Timeline for Alternative model structure on retractive spaces
Current License: CC BY-SA 3.0
10 events
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| Aug 23, 2013 at 2:42 | comment | added | John Klein | Peter, you were right: I found a reference. Theorem 5.3 in the paper of Intermont and Johnson gives the model structure of the kind you mentioned (weak equivalence = weak homotopy equivalence after applying global sections, a fibration = Serre fibration after applying global sections). They also show that if we use a local sections along all opens to define the weak equivalences/fibrations, then that also forms a model category as well. | |
| Aug 22, 2013 at 19:30 | comment | added | John Klein | Peter: I think I found where the real problem is: it's with the small object argument. The point is that when doing the gluing construction to factorize a map, one has to know that a map from $D^n$ into a certain direct limit of section spaces actually factors through a finite stage (more precisely, it's the section space over $X$ at stage $k$ given by taking the sections of the $k$-th iterated gluing construction) The problem is that there's no reason to believe that such a factorization should exist unless $X$ itself is compact. | |
| Aug 22, 2013 at 16:38 | comment | added | Peter May | Maybe the edit I just added to my answer may help. You may get a model structure, but not with the weak equivalences you started with. | |
| Aug 22, 2013 at 16:07 | history | edited | Peter May | CC BY-SA 3.0 |
Expanded answer in light of comments.
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| Aug 22, 2013 at 4:08 | comment | added | John Klein | Yes Peter. I know I have to generate a complete argument to convince you (as well as myself). | |
| Aug 22, 2013 at 2:58 | comment | added | Peter May | John, I think you should provide details rather than argue in public. I would be happy to have you be right, in fact I would be very happy, but this is an area where long experience shows that healthy skepticism is justified. One place to expect trouble is proving that your proposed acyclic cell complexes are in fact acyclic. That is not formal. Details or nothing please! | |
| Aug 21, 2013 at 21:07 | comment | added | John Klein | I haven't checked carefully, but I cannot see what goes wrong: let's define the fibrations as above. Then define the cofibrations via the lifting property. Then show that attaching $X \times D^n$ to an object $U$ along a morphism $X\times S^{n-1} \to U$ gives a cofibration $U \to U'$. This is easy I think. Induction shows that repeated sequential attaching of this kind forms a cofibration. Next one should show the mod-cat factorizations exist. It seems to me this can be done as usual, by the gluing construction with respect to each of the types of cofibrant generators (cf. Dwyer-Spalinski). | |
| Aug 21, 2013 at 18:54 | comment | added | Peter May | Presumably you mean Chapter 6. My recollection (from a decade ago) is that we gave up on your choice: obviously that would be the first thing one thinks of. You are starting from the adjunction in Example 2.1.8, and I'm pretty sure we discarded that idea for good reason. But it has been a while. Warning 6.1.8 may be relevant. | |
| Aug 21, 2013 at 17:53 | comment | added | John Klein | Peter: I can't see how Johann's structure does what I want. I would like the set of generating cofibrations to be (or at least contain) the inclusions $X \times S^{n-1} \subset X \times D^n$ and $X \times D^n \times 0\subset X\times D^n \times I$. If this is correct, then it seems to me that the fibrations should be morphisms $E \to B$ of $R(X)$ such that the induced map of section spaces $\text{sec}(E\to X) \to \text{sec}(B \to X)$ is a Serre fibration. Perhaps this doesn't work. By the way, I don't see objects of the form $X \times S^{n-1}$ appearing in your Chapter 7. | |
| Aug 21, 2013 at 17:23 | history | answered | Peter May | CC BY-SA 3.0 |