Timeline for Localic maps given by series
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 16, 2022 at 15:22 | comment | added | Andreas Blass | Thanks. I had missed the one-way implication. | |
| Jul 16, 2022 at 15:20 | comment | added | Simon Henry | @AndreasBlass : I've only put an implication in the case of universal quantification, not an equality. But unless I've missed something, this is enough to show that $f^*$ preserves convergence (in the sense that if something converge in the base topos then $f^*$ of it also converge, not in the sense that the proposition "$U_n$ converge to $x$" is preserved). | |
| Jul 16, 2022 at 14:57 | comment | added | Andreas Blass | I don't see that $f^*$ respects universal quantification over $\mathbb N$. Suppose $f:\text{Sets}\to\text{Sh}(\mathbb R)$ is the point $0\in\mathbb R$, and suppose the truth value of $P(n)$ is the open interval $(-1/n,1/n)$. Since all these intervals contain 0, each $f^*(P(n))$ is true. But the truth value of $(\forall x\in\mathbb N)\,P(n)$ is the empty subset of $\mathbb R$, and so $f^*$ of it is false. What am I missing here? | |
| Jul 16, 2022 at 10:10 | vote | accept | Valery Isaev | ||
| Jul 16, 2022 at 8:08 | history | answered | Simon Henry | CC BY-SA 4.0 |