Timeline for The center of $\mathbf{hTop}$
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| S Dec 12, 2022 at 0:01 | history | bounty ended | CommunityBot | ||
| S Dec 12, 2022 at 0:01 | history | notice removed | CommunityBot | ||
| S Dec 3, 2022 at 22:34 | history | bounty started | Martin Brandenburg | ||
| S Dec 3, 2022 at 22:34 | history | notice added | Martin Brandenburg | Canonical answer required | |
| Nov 25, 2022 at 10:32 | comment | added | Martin Brandenburg | @Tyrone Yes I am! | |
| Nov 24, 2022 at 9:27 | comment | added | Tyrone | Are you still interested in the classical homotopy category? I can show in this case that $\alpha_{S^1}$ cannot be constant. Achim's analysis below then forces it to be the identity. I have no idea how to produce nontrivial elements of the centre in this case | |
| Nov 22, 2022 at 21:45 | comment | added | Achim Krause | Here's a better description: Note that in the homotopy category, a morphism $S\to X$ for a discrete space $S$ is the same as a map $S\to\pi_0(X)$ of sets. In other words, $\pi_0$ is also right adjoint to the functor taking a set to the discrete space. (The usual adjunction goes the other way). The counit of this adjunction gives a natural transformation $\pi_0(X)\to X$, with $\pi_0(X)$ viewed as discrete space. We also have the usual natural transformation $X\to\pi_0(X)$, compose those to get a natural map $X\to X$. | |
| Nov 22, 2022 at 21:34 | comment | added | Martin Brandenburg | @AchimKrause Thank you! To be honest, I do not fully understand why your example satisfies naturality. | |
| Nov 22, 2022 at 13:40 | comment | added | Najib Idrissi | @Tyrone But there is a weak homotopy equivalence from $S^0$ to the sine curve, and $hTop$ being the localization, this map must have an inverse in $hTop$. | |
| Nov 22, 2022 at 13:05 | comment | added | Achim Krause | The weak homotopy category of all spaces is equivalent as a category to the homotopy category of CW complexes. You're right that the natural transformation I'm defining is not induced by something on the category of all topological spaces with continuous maps, but that doesn't matter, you can just work with CW complexes. (I'll admit that I was vage about what I mean by "space" in my comment) | |
| Nov 22, 2022 at 11:52 | history | edited | Martin Brandenburg | CC BY-SA 4.0 |
added 298 characters in body
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| Nov 22, 2022 at 11:49 | comment | added | Tyrone | @AchimKrause If you want to work with the weak homotopy category of all spaces, then you want path components (since connected components are not preserved by weak equivalence). But this construction cannot be made natural (the topologist's sine curve does not map nontrivially onto $S^0$, even though it is weakly equivalent to it) | |
| Nov 22, 2022 at 11:27 | answer | added | Achim Krause | timeline score: 7 | |
| Nov 22, 2022 at 11:11 | comment | added | Achim Krause | (This works in the homotopy category of CW complexes or Kan complexes, or equivalently in the weak homotopy category of all spaces, maybe not in the homotopy category of all spaces) | |
| Nov 22, 2022 at 11:09 | comment | added | Achim Krause | Here's a nontrivial element of the center: For each space $X$, choose a point in each connected component, and let $\alpha_X: X\to X$ be the constant map to the chosen point on each connected component. The required diagrams do commute up to homotopy. | |
| Nov 22, 2022 at 11:01 | history | edited | Martin Brandenburg | CC BY-SA 4.0 |
added 220 characters in body
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| Nov 22, 2022 at 10:55 | history | asked | Martin Brandenburg | CC BY-SA 4.0 |