Talk:Nerve complex

I rather dispute the 'clarity' of using more mathematical symbols here. In fact this really has served only to compress things (and add symbols that can't be read by my browser). Substituting symbols for words msut be used with care.

Charles Matthews 13:02, 10 Apr 2004 (UTC)

homotopy equivalence[edit]

Is it true that if all intersections are contractible then nerve is homotopy equivalent to original space? Tosha 17:40, 13 July 2006 (UTC)[reply]

Yes, it seems so. It's mentioned at [[1]], there without proof. -- Jakob.scholbach 06:01, 4 November 2006 (UTC) A proof is given in Hatchers Book on algebraic topology, chapter 4G. -- Jakob.scholbach 17:50, 6 November 2006 (UTC)[reply]

The homotopy equivalence result is not mentioned in the article. Instead, an example is given which implies that the geometric realization of the nerve is homeomorphic to the original space when the mentioned contractible conditions are met. It would be useful to mention the homotopy equivalence as well as which further assumptions are needed to ensure the stronger conditions of homeomorphic equivalence, and give references for proofs and details. Yasmar (talk) 22:12, 27 July 2009 (UTC)[reply]

I agree with Yasmar. The article should take some pains differentiate homeomorphic, homotopy equivalent, weakly homotopy equivalent. It seems most natural to me from what I've read that the nerve should be a simplicial complex; there are a variety of distinct ways that one can turn a simplicial complex into a simplicial set, each with their own subtleties.