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If space and quantity are in duality with one another, why consider spaces and quantities modelled on test spaces? What happened to test quantities?
As a guess, might we view the category opposite to the category of test spaces the category of test quantities? (E.g., the category of finitely presented rings could be considered a category of test quantities, perhaps.) Some categories might have more cause to be considered a category of test spaces, e.g., extensivity might be more reasonable for spaces than for quantities.
As a physicist I find it strange to say “quantity” for what shuld be called a distribution of quantities or even algebra of all possible distributions of qunatities. When one looks at observable in quantum mechanics, then one takes an expectation and one gets a number with some units, not a function. A function is a collection of values at many points, so it is a distrubution over a space. If you come in front of the audience of experimental physicists than this jargon of Lawvere will alienate you from the audience, it is my feeling.
In noncommutative geometry we talk about functions and observables on noncommutative spaces, never quantities, as far as I noticed so far. Quantity would be something like integral over the whole space of a distribution. We also talk about duality between spaces and algebras (algebras of functions); it is strange to call an algebra a “quantity”. In physics, a quantity has units.
Thanks for the comments. I can feel I’m only just touching on a network of issues I’d like to get clearer about, including what to make of the intensive/extensive quality distinction, and how all this fits with Urs here
we can ask if Spaces=presheaves “glue”. The failure of that is measured by cohomology.
we can ask if Quantities=co-presheaves “co-glue”. The failure of that is measured by homotopy.
I was going to make a comment like Zoran’s. The problem here is basically one of grammar. Yes, space and quantity are dual, but the term “space” refers to the entire space (not an individual point), while the term “quantity” refers to an individual quantity (not the entire algebra). One could speak of a duality between spaces and algebras, or (moving from the objects of study to the study itself) between geometry and algebra.
I agree with Todd; the category of finitely presented rings is a great example of a category of test algebras (or test algebras of quantities).
Concerning terminology: right, it does not really run smoothly. One could say “space and quantiTIES” or “point and quantity”.
Concerning Lawvere’s intensive/extensive: I still haven’t really looked into this.
OK, so where there is
Consider the category of test spaces = CartSp.
Then
spaces modeled on are generalized smooth spaces such as diffeological spaces;
quantities modeled on are smooth algebras (-rings),
the final ’quantities’ could be ’algebras of quantities’.
But there might also have been an account in terms of test algebras? E.g.,
Consider the category of test algebras = fpRings.
Then
spaces modeled on are ???
quantities (of algebras) modeled on are ???
I probably ought to be mumbling something about schemes and rings of some kind.
6 Why not algebras like the noncommutative geometry has accepted ? (Besides the main examples, if something is a quantity, then quantities have some sort of algebraic operations, and this remains if categorifies arbitrarily, I think)
7 One has to have a site not only a category to have a sensible answer (e.g. algebraic spaces).
Lawvere has talked about the intensive/extensive quality distinction for years, e.g.,
Extensive and intensive quantities Workshop on Categorical Methods in Geometry, Aarhus (1983)
(a proposal for a general axiomatization of homotopy/homology-like “extensive quantities” and cohomology-like “intensive quantities”)
and Anders Kock is working on extensive quantities. Maybe there’s a gem there like cohesiveness.
One of these days the great Archive for Mathematical Sciences & Philosophy will open and we’ll be able to hear many of Lawvere’s lectures including the series of three from 1989:
To some extent “intensive quantity” seems to be just another perspective on “space”. Sheaves of function algebras are the canonical example in both cases. (The space is the space that the functions take values in.)
This seems to harmonize with Anders Kock’s emphasis in Calculus of extensive quantities that he is concentrating on extensive quantities and disregarding the intensive one.
Also the elementary remarks at the beginning of Kock-Reyes SOME CALCULUS WITH EXTENSIVE QUANTITIES: WAVE EQUATION suggest that extensive/intensive duality is just another perspective on Isbell duality.
The genuine structural input in these articles by Kock seems to be that he imposes extra conditions on his extensive quantities, such as commutativity of sorts.
I guess the useful aspect here is to realize that the collection of all distribution spaces parameterized over all manifolds is, as a totality, analogous to a ring/algebra (beware: not a sheaf of them). At its root this is just the banality that assigning distributions to spaces is covariant. But then one can check what further properties this shares with algebras for an algebraic theory.
I should think about it some more. But not tonight.
To some extent “intensive quantity” seems to be just another perspective on “space”.
Good, that’s what I wondered:
If intensive quantities are contravariant functors from some category of spaces, won’t they resemble in some way spaces modelled on that category of spaces?
But Lawvere also believes it matters whether the target is a distributive or a linear category, as mentioned here.
The article quantity is currently promising “more” in a “minute” which to me seems to have lasted over five years now.
Do you think quantity should
To me, the merge and redirect seems the way to go for the time being. I would do that if a few others agree, to remind myself of how to do merges.
Yeah, to me merge-and-redirect seems reasonable here.
Okay, made “quantity” into “quantity > history” according to the HowTo. Did not merge anything into space and quantity, since there were hardly anything to merge.
While I was at it, I made unnaturally isomorphic history, too, since unnatural isomorphism supersedes it (and already had a redirect from the now-history article).
To unnatural isomorphism I guess you mean.
Thanks for pointing out. Yes, was a typo, but a harmful one (since it created a ready-made article-creater to an unnecessary non-existent article). Corrected it.
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