Timeline for The $(\infty, 1)$-category of all topological spaces, including the bad ones
Current License: CC BY-SA 4.0
14 events
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| Feb 7, 2022 at 16:04 | history | edited | Zhen Lin | CC BY-SA 4.0 |
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| Feb 5, 2022 at 23:53 | comment | added | Zhen Lin | After a night's sleep I realise that tensors/cotensors with $\Delta^n$ for all $n$ + conical colimits/limits implies all tensors/cotensors, and I'm sure that we do have conical colimits and limits, so if the folklore that $\textbf{Top}$ is not tensored/cotensored is true, then we can't even have tensors/cotensors with $\Delta^n$ (for some $n$). Gah. I should ask another question about this. | |
| Feb 5, 2022 at 15:59 | history | edited | Zhen Lin | CC BY-SA 4.0 |
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| Feb 5, 2022 at 15:47 | comment | added | Zhen Lin | @Tim Sorry, I just realised I wrote nonsense about $\mathcal{S}$ being reflective in $\mathcal{T}$. It is coreflective, for the reason you state, and morally also because the mixed model structure is a right Bousfield localisation of the Hurewicz model structure. Good point about totally disconnected spaces – $\textrm{Map} (X, Y)$ is discrete for totally disconnected $Y$, so there is a weird 1-category fully embedded in $\mathcal{T}$... | |
| Feb 5, 2022 at 15:43 | history | edited | Zhen Lin | CC BY-SA 4.0 |
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| Feb 5, 2022 at 15:30 | comment | added | Tim Campion | Some random thoughts: (1) Note that CW approximation shows that $\mathcal S$ is not only reflective in $\mathcal T$, but also coreflective. (2) For Question 4, note that totally disconnected spaces, viewed as objects of the simplicially-enriched category $\mathbf{Top}$, are already local with respect to $\Delta^1 \to \Delta^0$. So the 1-category of totally-disconnected spaces sits as a full reflective subcategory of $\mathcal T$. Totally disconnected spaces are not generated under colimits by any small full subcategory, so the same is true for $\mathcal T$. So $\mathcal T$ is not presentable. | |
| Feb 5, 2022 at 15:19 | history | edited | Zhen Lin | CC BY-SA 4.0 |
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| Feb 5, 2022 at 15:17 | comment | added | Tim Campion | @SimonHenry IIRC Strom's original construction worked with all topological spaces and had no issues. It was Cole's later re-proof of the theorem in the $k$-space setting which had issues. Subsequent proofs (e.g. Barthel and Riehl) have worked in $k$-spaces, are more conceptual, and also have no known issues. But there's no reason one can't work with Strom's original version in all topological spaces. | |
| Feb 5, 2022 at 15:15 | history | edited | Zhen Lin | CC BY-SA 4.0 |
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| Feb 5, 2022 at 15:10 | history | edited | Zhen Lin | CC BY-SA 4.0 |
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| Feb 5, 2022 at 14:53 | comment | added | Zhen Lin | I also realised there are some subtleties in what I claimed. I think there's no problem if I ask about a cartesian closed category of topological spaces instead but that does make the question less well motivated... | |
| Feb 5, 2022 at 14:33 | comment | added | Simon Henry | I just noted that the nLab (nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/…) seems to suggest (if I understand correctly) that it is not so clear that we still know how to prove that the Strom model structure exists without restricting to k-spaces though. I don't much about the issue and maybe I'm misinterpreting what the nLab say, but that might be important for some of the question you are asking... | |
| Feb 5, 2022 at 14:30 | comment | added | Simon Henry | The answer to question 1 should be yes assuming the stom model structure is simplicial (which I think is true and should be easy to check anyway) simply because every space is bifibrant in the strom model structure and in general the infinity categorical localization of a simplicial model category is equivalent to the simplicial category of bifibrant objects. | |
| Feb 5, 2022 at 11:55 | history | asked | Zhen Lin | CC BY-SA 4.0 |