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at vector bundle I have spelled out the proof that for $X$ paracompact Hausdorff then the restrictions of vector bundles over $X \times [0,1]$ to $X \times \{0\}$ and $X \times \{1\}$ are isomorphic.
It’s just following Hatcher, but I wanted to give full detail to the argument of what is now this lemma.
You didn’t want to give the argument for numerable bundles on arbitrary spaces? The proof is exactly the same.
Sorry, which statement are your referring to? The only place where I assumed something extra, namely paracompact Hausdorff, is this prop. For the proof of that I seem to need a partition of unity, no?
The statement in #1 generalises from arbitrary vector bundles on paracompact spaces to numerable bundles on arbitrary spaces. The proof uses the so-called stacking lemma, which is in Dold’s Algebraic Topology, section A.2. Numerable bundles are those that trivialise over an open cover with a subordinate partition of unity (so all bundles when on a paracompact space).
I see. So I was headed for the discussion of the classifying space, where I need all bundles. But feel invited to add this remark.
If you do so, notice that I am splitting off the material on toopological vector bundles from the main entry “vector bundle” to topological vector bundle. More on this in the next comment.
I have expanded the Idea-section at vector bundle a fair bit.
Then, in view of the recent disussion with Todd, I am splitting off an entry topological vector bunde for discussion of the standard topological stuff (no sheaf semantics etc.).
I moved over the corresponding material on Definition and properties. Then I polished the definition material at topological vector bundle, or at least two thirds of it. Need to interrupt for a moment.
Is there anywhere in QFT where vector bundles over some space $M$ which are Frobenius objects have some particular interpretation? I guess one could consider a non-extended 2d TQFT with target category the vector bundles over $M$, but I wonder if these have a more natural interpretation anywhere else.
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