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Let G be a Lie group and let BG be the smooth stack of principal G-bundles. Then we have a natural equivalence G≃ΩBG. If we apply the internal hom [S1,−] to the homotopy pullback diagram defining ΩBG (and if internal homs preserve homotopy pullbacks, something I need be reassured about) then we get a homotopy pullback diagram realizing the free loop space object ℒG=[S1,G] as the based loop space of ℒBG=[S1,BG]. On the other hand, since G is a smooth group, then so is ℒG and we can consider the smooth stack BℒG, whose based loop space is again ℒG.
What is the relation between BℒG and ℒBG? (if any)
From what I read in the lines after equation (3.2) in Konrad Waldorf’s Spin structures on loop spaces that characterize string manifolds I guess that for a connected G the two stacks BℒG and ℒBG should actually be equivalent, but I don’t clearly see this yet.
I guess simple-connectedness would play a role here, no?
Could be: Konrad Waldorf writes “Since Spin(n) is connected, LFM is a principal LSpin(n)-bundle over LM, see etc. etc.”, which is the sentence I derived my guess from (here M is a spin manifold, FM is its frame bundle (i.e., the total space of the associated principal Spin-bundle) and L stands for “loop space”). But since Spin(n) is actually simply connected (Konrad work with high enough n), it may well be that the sentence should actually have read “Since Spin(n) is simply connected…”.
I don’t know about the stacky-part, but in terms of Lie groups and their classifying spaces then L and B commute. Indeed, Map(X,−) and B commute. For a Lie group G, we have the fibration sequence G→EG→BG with EG contractible. Then this maps to Map(X,G)→Map(X,EG)→Map(X,BG) which is still a fibration sequence. Moreover, the contraction of EG maps to a contraction of Map(X,EG) whence the base of the fibration is BMap(X,G). This argument works with C∞-maps or with CW-complexes or whatever you require. There’s no requirement of connectivity or simple connectivity.
Hmm, and we should use the fact we can take EG to be a group, so that Map(X,EG)/Map(X,G) is perfectly well-defined as a homogeneous space and is a model for BMap(X,G). And shouldn’t this be just Map(X,BG)?
You don’t need it to be a group, just a principle LG-bundle.
Hi,
I guess the connectivity assumption is needed in order for Map(S1,BG) to be connected. If not, then it is only the conneceted component of the basepoint to be a BMap(S1,G). But connected should be enough, with no need of simple connectedness.
Yes, that the two stacks (∞-stacks) are equivalent follows directly from [S1,−] preserving all ∞-limits, which gives
Ω[S1,BG]≃[S1,ΩBG]≃[S1,G]and hence by definition of B as the inverse equivalence in
Grp(H)Ω←≃→BH*/≥1we have
[S1,BG]≃B[S1,G]as soon as [S1,BG] is connected, which it is precisely if for all n∈ℕ there is an essentially unique G-principal infinity-bundle on the circle times the germ of an n-disk .
(I am including some links just for the sake of eventual bystanders.)
I’ve thought about this some more, and the connectivity assumptions are there to ensure that Map(X,BG) is connected since it is the identity component of this which is BMap(X,G).
The reason being that a point of BMap(X,G) must have a lift to EMap(X,G)=Map(X,EG). Now a point of Map(X,BG) classifies a G-bundle over X and a lift of this to EG is a section, whence a trivialisation. Thus a point of Map(X,BG) is in BMap(X,G) if (and only if) the associated bundle is trivial, which occurs if and only if it is in the identity component of Map(X,BG).
Thus the general statement is that:
BMap(X,G)=Map0(X,BG)1 to 9 of 9