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In fact, there is a theorem by Danny Stevenson and David Roberts, extended a theorem by John Baez and Danny Stevenson that shows that large classes of principal oo-bundles, even, do have classifying topological spaces in this sense.
Wow! I haven’t thought about this in a long time. I didn’t realise I had my name to such a theorem (not that I mind). Danny told me he’d send some stuff soon, but that was a few weeks ago (I don’t mind - he’s a busy man).
Could anyone (Urs?) tell me what the theorem is? :D
Edit: For historical completeness (and because I forgot to put it in earlier) the quote is from the section “…because they have automorphisms” in the page moduli space.
Could anyone tell me (Urs?) what the theorem is?
Under mild assumptions on your topological 2-group $G$, there is a topological space $\mathcal{B}G$ such that for a sufficiently well behaved topological space $X$, homotopy classes of continuous maps $X \to \mathcal{B}G$ are in bijection with Cech cohomology with values in $\mathbf{B}G$ on $X$.
Proven in that article by John and Danny. Back when they wrote this I kept talking about it with Danny, and he kept mentioning how some of the technical details in the proof relate to your thesis. I forget what exactly.
Under mild assumptions on your topological 2-group G…
oh, yes, I know that one. I’m not sure if my thesis still relates to what Danny thinks it does. We did do a bit of work on it together, but I changed direction a number of times. You’ll have to check with him if my name is still warranted on this result, because I’m not sure what the state of play is now.
You’ll have to check
No David, sorry, I am busy. You’ll check. :-)
Yeah, it’s not a high priority. I didn’t mean to imply that you personally needed to do it. I’ll ask and report back.
But don’t hasten to sell yourself under value. Whatever the state of the art is now, on Danny’s side, your thoughts back then had an impact on this result, and I think it is right that the page mentions this.
(By the way, I suppose we are talking about the page principal infinity-bundle??)
Ah, by the way, I have been vaguely trying to think about the issue here in more abstract terms:
in the language that I have learned to like, the situation here can be described as follows:
we are in an $(\infty,1)$-topos such as that of Lie infinity-groupoids $Sh_{(\infty,1)}(CartSp)$. The key point is that it is a locally contractible (infinity,1)-topos, which means that we have a canonically defined geometric realization functor, namely
$\Pi : Sh_{(\infty,1)}(CartSp) \to \infty Grpd$.
So for $X$ any object (a manifold, say), $G$ any smooth $n$-group and $\mathbf{B}G$ its delooping, we have the nonabelian $G$-cohomology of $X$ as
$H_{smooth}(X,\mathbf{B}G) := \pi_0 \mathbf{H}(X, \mathbf{B}G)$classifiyg smooth $G$-principal $\infty$-bundles.
But now we can apply $\Pi$. This sends the manifold $X$ to its underlying topological space (up to weak homotopy equivalence) and if $\mathbf{B}G$ is given by a simplicial manifold or simplicial diffeological space, then $\Pi$ sends that to the corresponding geometric realizaiton $\Pi (\mathbf{B}G) = \mathcal{B}G$ as a simplicial topological space.
So we get a map
$H_{smooth}(X, \mathbf{B}G) = \pi_0 \mathbf{H}(X,\mathbf{B}G) \stackrel{\pi}{\to} [X, \mathcal{B}G]$which is the image of $\Pi$ on the homotopy categories.
So the question that you, Danny and John looked at is, from this point of view: when is $\pi_0 \Pi$ – the image of $\Pi$ on the homotopy category – full and faithful?
Possibly this is just a weirdly abstract reformulation of the obvious, but I thought maybe if one looked at it from an large enough abstract distance like this, there’d be some general useful things to say about this.
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