that effectively make you say "principal 2-bundle" and no longer say "gerbe".
So I expect a general 2-bundle to be a gerbe over some suitable cover, like the total space of a principal bundle.
Indeed. There is described in the second section at string structure. In my last article with Stasheff and sati we call this "local semi-trivialization": instead of pzulling back the 2-bundle to its own total space where it trivializes, one pulls it back just to a covering spaces, where it jusgt b ecomes a bit simpler.
At least in my set-up, it's a bit simpler:
Even simpler?!
Can you write into some entry precisely what you have in mind?
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