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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeJan 30th 2012
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexHi Andrew, (Copying text of email, for reference) This is just to continue our discussion on MO about path spaces, <http://mathoverflow.net/questions/85960/induced-map-on-path-manifolds-is-it-a-submersion> specifically the problem of the spaces of open paths. The reason I want to do this is I want to show that the set of (smooth) functors $$C(U)\to X$$ is a Frechet Lie groupoid, for $X$ a Lie groupoid, and $C(U)$ the Cech groupoid associated to a finite open cover $U$ of $S^1$. This cover is given by open intervals $I_1, \ldots I_n$. This set of functors is given by an iterated pullback $$Hom(I_1,X_0) \times_{Hom(I_{12},X_0)} Hom(I_{12},X_1) \times_{Hom(I_{12},X_0)} Hom(I_2,X_1) \times\ldots$$ where $I_{jk} = I_k \cap I_k$ and so on the face of it, we need to define smooth manifold structures on these sets of open paths. However, I believe these open paths extend, in a neighbourhood of each open endpoint, to a smooth map from a closed interval (perhaps after embedding the chart near the endpoint into $\mathbb{R}^n$). This is what you indicated is needed to define the smooth structure, so I'm happy that that part seems to work. Then we need to know that the maps involved in the above pullback are submersions, which was the content of my question. I think this should be true, but you indicated some issues. When you said you'd 'need some time to work out the details', does that mean you will work out the details? (Please don't take that as being nosey or demanding, I'm just checking and I can't think how to write it in a diplomatic way). If you like, give me a hint and I will think about it also. -- All this links with the purpose behind my MO question <http://mathoverflow.net/questions/79256/ind-frechet-manifolds> where I think I want to consider the disjoint union (yes!) of Frechet Lie groupoids with index set the set of (certain) finite subsets of the dyadic rationals in $[0,1]$. Such finite subsets ('partitions') encode the finite covers of $S^1$ as above, and those are the Frechet Lie groupoids I want to glom together to form a 'loop space' of the Lie groupoid. Actually the set above is the manifold of objects of a Frechet Lie groupoid (hypothetically), the manifold of arrows is formed in an entirely analogous way. I think I can legitimately take the disjoint union, because the glue that holds nearby partitions together is provided by the manifold of arrows rather than the manifold structure on the set of objects. One positive side to this approach rather than the approach I last emailed you about is that we should recover (up to equivalence) the standard manifold of loops of a manifold, should the Lie groupoid in question actually be a manifold.

   Hi Andrew, (Copying text of email, for reference)

   This is just to continue our discussion on MO about path spaces,

   http://mathoverflow.net/questions/85960/induced-map-on-path-manifolds-is-it-a-submersion

   specifically the problem of the spaces of open paths.

   The reason I want to do this is I want to show that the set of (smooth) functors

   $$C(U)\to X$$

   is a Frechet Lie groupoid, for $X$ a Lie groupoid, and $C(U)$ the Cech groupoid associated to a finite open cover $U$ of $S^1$. This cover is given by open intervals $I_1, \ldots I_n$.

   This set of functors is given by an iterated pullback

   $$Hom(I_1,X_0) \times_{Hom(I_{12},X_0)} Hom(I_{12},X_1) \times_{Hom(I_{12},X_0)} Hom(I_2,X_1) \times\ldots$$

   where $I_{jk} = I_k \cap I_k$ and so on the face of it, we need to define smooth manifold structures on these sets of open paths.

   However, I believe these open paths extend, in a neighbourhood of each open endpoint, to a smooth map from a closed interval (perhaps after embedding the chart near the endpoint into $\mathbb{R}^n$). This is what you indicated is needed to define the smooth structure, so I’m happy that that part seems to work.

   Then we need to know that the maps involved in the above pullback are submersions, which was the content of my question. I think this should be true, but you indicated some issues. When you said you’d ’need some time to work out the details’, does that mean you will work out the details? (Please don’t take that as being nosey or demanding, I’m just checking and I can’t think how to write it in a diplomatic way). If you like, give me a hint and I will think about it also.

   –

   All this links with the purpose behind my MO question

   http://mathoverflow.net/questions/79256/ind-frechet-manifolds

   where I think I want to consider the disjoint union (yes!) of Frechet Lie groupoids with index set the set of (certain) finite subsets of the dyadic rationals in $[0,1]$. Such finite subsets (’partitions’) encode the finite covers of $S^1$ as above, and those are the Frechet Lie groupoids I want to glom together to form a ’loop space’ of the Lie groupoid. Actually the set above is the manifold of objects of a Frechet Lie groupoid (hypothetically), the manifold of arrows is formed in an entirely analogous way.

   I think I can legitimately take the disjoint union, because the glue that holds nearby partitions together is provided by the manifold of arrows rather than the manifold structure on the set of objects.

   One positive side to this approach rather than the approach I last emailed you about is that we should recover (up to equivalence) the standard manifold of loops of a manifold, should the Lie groupoid in question actually be a manifold.

2. • CommentRowNumber2.
   • CommentAuthorAndrew Stacey
   • CommentTimeJan 30th 2012
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexCross-posted!

   Cross-posted!