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If can make valuations into functors, then Monotonicity is just the functor action on the morphism. However the functor would have to map to $([0,\infty], \ge)$ if we want to include $+$ as a monoidal product so the functor is contravariant?
Then a valuation on a category (think frame) is a presheaf $F : X^{op} \to V$ where $V$ is the monoidal category of positive real numbers and infinity pointing to the smaller one (the category that comes up in Lawvere metric spaces) with addition as the monoidal product. Subject to the extra condition that $F(x) \otimes F(y) = F(x \prod y) \otimes F(x \coprod y)$.
I don’t know if this is useful to think about in this way. Oh and continuity is quite literally continuity of the presheaf. The extra condition is somewhat bothersome however, I can’t think of anywhere that comes up in category theory.
@Alizter: concerning the extra condition, especially how it comes up in other contexts in category theory, see this MO discussion initiated by David Spivak.
@Tobias oh that is very nice! Thank you very much for the link!
The current statement of Theorem in Section 6 does not match Alvarez-Manilla’s Theorem 3.27, which talks about τ-smooth Borel measures and does not mention regularity.
On the other hand, locally compact sober spaces are regular, so this case appears to be redundant anyway. Some clarification appears to be necessary.
As a side remark, turning “here” into references does not appear to be nLab’s style.
I’ve seen “here” pointing to references more than a few times in the nLab. I think it’s fine.
Thanks for the alert. I see that also the actual publication data was missing, and formating of theorems and journal links was bad. Have touched all this now.
I suggest to uniformly code references close to this:
{#LastNameYear} FirstName LastName, _Title_, Journal Volume Issue, Year ([arXiv:Identifier](arXivURL), [doi:Identifier](doiURL))
and to refer to it from the text as
see [LastName Year, Theorem X](#LastNameYear)
And for those who can be bothered: Once you have added any refererence item with complete and well-formatted publication data, just copy-and-paste it over to the author’s nLab pages under “Selected writings”, with a pointer back to the pertinent entries.
If we do this consistently for ten years, we’ll have created the most useful online data base.
@Dmitri Pavlov Apologies, I tried to condense two statements that were present in the page without checking that they indeed corresponded to what was in the original paper. Is it fixed now, or is the information still incorrect?
@Urs: I can do that, but what do you mean by “pointer back to the pertinent entries”?
He means, I think, to hyperlink back to the nLab mathematics page where the work is mentioned, from the nLab’s page for the work’s author.
Yes. Paolo, just have a look at any example, for instance Cumrun Vafa – Selected writings
Let me cite all three theorems involved.
All 3 theorems talk about extending locally finite continuous valuations to measures.
Theorem 3.23 in his PhD thesis says that such valuations on regular Hausdorff (i.e., T3) spaces extend uniquely to regular τ-smooth Borel measures.
Theorem 3.27 in his PhD thesis says that such valuations on locally compact sober spaces extend uniquely to τ-smooth Borel measures.
He defines locally compact spaces as those spaces for which compact subsets form a fundamental system of neighborhoods.
Theorem 4.4 in his paper says that such valuations on regular (but not necessarily Hausdorff) spaces extend uniquely to τ-smooth Borel measures.
Theorem 4.12 in his paper is the same as Theorem 3.27 in his PhD thesis.
I guess it is important here that local compactness is used in the weaker form, since the stronger form implies regularity, which would make the second theorem redundant.
Thank you!
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