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made more explicit (here) the back-link to topological vector bundle for classical concordance for topological vector bundles
added pointer to:
added pointer to
and cross-linked with shape via cohesive path ∞-groupoid
In this newly added proposition, why is the map X→X⨯[0,1] a relative cell complex? (Claimed in Item 1.)
I see how to prove it when X is cofibrant, but how do you prove it for a general topological space X?
True, thanks for catching.
This is being undecided whether to lift using the Serre or Hurewicz. Either to add the condition that the bundle be numerable, or else that $X$ be CW.
Will edit. But to get some dinner first now…
Where the proof (that concordance classes of principal bundles are iso classes) invokes the identification of principal bundle morphisms with global certain global sections, I have added (here) a diagram that shows this correspondence.
In the given case, it is easy to see this correspondence by analytic inspection: It’s trivial on underlying point sets, and continuity is readily verified.
But I would like to state this more abstractly, so that the argument can be internalized into other categories.
In the diagram that I have added (here) the step from the left to the right is just the universal property of quotient.
But from right to left one needs to pull back (along the dashed morphism) and use some preservation of coequalizer diagrams under pullback. The dashed map should be a closed embedding, hence a regular mono (in $cgHaus$), so that the pullback along it should preserve quotients. Something like this.
I have now added (here) a more general-abstract argument for the (superficially trivial-seeming) lemma that isomorphisms of principal bundles are in bijection to sections of their quotiented fiber product.
This needs regularity of the ambient category, and so I have now added the assumption to the ambient Proposition that we work in compactly generated weak Hausdorff spaces.
The point of the exercise is for the argument (that concordance classes are iso-classes) to work internally in more generality. For example, I think it follows verbatim for equivariant bundles, now by lifting using the fine model structure on topological G-spaces of the classical model structure.
Will type that out next, but first some breakfast.
[ edit: full details are a little lengthy – I have typed them out in a local pdf, but not included in the nLab page ]
Continuing in the spirit of looking at the lifting problem here internal to other ambient categories, the variant of the statement for smooth/diffeological bundles might be a fun application of the model structure(s) on diffeological spaces.
But does it work?
By Prop. 4.28, Prop. 4.30 in Christensen & Wu 2013 every diffeological principal bundle is a fibration in their sense, and, by their Prop. 4.26, their fibrations are stable under pullback, which implies that the right vertical morphism in the lifting diagram here would a be fibration in their sense.
But I don’t see that they have results which could be quoted to see that the left vertical map is an acyclic cofibration; and even if it is they may not provide the implication that the lifting exists, since they don’t establish a full model structure. (Of course they don’t claim to, I am just scanning to see what could be used here.)
Next I am trying to see if Kihara’s model structure could be put to use here. But from arXiv:1702.04070 it looks like it is/was open which smooth manifolds are cofibrant in his model structure.
So this may be fiddly. But if anyone knows/sees whether the lifting problem here works in diffeological spaces and for diffeological principal bundles, drop a note.
Re #10: Both questions have a positive answer.
This is something I intend to write up this winter (in December, I hope).
The main points that I plan to include: the projective model structure exists, the map X→X⨯[0,1] is a cofibration if X is cofibrant, and smooth manifolds are cofibrant.
Thanks for the remark. That sounds good!
(Now I remember we may have talked before about you working on such a model structure, but I may have forgotten what you may have said there.)
To just go on in this wishful direction for a moment:
If next we had a proper-equivariant version of the model structure on diffeological spaces (i.e. on presheaves of diffeological spaces over the orbit category of a, let’s say, finite group $G$), such that for $X$ a smooth $G$-manifold the morphism $X \to X \times [0,1]$ were still a cofibration, then we’d know that also for (properly equivariantly locally trivial) diffeological equivariant principal bundles their concordance classes coincide with their isomorphism classes. Which would then imply the classification theorem for smooth equivariant principal bundles…
(This kind of result is the first real application of the lifting properties in would-be model structures on diffeological spaces that I am running into. It’s interesting.)
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