Timeline for Example of a saturated class of morphisms which is not _obviously_ saturated?
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12 events
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| Feb 9, 2021 at 22:37 | comment | added | Tim Campion | I wonder if one could use the fact that at least for "many" spaces, a map $X \to Y$ is a weak homotopy equivalence iff the induced map $X \times \mathbb R^\infty \to Y \times \mathbb R^\infty$ is a homeomorphism to give a direct proof that the weak homotopy equivalences are saturated. I would find this very amusing if it worked! | |
| Jan 2, 2015 at 22:09 | vote | accept | Tim Campion | ||
| Dec 31, 2014 at 4:54 | answer | added | Mike Shulman | timeline score: 6 | |
| Dec 31, 2014 at 4:45 | comment | added | Mike Shulman | In my opinion, the widespread belief that basepoint issues are formalities is one of the more insidious illusions fostered by traditional algebraic topology. (-: Other contexts in which the subtlety of basepoints is noticeable include equivariant homotopy theory and fixed-point theory; in both cases a choice of consistent basepoint would have to be a fixed point, but often the existence or nonexistence of a fixed point is one of the questions we want to deploy algebraic topology to answer! | |
| Dec 31, 2014 at 2:05 | comment | added | Ben Wieland | I think that simple homotopy equivalences satisfy 2-out-of-3, but are not saturated in the category of finite complexes. | |
| Dec 30, 2014 at 21:55 | comment | added | Tim Campion | Maybe this is not so obvious. Is this really the motivating example? If so, I find it surprising because unlike the difference between stable and unstable phenomena, basepoint issues are usually regarded as formalities: I've never seen so much hinge on basepoints before. | |
| Dec 30, 2014 at 19:26 | comment | added | Tim Campion | I've just noticed that the basepoint-dependence of the homotopy groups means that there isn't one single functor of homotopy groups out of $\mathsf{Top}$ which can be used to make the saturation of weak equivalences in $\mathsf{Top}$ obvious in the way I described above. However, I would still maintain that saturation of the weak equivalences in $\mathsf{Top}$ follows almost immediately from the definitions, and so does not serve as an answer to my question. | |
| Dec 30, 2014 at 19:15 | history | edited | Tim Campion | CC BY-SA 3.0 |
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| Dec 30, 2014 at 17:19 | history | edited | Tim Campion | CC BY-SA 3.0 |
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| Dec 30, 2014 at 17:08 | history | edited | Tim Campion | CC BY-SA 3.0 |
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| Dec 30, 2014 at 17:00 | history | edited | Tim Campion |
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| Dec 30, 2014 at 16:29 | history | asked | Tim Campion | CC BY-SA 3.0 |