Timeline for Internal characterization of topos validity
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Dec 26, 2016 at 15:18 | comment | added | Andrej Bauer | To bolster @მამუკაჯიბლაძე's remark: it's more or less a mistake to consider only global points $1 \to \Omega$ as possible values in a valuation. The correct thing is to consider global points, i.e., morphisms $A \to \Omega$ for arbitrary $A$. Then things will work out correctly, including the equivalence you seek. Under special circumstances, one might be able to restrict to less general domains of points, for instance to $1$ in a well-pointed category or to the representables in a sheaf topos. In other words, pick up a newer book; Colin McLarty's "Elementary toposes" is not too daunting. | |
| Dec 25, 2016 at 21:14 | comment | added | მამუკა ჯიბლაძე | I agree that discussion in 7.4 is sort of vague, but what you say is nowhere stated there; on the contrary, a counterexample is given, for $\varphi(x)=x\lor\neg x$ in $M$-sets, where $M$ is a monoid which is not a group. In this topos $\Omega$ has only two elements, which implies that $\varphi$ is valid there in Goldblatt's sense. However the above diagram does not commute for $\varphi$ - the topos is not Boolean. Such pathologies are avoided in Kripke-Joyal semantics, where variables are interpreted by identity morphisms, and then the statement about the diagram becomes trivially true. | |
| Dec 25, 2016 at 20:11 | history | edited | fosco | CC BY-SA 3.0 |
I'm the bra-ket fairy
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| Dec 25, 2016 at 19:25 | review | First posts | |||
| Dec 25, 2016 at 19:53 | |||||
| Dec 25, 2016 at 19:21 | history | asked | user102845 | CC BY-SA 3.0 |