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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeMay 20th 2011

I’m sure I’ve asked about the intensive/extensive distinction before, (yes here), but how does it fit in with the discussion on the page Space and Quantity? Quantity as a copresheaf there seems to concern extensive quantity. My query was sparked off by reading Anders Koch’s Calculus of extensive quantities, where he writes

Quantities of a given type distributed over a given space (say distributions of smoke in a given room) may often be added, and multiplied by real scalars – ideally, they form a real vector space. Lawvere stressed that the dependence of such vector spaces on the space over which the quantities in question are distributed, should be taken into account; in fact, the dependence is functorial. The viewpoint leads to a distinction between two kinds of quantities: the functorality may be covariant, or it may be contravariant: In this context, the covariant quantity types are called extensive quantities, and the contravariant ones intensive quantities. This usage is an attempt to put mathematical precision into the use of these terms in classical philosophy of physics. Mass distribution is an extensive quantity; mass density is an intensive one. Lawvere observed that extensive and intensive quantites often come in pairs, with a definite pattern of mutual relationship, like the homology and cohomology functors on the category of topological spaces.

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?

Also on that page, where it has

(That this is an adjunction can be understood as a special case of abstract Stone duality induced by a dualizing object.),

the intention is to refer to Paul Taylor’s version of Stone duality?

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeMay 20th 2011
• (edited May 20th 2011)

Maybe that’s what’s meant by “Space of Quantity” in this article. Unfortunately it first appears on a page (p. 19) that can’t be read, but there is something on p. 20. It sounds like it’s the nature of the target category that matters. Contravariant to a linear category - intensive quality; contravariant to a distributive category - generalized space? Shouldn’t there then be a fourth thing: covariant to a distributive category - extensive quantity; contravariant to a linear category - ????

Limited access is annoying, but I have no access to ’Unity and identity of opposites in calculus and physics’, Applied Categorical Structures, Volume 4, Numbers 2-3 / June 1996.

• CommentRowNumber3.
• CommentAuthorTobyBartels
• CommentTimeMay 24th 2011

Yes, Abstract Stone Duality is Paul Taylor’s theory. One of his key points (at least as he sees it) is the ambimorphic nature of Sierpinski space.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeMay 24th 2011

But why is Taylor’s theory being singled out in Space and Quantity: Details when it just speaks of a duality between generalized spaces and generalized quantities?

(That this is an adjunction can be understood as a special case of abstract Stone duality induced by a dualizing object.)

Isn’t ’abstract’ here being used in the sense of ’more general than ordinary’ Stone duality?

• CommentRowNumber5.
• CommentAuthorTobyBartels
• CommentTimeMay 24th 2011

Oh OK, I don’t know what’s intended by the writer there (even if I wrote it), just that the adjunction is indeed important to Taylor.

• CommentRowNumber6.
• CommentAuthorDean
• CommentTimeNov 25th 2019

Here is the part that was missing on page 19: “A further determination is suggested by the idea “space of quantity” which lies at the base of (not only cartesian coordinatizing but also) calculus of variations and functional analysis” (Page 19). So an example would be a topological module.