Timeline for Pointed Hurewicz model structure
Current License: CC BY-SA 2.5
16 events
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| Jan 25, 2020 at 10:06 | comment | added | Harry Gindi | @SebastianGoette I don't know of any counterexamples, but what I wrote in my answer ten years ago was that there is no reason to expect this category to carry such a model category structure. The fact that Strøm's theorem works in the unbased setting is extremely surprising. The original question notes that Strøm was unable to replicate his result in the pointed case. I haven't read his paper, but I would guess that is a nice place to look for explicit counterexamples, if they exist. | |
| Jan 25, 2020 at 9:59 | comment | added | Sebastian Goette | Thanks for your explanations. Of course, in the model category language, everything is clear. It is also clear from the above that the pointed homotopy category can be constructed by taking well-pointed spaces as objects, and pointed maps up to pointed homotopy as morphisms. What is not clear to me: what happens if we take all pointed spaces and pointed maps up to pointed homotopy? How different will this category be from the homotopy category? Are there "nice" counterexamples to illustrate the difference? | |
| Jan 25, 2020 at 9:49 | comment | added | Harry Gindi | @SebastianGoette PS It is called the 'generalized Whitehead theorem' because the original Whitehead theorem says that every weak homotopy equivalence between CW complexes (fibrant-cofibrant objects of the Quillen-Serre model structure on CGWH spaces) is homotopic to an honest homotopy equivalence. | |
| Jan 25, 2020 at 9:46 | comment | added | Harry Gindi | @SebastianGoette Also, I don't know what you're getting at with the question about objects that are homotopy equivalent to cofibrant objects. The main point here is that a pointed map whose underlying map is a homotopy equivalence, between fibrant-cofibrant objects is going to be homotopic to a homotopy equivalence in the model category-theoretic sense, which in this case means that it is pointed-homotopic to a pointed-homotopy equivalence. Which is a bit confusing, I guess, but it makes perfect sense if you work a bit with model categories. | |
| Jan 25, 2020 at 9:36 | comment | added | Harry Gindi | @SebastianGoette The statement of the generalized Whitehead theorem is essentially this: If $X$ is cofibrant and $Y$ is fibrant, then $C(X,Y)/\sim=\operatorname{Ho}(C)(X,Y)$. Here the relation $\sim$ means homotopy in the model category theoretic sense, which coincides in this particular case with based homotopy of maps. | |
| Jan 25, 2020 at 9:30 | comment | added | Sebastian Goette | And are there not well-pointed spaces that are homotopy equivalent, but not pointed homotopy equivalent to cofibrant objects? In other words, is the "correct" homotopy category equivalent to the "naive" one, where one takes all objects, and morphisms modulo pointed homotopy? | |
| Jan 25, 2020 at 9:20 | comment | added | Sebastian Goette | What is the "generalized Whitehead Theorem" you are referring to? Do you have a nice reference for that? | |
| Dec 3, 2010 at 16:48 | history | edited | Dmitri Pavlov | CC BY-SA 2.5 |
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| Dec 1, 2010 at 12:19 | vote | accept | Jeff Strom | ||
| Dec 1, 2010 at 9:43 | comment | added | Harry Gindi | I only included the proof sketch above because David's comment below made me go back and check it. | |
| Dec 1, 2010 at 5:58 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| Dec 1, 2010 at 5:14 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| Dec 1, 2010 at 5:08 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| Dec 1, 2010 at 4:57 | history | undeleted | Harry Gindi | ||
| Dec 1, 2010 at 4:33 | history | deleted | Harry Gindi | ||
| Dec 1, 2010 at 4:33 | history | answered | Harry Gindi | CC BY-SA 2.5 |