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created nerve theorem
linked to it from homotopy groups in an (infinity,1)-topos, where it had implicitly been mentioned before, but not made explicit.
Apart from stating the theorem, I wrote a section that explains what’s going on from the nPOV. As far as I can see, at least.
For which classes of paracompact spaces is one assured of the existence of a good cover?
Manifolds, I presume, and CW complexes. What else?
The nerve theorem is obviously a wonderful result in view of hoped-for higher-dimensional van Kampen theorems.
For which classes of paracompact spaces is one assured of the existence of a good cover?
I was hoping my local expert on good covers would have helped me with that by now, but he didn’t yet. But let’s try to sort this out.
At least meanwhile I have polished and expanded the proof on my personal web that relates the nerve theorem to the left adjoint of the constant $\infty$-stack functor here.
Just to expand on the case of manifolds for a moment: any paracompact manifold admits a Riemannian metric, and for any point in a Riemannian manifold there is a geodesically convex neighborhood (any two points in the neighborhood are connected by a geodesic in the neighborhood; see for example the remark after lemma 10.3 in Milnor’s Morse Theory, page 59). That should do the trick, since it is immediate that a nonempty intersection of two geodesically convex regions is also geodesically convex, hence contractible.
I’d have to think a bit to convince myself of the case for CW complexes.
Thanks, Todd. I moved that remark to here for the moment. But will have to call it quits now. See you tomorrow!
I added what is hopefully a more or less correct proof that CW complexes admit good covers to good open cover (under Properties).
Thanks, Todd. Very nice!
I added formal Proposition/Proof environments.
For completeness, I also added the statement of the corollary that hence the category of paracompact manifolds and various of its subcategories admit coverages by good open covers.
I added several references to the nerve theorem article.
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