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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeJan 21st 2011
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexTo Urs, if you are listening, or anyone else who would like to know: there should be a dense sub-site of $Top_{loc.contr.}$ where the objects are the contractible topological spaces, and covers are open covers as per usual. This should extend the cases of smooth manifolds and topological manifolds. The sub-site on a locally contractible space consisting of the contractible open sets would then be the induced coverage from the big coverage on $Top_{loc.contr}$. It would be interesting to know if this was a small-generated site, but I doubt it (well, I'm not sure). More interesting perhaps is if we focus on the analogous subcategory of CGWH, this would remove a number of potentially pathological examples which drive up the 'size' of the dense sub-site.

   To Urs, if you are listening, or anyone else who would like to know:

   there should be a dense sub-site of $Top_{loc.contr.}$ where the objects are the contractible topological spaces, and covers are open covers as per usual. This should extend the cases of smooth manifolds and topological manifolds. The sub-site on a locally contractible space consisting of the contractible open sets would then be the induced coverage from the big coverage on $Top_{loc.contr}$.

   It would be interesting to know if this was a small-generated site, but I doubt it (well, I’m not sure). More interesting perhaps is if we focus on the analogous subcategory of CGWH, this would remove a number of potentially pathological examples which drive up the ’size’ of the dense sub-site.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 21st 2011
   • (edited Jan 21st 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHi David, I had been thinking about this briefly. Something like this should support a cohesive $\infty$-topos of "locally contractible topological $\infty$-groupoids", somewhat more general than the Euclidean-topological $\infty$-groupoids. I suppose the only thing to take care of is some bound on the site to keep the site small, as you indicate. I won't try to make suggestions about that now, since it is past my bedtime...

   Hi David,

   I had been thinking about this briefly. Something like this should support a cohesive $\infty$-topos of “locally contractible topological $\infty$-groupoids”, somewhat more general than the Euclidean-topological $\infty$-groupoids.

   I suppose the only thing to take care of is some bound on the site to keep the site small, as you indicate. I won’t try to make suggestions about that now, since it is past my bedtime…

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeJan 22nd 2011
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI now know that without some choice of universe or other mechanism to bound the size of the contractible spaces, the site of contractible topological spaces is not small generated, neither is that of contractible CGWH spaces. 1) Any set can be given a topology in which it is a contractible space, and in fact any set is the basis set for a locally convex topological vector space (which is contractible), so we can't even restrict to 'locally locally-convex-vector-space' 2) Any set is the basis set for a Hilbert space (real, say), which is compactly generated and weak Hausdorff, and contractible. Trivial results, I know, but they deserve to be recorded. And before you pre-empt me, I'd put these on the nLab if I wasn't just about to rush off. :)

   I now know that without some choice of universe or other mechanism to bound the size of the contractible spaces, the site of contractible topological spaces is not small generated, neither is that of contractible CGWH spaces.

   1) Any set can be given a topology in which it is a contractible space, and in fact any set is the basis set for a locally convex topological vector space (which is contractible), so we can’t even restrict to ’locally locally-convex-vector-space’

   2) Any set is the basis set for a Hilbert space (real, say), which is compactly generated and weak Hausdorff, and contractible.

   Trivial results, I know, but they deserve to be recorded. And before you pre-empt me, I’d put these on the nLab if I wasn’t just about to rush off. :)

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 22nd 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am wondering what one can say generally about topological vector spaces arising as Ind-objects of "small" objects. I found <a href="http://www.analg.ulg.ac.be/jps/rec/hsbs.pdf">this article</a> here with something related. Can anyone say more? I don't have time now, need to quit. More later.

   I am wondering what one can say generally about topological vector spaces arising as Ind-objects of “small” objects. I found this article here with something related. Can anyone say more?

   I don’t have time now, need to quit. More later.