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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 10th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI think there are some entries here which could be improved. [[smooth manifold]] talks of manifolds whose transition functions are smooth. So clicking on [[manifold]] one might expect to find out what a transition function is quite quickly. But it adopts the pseudogroup approach, so this takes a time. If this approach is preferred, why not say that a smooth manifold is a $G$-manifold for the specific pseudogroup $G$? Also there are a couple of discussions at [[manifold]], which might be removed if people agree. [[atlas]] need improving too. It's a stub which blasts straight into the highest $(\infty, 1)$-version.

   I think there are some entries here which could be improved.

   smooth manifold talks of manifolds whose transition functions are smooth. So clicking on manifold one might expect to find out what a transition function is quite quickly. But it adopts the pseudogroup approach, so this takes a time. If this approach is preferred, why not say that a smooth manifold is a $G$-manifold for the specific pseudogroup $G$?

   Also there are a couple of discussions at manifold, which might be removed if people agree.

   atlas need improving too. It’s a stub which blasts straight into the highest $(\infty, 1)$-version.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 11th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexHow should one think about the relationship between the pseudogroup approach to [[manifolds]] and the sheaf approach, expressed, e.g., at [[hole argument]]: >what really does characterize the manifold underlying a spacetime is its collection of all probes by the test spaces $\mathbb{R}^n$, i.e. by all morphisms $\mathbb{R}^n \to X$ in Diff and their relation among each other. I know the latter is a broader definition. But how to relate the former as a type of the latter?

   How should one think about the relationship between the pseudogroup approach to manifolds and the sheaf approach, expressed, e.g., at hole argument:

   what really does characterize the manifold underlying a spacetime is its collection of all probes by the test spaces $\mathbb{R}^n$, i.e. by all morphisms $\mathbb{R}^n \to X$ in Diff and their relation among each other.

   I know the latter is a broader definition. But how to relate the former as a type of the latter?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 11th 2013
   • (edited Nov 11th 2013)
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   Author: Urs
   Format: MarkdownItexIn the style of schemes, a manifold is a sheaf with the property that it has an étale cover by prescribed test spaces. This implies that the test spaces are glued with each other via transition functions obtained by comparing the embeddings of any two charts, and this is the "pseudogroup" description.

   In the style of schemes, a manifold is a sheaf with the property that it has an étale cover by prescribed test spaces. This implies that the test spaces are glued with each other via transition functions obtained by comparing the embeddings of any two charts, and this is the “pseudogroup” description.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 11th 2013
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   Author: David_Corfield
   Format: MarkdownItexOK, thanks.

   OK, thanks.