Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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.
added hyperlink to
Added more explicit pointer to McCord 67, Thm. 2.
also to the page number in Weil 52, p. 141
I fixed the section number in a reference to Weil’s paper that I added once: it’s Section 5, not 6. Weil also has a relative version formulated as a Corollary: if two maps f,g send every x∈X to some U_i⊂Y, then f and g are homotopic.
1 to 12 of 12