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.

]]>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??)

]]>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.

]]>You’ll have to check

No David, sorry, I am busy. You’ll check. :-)

]]>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.

]]>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.

]]>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.

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