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Parametrized geometric theories


Steve.Vickers
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[Question to Olivia]

Can you say more about the problem of parameters?

On the face of it, geometric logic deals with parameters well, if the parameters are points of a space [T] (ie models of a geometric theory T).

If we then define a space parametrized by p:[T], ie a geometric theory T’(p), then that is an internal geometric theory/site in S[T]. We can form a semidirect product T’’ = T x| T’, and then get a bundle [T’’] -> [T] such that the fibre over any point p is [T’(p)]. (Generalized point; generalized fibre = pullback.) [T’’] can be though of as Σ_{p:[T]} [T’(p)], as long as you understand that Σ is not a coproduct of spaces. 

Ming Ng’s talk tomorrow is about trying to do this when T is a theory of places (on Q), and T’(p) is the theory of points of the completion for place p. It’s pretty clear how to do that for absolute values p, so the interesting mathematics lies in factoring out equivalence to get places.

You mentioned an issue of declaring “constant” sorts, but that’s fairly painless if you exploit the ability to declare derived sorts in a geometric theory. It’s roughly what you see in the Elephant B4.2.7 account of geometric theories, and in “AUs and classifying toposes” I showed how to exploit it in a base-independent way by using the AU constructions to approximate colimits.

All that assumes, of course, that T’(p) is constructed geometrically from (the generic model of) T. The big challenge to geometric reasoning is whether it can deal with the topos applications where at present the sites we have are not defined geometrically.

Steve.


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Olivia Caramello
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I was indeed referring to the implicit, but not explicit, presence of parameters in the classical formalization of geometric logic (the one modelled on the usual presentation of first-order logic), and to the (straightforward) possibility of explicitly adding them without affecting the expressiveness of the logic.  

I cannot really say much more on the subject of parameters since this is the subject of work-in-progress with Riccardo Zanfa, which we have not yet completed; of course, if you are interested, we can send you our work as soon as it is ready.

In any case, I look forward to Ming Ng's talk tomorrow.


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Steve.Vickers
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@ocaramello

Thanks, yes I’d love to see your work with Zanfa when you have it complete.

However, I sensed in this morning’s discussion that there was a recognised question about parameters, so what I’m really interested in for the moment is getting a better understanding of what the question is.

Is it simply that if (if!) we could define all our sites geometrically, then everything would be fine, or is there a perceived deeper problem?

In my own work I have seen how geometricity promises to give a rather neat account of bundles: by defining the geometric transformation base point |-> fibre you define the whole bundle, and I’m not sure how widely understood that is.

Steve 


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Olivia Caramello
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It would really be premature for me to discuss about the issues you mention before our work is completed. We will be happy to get back to you once it is!


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Steve.Vickers
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@ocaramello 

Olivia - I think you misunderstand my question.

I’m trying to catch up on the common understanding that seemed to underlie the panel discussion round Monday’s talks. You had some remarks regarding parametrization and relative geometric geometry, and others in the discussion seemed to know what you were talking about. (A lot of it may have referred to the liquidity parameter p of Scholze’s talk, which unfortunately I missed.)

We know well that bundles, understood as parametrized spaces, can be handled nicely by relative geometric reasoning, provided the sites involved are constructed geometrically out of the parameter. Sadly for this approach, Scholze’s sites are not described geometrically. Is that any part of what the discussion was referring to?


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Olivia Caramello
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Posted by: steve-vickers
Is that any part of what the discussion was referring to?

No, as far as I remember.


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Steve.Vickers
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@ocaramello 

Frustrating. I didn’t understand it and you can’t remember it. Will the discussions be included in the recordings?


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Olivia Caramello
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No, the recordings will only include talks.


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