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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeApr 13th 2025
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   Author: DavidRoberts
   Format: MarkdownItexIt is a nearly classical fact that for the simplicial manifold $Y^{[\bullet+1]}$ coming from a surjective submersion $Y\to M$, the complex $\Omega^k(Y^{[\bullet+1]})$ is exact. It occurred to me that this should probably be true if we replace this 1-truncated hypercover by a $n$-truncated hypercover, or even a general one. It might even be just exact in a range, rather than exact everywhere, and if so, this would be good to have a reference for, or otherwise get a proof pinned down.

   It is a nearly classical fact that for the simplicial manifold $Y^{[\bullet+1]}$ coming from a surjective submersion $Y\to M$, the complex $\Omega^k(Y^{[\bullet+1]})$ is exact. It occurred to me that this should probably be true if we replace this 1-truncated hypercover by a $n$-truncated hypercover, or even a general one. It might even be just exact in a range, rather than exact everywhere, and if so, this would be good to have a reference for, or otherwise get a proof pinned down.

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 14th 2025
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   Author: Dmitri Pavlov
   Format: MarkdownItexThe hyperdescent property for Ω^k follows from the hypercompleteness of the site of smooth manifolds (see, e.g., Amabel–Debray–Haine) and the ordinary Čech descent property (as shown in Bott–Tu, for example).

   The hyperdescent property for Ω^k follows from the hypercompleteness of the site of smooth manifolds (see, e.g., Amabel–Debray–Haine) and the ordinary Čech descent property (as shown in Bott–Tu, for example).

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeApr 14th 2025
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   Author: DavidRoberts
   Format: MarkdownItex@Dmitri Thanks! I thought it would be something known from abstract properties, and that you would probably know it :-)

   @Dmitri Thanks! I thought it would be something known from abstract properties, and that you would probably know it :-)