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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeJul 4th 2013
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   Author: TobyBartels
   Format: MarkdownItexTreating [[locally path-connected spaces]] as [[nice topological spaces]], we see that nice [[path-connected spaces]] are the same as nice [[connected spaces]], and the definition of the latter is more elementary (in point-set topology) than the former. Then nice [[simply connected spaces]] are the same as nice [unicoherent spaces](https://en.wikipedia.org/wiki/Unicoherent_space), which are again more elementary. This should continue for the entire hierarchy of $n$-[[n-connected space|connected spaces]], so I wrote something there about it.

   Treating locally path-connected spaces as nice topological spaces, we see that nice path-connected spaces are the same as nice connected spaces, and the definition of the latter is more elementary (in point-set topology) than the former. Then nice simply connected spaces are the same as nice unicoherent spaces, which are again more elementary. This should continue for the entire hierarchy of $n$-connected spaces, so I wrote something there about it.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJul 5th 2013
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   Author: Mike Shulman
   Format: MarkdownItexSounds like the Cech-y definition of the fundamental $\infty$-groupoid.

   Sounds like the Cech-y definition of the fundamental $\infty$-groupoid.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 5th 2013
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   Author: Urs
   Format: MarkdownItexhence the _[[nerve theorem]]_.

   hence the nerve theorem.

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeJul 5th 2013
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   Author: TobyBartels
   Format: MarkdownItexYes, that certainly seems to be related, although I don't know how to make it match up precisely.

   Yes, that certainly seems to be related, although I don’t know how to make it match up precisely.

5. • CommentRowNumber5.
   • CommentAuthorTobyBartels
   • CommentTimeJul 6th 2013
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   Author: TobyBartels
   Format: MarkdownItexI have removed that section as wrong. I was pretty sure that I had read about an elementary point-set equivalent of simple connectedness for nice topological spaces, and I found unicoherence, which seemed reasonable, but it doesn't work. (Specifically, the projective plane is unicoherent but not simply connected.) I can keep searching, or maybe I hallucinated this idea, or maybe we should extract something correct from the nerve theorem. But I don't have a correct replacement for now.

   I have removed that section as wrong. I was pretty sure that I had read about an elementary point-set equivalent of simple connectedness for nice topological spaces, and I found unicoherence, which seemed reasonable, but it doesn’t work. (Specifically, the projective plane is unicoherent but not simply connected.) I can keep searching, or maybe I hallucinated this idea, or maybe we should extract something correct from the nerve theorem. But I don’t have a correct replacement for now.