Timeline for What are the algebras for the ultrafilter monad on topological spaces?
Current License: CC BY-SA 4.0
15 events
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Dec 21, 2019 at 19:46 | comment | added | Ivan Di Liberti | Is there any connection between these guys you are studying and the bitopological description of Priestley spaces? | |
Dec 18, 2019 at 18:07 | comment | added | Tim Campion | @LSpice Eh, you can view it as just being like a prime. It started because I was thinking of a $\beta$ structure in terms of the structure map $\xi: \beta X \to X$, but then the formulation I arrived at didn't mention this map at all. So it's kind of a relic. | |
Dec 18, 2019 at 18:06 | comment | added | LSpice | Is the $\xi$ in $\tau^\xi$ just punctuation giving a new name, like a prime $\tau'$, or does it have some extra meaning? | |
Dec 18, 2019 at 18:05 | history | edited | LSpice | CC BY-SA 4.0 |
Name of [here], and \operatorname's
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Dec 18, 2019 at 16:24 | answer | added | Andrea Marino | timeline score: 2 | |
Dec 18, 2019 at 14:40 | comment | added | Tim Campion | @AndrejBauer It certainly looks like that, which is partly why I'm hopeful that it's a condition which may have been studied before... | |
Dec 18, 2019 at 14:39 | comment | added | Tim Campion | @AndreaMarino I'm confused -- surely you're not saying that for any topology $\tau$ there is at most one $\tau^\xi$ which defines a $\beta$-space (since any infinite indiscrete space is a counterexample)? I'd be very interested to hear more about this! | |
Dec 18, 2019 at 12:23 | comment | added | Andrea Marino | Well I found a pretty indirect characterization. There exist an explicitly constructible refinement $\tau'$, and $\tau$ defines a $\beta$ space iff $\tau'$ is compact and locally compact. If you want I can post it, but I think we can do better. A necessary condition though is that $X$ must be also locally compact . | |
Dec 18, 2019 at 8:19 | comment | added | Andrej Bauer | Condition (3) looks like some sort of regularity of $\tau$ relative to $\tau^\xi$, for if $\tau = \tau^\xi$ it's just ordinary regularity, isn't it? | |
Dec 18, 2019 at 1:42 | comment | added | Tim Campion | @AndreaMarino I agree! I think it's worthwhile to revisit some of these classic things from time to time. Manes' theorem, in particular, is a gem which deserves to be more widely known. The proof -- once you know what the Stone-Cech compactification of a discrete space is -- is an easy, fun application of the Beck Monadicity Theorem. And there's a whole cottage industry of extensions of these ideas, starting with a description of an arbitrary topological space as a kind of "lax algebra" for the ultrafilter monad. | |
Dec 18, 2019 at 1:12 | comment | added | Andrea Marino | This is so cool. I missed good point topology problems :) | |
Dec 18, 2019 at 0:06 | history | edited | Martin Sleziak |
added the (monads) tag - feel free to rollback my edit if I missed something and the tag is not suitable here
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Dec 17, 2019 at 23:43 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 229 characters in body
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Dec 17, 2019 at 18:43 | history | edited | Tim Campion | CC BY-SA 4.0 |
deleted 59 characters in body
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Dec 17, 2019 at 18:35 | history | asked | Tim Campion | CC BY-SA 4.0 |