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Paolo Perrone (Jun 04 2020 at 19:16):Hey all,
This is the discussion thread of David Spivak's talk, "Internal probability valuations".
The talk, besides being on Zoom, is livestreamed here: https://youtu.be/Ur6vxdRKr9g
Paolo Perrone (Jun 04 2020 at 19:22):Date and time: Sunday, 7 Jun, 16h UTC.
Bas Spitters (Jun 07 2020 at 15:01):@Tobias Fritz Here's the paper I mentioned:
Integrals and Valuations (with Thierry Coquand)
http://www.logicandanalysis.org/index.php/jla/article/view/174/66
as Steve Vickers has observes the construction is geometric, and thus is computed pointwise:
https://www.cs.bham.ac.uk/~sjv/Riesz.pdf
Bas Spitters (Jun 07 2020 at 15:04):https://ncatlab.org/nlab/show/valuation+%28measure+theory%29
Bas Spitters (Jun 07 2020 at 15:13):Steve considers valuations that preserve directed joins. We formulate continuity by using the presentation of the compact regular locales, but the same ideas are very similar, as we also saw when we did it synthetically (my talk today).
Paolo Perrone (Jun 07 2020 at 15:55):Hi! We start in 5 minutes.
Bas Spitters (Jun 07 2020 at 16:14):One reason internal valuations are important is that they correspond to quantum states on C*-algebras.
Bohrification (with Chris Heunen and Klaas Landsman)
http://arxiv.org/abs/0909.3468
Tobias Fritz (Jun 07 2020 at 16:16):Right, thanks for the pointer! That makes me realize that the Bohr topos in which Bohrification takes place has a very similar flavour to the hyperspace topos that David is talking about right now (implicitly)
Robert Furber (Jun 07 2020 at 16:47)::clap:
Robert Furber (Jun 07 2020 at 16:54):Steve used presentations throughout that paper.
Bas Spitters (Jun 07 2020 at 17:09):@Robert Furber thanks for reminding me. Yes, the presentation is preserved by pullbacks of geometric morphisms, and this is enough to compute the points.
Sam Staton (Jun 07 2020 at 20:40):@Tobias Fritz, what's the hyperspace topos?
Tobias Fritz (Jun 07 2020 at 21:07):By "hyperspace topos" I mean the topos of sheaves on the space of compact subsets of a Hausdorff space , with topology generated by the sets of compacts contained in a given open of . The main case of interest to us is , in which we want to interpret internal (continuous) valuations as a generalization of stochastic processes.
Paolo Perrone (Jun 08 2020 at 01:03):Hi all! Here's the video.
https://youtu.be/m9aaPZXiJpQ
Bas Spitters (Jun 08 2020 at 08:31):@Tobias Fritz The Vietoris construction is constructive, and even geometric when phrased appropriately. Is that relevant? Internally, one could look at G(V(X)) for a locale X for example.
Bas Spitters (Jun 08 2020 at 12:27):@Tobias Fritz @David Spivak In this paper where we use geometric mathematics to compute the interpretation of the Gelfand spectrum in Bohrification. We observe that the same works for valuations on a compact regular locale (i.e. Radom measure). Would that be enough for your setting?
Tobias Fritz (Jun 08 2020 at 22:51):Since David doesn't seem to be here let me try to comment instead. All of those works are indeed closely related to what we're doing, and we still need to read those papers in more detail and discuss how we can optimally make use of those existing results.
I don't think that we can expect our internal space to be locally compact. But if I understood correctly, in order for Steve Vickers' geometricity argument to apply to continuous valuations, it also doesn't need to be locally compact, right?
Section 3.5 of your Gelfand spectra paper is conceptually closely related to what we're doing. If I understand correctly, you've observed that an AQFT can be formulated as an internal C*-algebra on a suitable topos of copresheaves on spacetime regions. Very roughly, we're doing an analogous thing for Euclidean QFTs by writing them as internal "probability spaces" in a topos of presheaves on spacetime (plus a mild sheaf condition).
Bas Spitters (Jun 09 2020 at 07:34):Indeed, the local compactness is only used to show that the frame is preserved. In general, only the presentation of the frame is preserved, and this is usually enough.
Bas Spitters (Jun 09 2020 at 07:48):In that paper we are not doing AQFT, only Bohrification (approximating a quantum system by commutative subparts). There is work by Joost Nuijten about Bohrification and AQFT, but I'm not sure whether it is relevant to your work.