Timeline for What are compact objects in the category of topological spaces?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jun 5, 2021 at 20:28 | comment | added | Z. M | @JiříRosický What are cocompact objects in the category of topological spaces, i.e. compact object of the opposite category of topological spaces? | |
| Aug 18, 2018 at 1:26 | comment | added | Todd Trimble | Thanks for your question and answer. I no longer recall precisely what I was thinking those years ago at the Cafe, but currently I'm still hoping a more positive answer can be given under stronger hypotheses, either restricting the spaces or restricting the maps (e.g. closed continuous maps) or both. For example, we know that $Top(X, -)$ preserves colimits of $\omega$-chains of closed inclusions between $T_1$ spaces, if $X$ is limit point compact. Such hypotheses are rather stronger than I'd like, but maybe there are weaker hypotheses that would satisfy my curiosity. | |
| Dec 18, 2017 at 15:59 | comment | added | R. van Dobben de Bruyn | This argument actually shows much more: the topology on $X$ can be recovered categorically. In particular, every equivalence $F \colon \operatorname{\underline{Top}} \to \operatorname{\underline{Top}}$ is isomorphic to the identity. | |
| Dec 18, 2017 at 15:23 | comment | added | R. van Dobben de Bruyn | @PhilippeGaucher: ah, and a way to find the Sierpiński space $S$ categorically is as the unique two-point space for which the transposition is not continuous (this is a purely categorical characterisation). Then the set of open subsets of $X$ can be found as the set of morphisms $X \to S$, and you can also find the inclusion order on this set from this characterisation. | |
| Dec 18, 2017 at 9:31 | comment | added | Philippe Gaucher | @მამუკაჯიბლაძე A space $X$ is compact if and only if it is a compact object in the category of open subsets of $X$: it's Proposition 2.6 of ncatlab.org/nlab/show/compact+space. | |
| Dec 18, 2017 at 3:06 | comment | added | Dmitri Pavlov | A related question: I've always wondered what the compact objects in the category of locales are. The proof above fails in the case of locales already in the first step (Lemma): there is no such thing as an indiscrete locale. | |
| Dec 17, 2017 at 23:19 | comment | added | R. van Dobben de Bruyn | @მამუკაჯიბლაძე this was originally the reason I started looking into this. I think it's an interesting enough question that you can ask it on MO, especially because of its relationship to other existing questions like this one. | |
| Dec 17, 2017 at 17:12 | comment | added | მამუკა ჯიბლაძე | A natural question arises then (whether it deserves a separate MO entry I don't know) - can compact spaces be characterized by an abstract categorical property inside Top? | |
| Dec 17, 2017 at 16:35 | comment | added | R. van Dobben de Bruyn | @JiříRosický: Ah, that's great; thanks for the references. I strongly suspected that this was known already, but I was unable to find it (because I was searching for the wrong words). If you add this as an answer, I would be happy to accept. | |
| Dec 17, 2017 at 12:25 | comment | added | Jiří Rosický | This is due to Gabriel and Ulmer, Lokal prasentierbare Kategorien (see 6.4). It is also in my book with Adámek Locally Presentable and Accessible Categories (see 1.2(10)). | |
| Dec 16, 2017 at 23:13 | history | answered | R. van Dobben de Bruyn | CC BY-SA 3.0 |