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Hi All!
My name is Dean. I am new here- so nice to meet you all! I was glad to join since it seems many of the people here are those whose posts I enjoy reading all around nlab, mathoverflow, n-category cafe, and the like.
I’m looking for some clarification on several questions about intensive and extensive properties. These terms have been around for a while in physics, but Lawvere gave them a categorical meaning in his paper “Categories of Space and of Quantity”. This is an overall theme in his work, towards foundation for continuum physics without analytic complications. In physics, an extensive property (like mass or volume) is a property of a physical system that grows with the size of the system, and an intensive property (like density) is essentially one which can be defined locally.
If we think of the domain space as a category of spaces, we can think of an extensive property as a function which sends disjoint unions to sums. This matches the intuition that it grows with the size of the body. As I understand it, Lawvere generalized the target of this function to a linear category (one where coproduct and product are isomorphic - this puts a monoid structure on the hom sets). This is a sort of categorification so that product/coproduct act like sum. An intensive property is defined as a contravariant functor which sends coproducts to products, and where the target category has a monoidal structure. Actually, don’t quote me on that, since I am not confident in my reading of Lawvere’s philosophical terminology. This is all explained in “Categories of Space and Quantity”.
The example I like to think of is de Rham cohomology (intensive) and singular homology (extensive) on manifolds. These have a pairing, so that they integrate against each other, and cohomology has a ring structure (which I am thinking corresponds to the monoidal/multiplicative structure Lawvere speaks of).
Another example comes from Gelfand+duality. The (contravariant) functor $\text{CH} \rightarrow \text{top-} \mathbb{R} \text{-alg}$ from compact hausdorff spaces to topological $\mathbb{R}$-algebras sending a topological space $X$ to $[X, \mathbb{R}]_{\text{top}}$ (functions of topological spaces) is an intensive property, and the (covariant) functor $\text{CH} \rightarrow \text{top-} \mathbb{R} \text{-mod}$ sending $X$ to $[[X, \mathbb{R}]_{\text{top}}, \mathbb{R}]_{\mathbb{R} \text{-mod} }$ is an extensive property. Note that $[[X, \mathbb{R}]_{\text{top}}, \mathbb{R}]_{\mathbb{R} \text{-mod} }$ is a certain set of measures on $X$. We can then integrate a measure against a function.
My question is, is there any reason why we would only use linear categories in the targets here? I am probably interested in keeping this “product preserving” condition, lest we be considering literally all functors. But say we replace the target with Set or some distributive category again. Would we perhaps get some concept similar to space (for the contravariant case) and algebraic system (for the covariant case). I had in mind finite product preserving functors from CartSp to Set. This is covariant, yet product preserving and not coproduct preserving, so alas I suspect I am drawing connections where there aren’t any.
Another question: Lawvere also created a theory of intensive/extensive quality. See here quality+type. For an example from physics, I think of liquid/solid/gas (qualities) vs. temperature/pressure (quantities). These seem to stand in a sort of duality, though I can’t see if it is reflected at the formal level. Thoughts?
I would love any references on this topic, by the way!
Hi Dean, just a comment for the moment that we have a page intensive or extensive quantity, if you haven’t come across it yet. And there’s an associated discussion page here.
See also another nForum discussion here.
Thanks, David, I’ll check all those out. Should I move this to the chat corresponding to “intensive or extensive quantity”?
Also, I think what I was looking for may be “extensive and intensive quality”, but I have to read Lawvere’s 2007 paper “Axiomatic Cohesion” to see.
Since the name intensive or extensive quantity didn’t match that other discussion thread (’or’ rather than ’and’), I’ve just generated a new one, so now any commented modification to the page appears there. That would be a better place to talk.
Note that my second link in #2 is talking about qualities.
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