Timeline for The formal p-adic numbers
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16 events
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| Aug 14, 2015 at 7:54 | comment | added | Simon Henry | ... They where used a lot in the works (Burden,Mulvey,Banachewski, Spitters, Coquand and others...) on Banach space/Hilbert space/C^* algebra internal to toposes that culminated with the proof of the constructive Gelfand duality. For example Mulvey,Banachewski "A globalisation of the Gelfand duality theorem" or older: Mulvey,Burden "Banach spaces in categories of sheaves". | |
| Aug 14, 2015 at 7:47 | comment | added | Simon Henry | That is indeed a different notion. What I call a Cauchy approximation is a sequence $(A_n)$ of subset of $X$ (your metric space) such that $A_{n+1} \subset A_n$, the diameter of $A_n$ is smaller than $\frac{1}{n}$, and each $A_n$ is inhabited. Two Cauchy approximations are equivalent if for all $n$ there exists $a \in A_n$ and $a' \in A_n$ such that $d(a,a') < 1/n$. And you can define the completion as the set of equivalence class of Cauchy approximation. Those are equivalent to Cauchy filter in topos logic: each Cauchy Filter is equivalent to one generated by a Cauchy approximation. | |
| Aug 14, 2015 at 2:22 | comment | added | Mike Shulman | Very nice, thanks! I'm curious about this notion of "Cauchy approximation" you mention; in chapter 11 the HoTT Book uses something called a "Cauchy approximation" (the choice to use those rather than Cauchy sequences was due to @AndrejBauer), but I was under the impression that the difference between those and Cauchy sequences was only cosmetic. Is your notion of "Cauchy approximation" different? | |
| Aug 14, 2015 at 2:17 | vote | accept | Mike Shulman | ||
| S Aug 13, 2015 at 10:43 | history | suggested | CommunityBot | CC BY-SA 3.0 |
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| Aug 13, 2015 at 10:30 | review | Suggested edits | |||
| S Aug 13, 2015 at 10:43 | |||||
| Aug 13, 2015 at 9:18 | comment | added | David Roberts♦ | Yes, and I'm pleasantly surprised the sequential completion is the right thing in general, and youtu.be/RK5n-X-Jlbk?t=2m1s :-) | |
| Aug 13, 2015 at 8:09 | history | edited | Simon Henry | CC BY-SA 3.0 |
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| Aug 13, 2015 at 8:02 | comment | added | Simon Henry | Well the identify map from the locale $\mathbb{Q}_p$ to the locale $\mathbb{Q}_p$ will not be locally constant. My paragraph about locally constant function only works for locally connected locales and was there only to show that it is not absurd that the sequential completion coincide with the sheaf of continuous functions. | |
| Aug 13, 2015 at 7:59 | comment | added | David Roberts♦ | No, I meant continuous functions with values in Q_p. But that is, I see, only the case for locally connected spaces. In general one can have nasty spaces (but perhaps using locales things are tempered somewhat). | |
| Aug 13, 2015 at 7:55 | comment | added | Simon Henry | You mean for the sequential construction of real numbers ? I've learned this from Peter Johnstone's Elephant (example D4.7.12). The argument is fairly general and I think work for any sequential completion of a metric set. | |
| Aug 13, 2015 at 7:50 | history | edited | Simon Henry | CC BY-SA 3.0 |
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| Aug 13, 2015 at 7:49 | comment | added | David Roberts♦ | Thanks, Simon. I had a feeling that the functions would be locally constant, but couldn't any way prove it. | |
| Aug 13, 2015 at 7:40 | history | edited | Simon Henry | CC BY-SA 3.0 |
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| Aug 13, 2015 at 7:34 | history | edited | Simon Henry | CC BY-SA 3.0 |
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| Aug 13, 2015 at 7:24 | history | answered | Simon Henry | CC BY-SA 3.0 |