Timeline for Dualizing the notion of topological space
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43 events
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| Jun 22, 2021 at 7:02 | comment | added | Martin Brandenburg | In Large limit sketches and topological space objects (Section 7) I suggest a different notion of cotopological space, which makes sense in any cocomplete category. It is a formal dualization of an internal notion of a topological space in a complete category. Whereas the OP here suggests to use open sets, I work with convergent nets. | |
| Feb 13, 2020 at 10:45 | comment | added | Ivan Di Liberti | I was thinking... given a topological space $(X, \{i_j\}_{j \in J})$ we can define a cotopological space over the set $X$ using the $0$-dimensional Grothendieck construction, can't we? In fact to a map $i_j: U_j \to X$ corresponds a map $i^{-1}_j: X \to \mathcal{P}(U_j)$. Unfortunately $i^{-1}_j$ is not an epi in general, but maybe this can be fixed. | |
| Nov 23, 2019 at 16:30 | vote | accept | CommunityBot | moved from User.Id=30211 by developer User.Id=481663 | |
| Nov 23, 2019 at 15:55 | history | edited | YCor | CC BY-SA 4.0 |
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| Nov 23, 2019 at 11:58 | answer | added | Vadim Alekseev | timeline score: 18 | |
| Oct 4, 2018 at 20:16 | comment | added | PrimeRibeyeDeal | No one made the obligatory joke that maps of cotopological spaces are "ntinuous maps"? | |
| Dec 25, 2017 at 15:30 | comment | added | მამუკა ჯიბლაძე | Although a trivial reformulation, the following might help on the intuitive level: using a well known duality, the category of cotopological spaces is dually equivalent to the category of very special kind of topological Boolean algebras: complete atomic Boolean algebras (caBa) equipped with a topology having the property that every open subset is in fact a complete atomic subalgebra. This might be also interesting since in its turn the category of caBa's is equivalent to the category of compact Hausdorff Boolean algebras. | |
| Apr 17, 2017 at 22:37 | comment | added | user30211 | @Ivan: Perhaps not because we consider here the relation generated by several relations, not the union of relations (which is not in general a relation). | |
| Apr 17, 2017 at 21:33 | comment | added | Ivan Di Liberti | Maybe I am wrong but isn't a cotopology on $X$ a (quite special) topology on $X \times X$? | |
| Mar 28, 2017 at 21:36 | history | edited | user30211 | CC BY-SA 3.0 |
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| Mar 28, 2017 at 21:22 | comment | added | LSpice | @GarlefWegart, I'm sorry that I'm not sure what to make of the question (probably my fault). I guess, but am too tired to see that I'm not saying something stupid, that a Grothendieck topology can be extended from the opposite of the category of quotient objects of $X$ to $\mathrm{Set}^{\mathrm{op}}$ in a trivial way, but I don't know if this is a useful or interesting thing to do. | |
| Mar 27, 2017 at 21:13 | comment | added | Gerrit Begher | @LSpice: But if so, it is a particular Grothendieck Topology on $Set^{op}$, right? | |
| Mar 26, 2017 at 20:18 | comment | added | Włodzimierz Holsztyński | In my opinion, the topological (cotopological) axiomatization should not mention the notion of a subobject (quotient object). | |
| Mar 26, 2017 at 12:22 | comment | added | user95282 | A cotopology is a set of equivalence relations. Does this lead to any interesting questions in the theory of Borel equivalence relations? [ A.S. Kechris: "New directions in descriptive set theory", Bulletin of Symbolic Logic 5(2) (1999), 161–174; V. Kanovei: "Borel equivalence relations", AMS 2008 ] | |
| Mar 26, 2017 at 10:33 | comment | added | HeinrichD | Crosspost math.stackexchange.com/questions/2202492 | |
| Mar 26, 2017 at 3:42 | comment | added | zibadawa timmy | I'll note that a cursory google search suggests that the term "cotopological space" already exists, and does not refer (at least in any obvious way) to a categorified dualization. Rather it refers to some topology which has certain compatibilities with a given one. | |
| Mar 26, 2017 at 3:13 | history | edited | Tom Goodwillie | CC BY-SA 3.0 |
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| Mar 26, 2017 at 1:44 | history | edited | user30211 | CC BY-SA 3.0 |
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| Mar 25, 2017 at 23:16 | comment | added | LSpice | Isn't this literally just a Grothendieck topology on the opposite of the category of quotient objects of $X$? | |
| Mar 25, 2017 at 21:57 | history | edited | user30211 | CC BY-SA 3.0 |
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| Mar 25, 2017 at 21:31 | history | edited | user30211 | CC BY-SA 3.0 |
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| Mar 25, 2017 at 20:59 | history | edited | user30211 | CC BY-SA 3.0 |
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| Mar 25, 2017 at 20:48 | history | edited | user30211 | CC BY-SA 3.0 |
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| Mar 25, 2017 at 20:41 | comment | added | nfdc23 | @PaulSiegel: The criterion of being guided by interesting examples is apt much beyond grants and thesis topics. Although metric spaces give rise to co-topological spaces (the metric spaces are not themselves examples), the equivalence relation $R_{\epsilon}(x)$ of identifying all points outside of an open ball seems pathological (so likewise for things made from such far-out equivalence relations). So it remains unclear what are interesting examples. I realize it is a matter of taste, but what are some great MO questions whose motivation is shakier than "dualizing is a natural thing to do"? | |
| Mar 25, 2017 at 20:24 | history | edited | user30211 | CC BY-SA 3.0 |
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| Mar 25, 2017 at 19:49 | history | edited | user30211 | CC BY-SA 3.0 |
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| Mar 25, 2017 at 19:19 | comment | added | Paul Siegel | @nfdc23 Those considerations are appropriate if one is applying for a grant or helping a student find a thesis problem, but there are many great MO questions whose original motivation was shakier than "dualizing is a natural thing to do". Also, the OP has produced a fairly rich supply of examples, namely all metric spaces. Odds are this construction will produce something familiar - bornological spaces, perhaps? - and my curiosity is piqued, at least. | |
| Mar 25, 2017 at 18:52 | history | edited | user30211 | CC BY-SA 3.0 |
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| Mar 25, 2017 at 18:52 | comment | added | nfdc23 | Tom Goodwillie is pointing out something important: one should have good examples to motivate why one would introduce or explore some concept or else there is nothing to guide what is being done. That "dualizing is a natural thing to do" is not by itself a compelling reason to think about something: it should illuminate a concept of prior interest or shed light on something that is hard to understand or something like that (e.g., affine group schemes genuinely illuminate things, but not because of being a "dual" concept). Otherwise the concept has no reason to be productive vs. a dead end. | |
| Mar 25, 2017 at 18:32 | comment | added | Gro-Tsen | Yet another way to dualize would be this: instead of defining the topology with open or closed sets, do it with both simultaneously: consider pairs of arrows $(U\to X,F\to X)$ which write $X$ as a set coproduct of $U$ and $F$, and write the axioms for topology on such pairs. Now dualize and consider pairs $(X\to D,X\to T)$ making $X$ into the set product of $D$ and $T$ and try to write axioms for those. I really don't know which is most natural. | |
| Mar 25, 2017 at 18:26 | history | edited | user30211 | CC BY-SA 3.0 |
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| Mar 25, 2017 at 18:20 | comment | added | user30211 | Thanks for your comment. In fact analogizing from the closed set definition of a topological space (I believe) would result in replacing joins with meets in the note above (if this is not clear I can be more detailed). Perhaps there is a correspondence between these. I am aware of a definition of a topology in terms of a boundary operator (which is symmetrical). | |
| Mar 25, 2017 at 18:15 | history | edited | user30211 | CC BY-SA 3.0 |
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| Mar 25, 2017 at 18:14 | comment | added | Gro-Tsen | I like your idea a lot, but keep in mind that a topology can just as well be defined by closed sets, and if you dualize on the closed sets, you get something presumably very different, because one place where the symmetry breaks down horribly is that a subobject has a complement whereas a quotient object does not have a canonical transversal (or whatever that would be). One can also define topologies using closure axioms and maybe that would also give some interesting dual notion. | |
| Mar 25, 2017 at 17:07 | comment | added | Tom Goodwillie | Cool. So now we can define "cocontinuous map", "quotient cotopology", "subcospace cotopology". And maybe some coseparation axioms. And work out what colimits and limits in this category are. Oh, and think of some good examples. | |
| Mar 25, 2017 at 15:08 | history | edited | user30211 | CC BY-SA 3.0 |
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| Mar 25, 2017 at 15:02 | history | edited | user30211 | CC BY-SA 3.0 |
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| Mar 25, 2017 at 14:51 | comment | added | user30211 | The coproduct of quotient objects is defined via universal property in the category of quotient objects of a particular object. In the case of a set, we can view quotient objects as equivalence relations on the set in question. Their coproduct is then the equivalence relation generated by two equivalence relations. Their product is the intersection of equivalence relations. | |
| Mar 25, 2017 at 14:46 | comment | added | user30211 | Sorry I should have dualized that too. I meant $X \rightarrow \{ * \}$. I have no motivation besides that dualizing is a natural thing to do. | |
| Mar 25, 2017 at 14:43 | history | edited | user30211 | CC BY-SA 3.0 |
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| Mar 25, 2017 at 14:37 | comment | added | nfdc23 | What is "$X \rightarrow \emptyset$"? What is a co-product of quotient objects (e.g., of copies of the identity map)? Is there some motivation for asking about this? | |
| Mar 25, 2017 at 14:25 | review | First posts | |||
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| Mar 25, 2017 at 14:21 | history | asked | user30211 | CC BY-SA 3.0 |