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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 24th 2017
   • (edited May 24th 2017)
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   Author: Urs
   Format: MarkdownItexat _[[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](https://ncatlab.org/nlab/show/vector+bundle#CoverForProductSpaceWithIntrval).

   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.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 24th 2017
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   Author: DavidRoberts
   Format: MarkdownItexYou didn't want to give the argument for numerable bundles on arbitrary spaces? The proof is exactly the same.

   You didn’t want to give the argument for numerable bundles on arbitrary spaces? The proof is exactly the same.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 24th 2017
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   Author: Urs
   Format: MarkdownItexSorry, which statement are your referring to? The only place where I assumed something extra, namely paracompact Hausdorff, is [this prop](https://ncatlab.org/nlab/show/vector%20bundle#ConcondanceOfTopologicalVectorBundles). For the proof of that I seem to need a partition of unity, no?

   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?

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 25th 2017
   • (edited May 25th 2017)
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   Author: DavidRoberts
   Format: MarkdownItexThe 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]( https://books.google.com.au/books?id=PxrCQAAQBAJ&pg=PA361&lpg=PA361&dq=stacking+lemma+algebraic+topology+dold&source=bl&ots=wEqpYlAKkd&sig=5ETm3PbSHVRIcWb1YSJpFz1FDCY&hl=en&sa=X&ved=0ahUKEwigjO-V6InUAhWHsJQKHStOA3EQ6AEICjAA#v=onepage&q=stacking%20lemma%20algebraic%20topology%20dold&f=false). Numerable bundles are those that trivialise over an open cover with a subordinate partition of unity (so all bundles when on a paracompact space).

   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).

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 25th 2017
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   Author: Urs
   Format: MarkdownItexI 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 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.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMay 25th 2017
   • (edited May 25th 2017)
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   Author: Urs
   Format: MarkdownItexI 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.

   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.

7. • CommentRowNumber7.
   • CommentAuthorperezl.alonso
   • CommentTimeMar 20th 2024
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   Author: perezl.alonso
   Format: MarkdownItexIs 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.

   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.