Lawvere's 'Categories of space and of quantity" - the projection formula

Question

I'm trying to read bits and pieces of W. Lawvere's Categories of Space and of Quantity, and, as usual have lots of questions. Page numbers will refer to the number printed on the corners of the pages.

The article is pretty dense (for me), so I won't paraphrase much. An extensive quantity is, broadly speaking, a "quantity of space", and an intensive quantity is a "ratio" between extensive ones. For instance, mass and volume are extensive (measures), while density is intensive (function). [See last paragraph of page 18.]

For a variety of reasons, "ratios" are harder to formalize than "multiplications", so the author posits the relationship between intensive and extensive quantities should be seen as an action of intensive quantities on extensive ones which produce more extensive ones. For instance, density acting on volume to give mass.

An excerpt from the bottom of page 21 and the top of page 22 (boldface is added by me):

$\;\;\;$The common spatial base of extensive and intensive quantities also supports the relation between the two, which is that the intensives act on the extensives. For example, a particular density function acts on a particular volume distribution to produce a resulting mass distribution. Thus it should be possible to "multiply" a given extensive quantity on a certain space by an intensive quantity (of appropriate type) on the same space to produce another extensive quantity on the same space. The definite integral of the intensive quantity "with respect to" the first extensive quantity is defined to be the total of this second resulting extensive quantity. This action (or "multiplication") of the contravariant on the covariant satisfies bilinearity and also satisfies, with respect to the multiplicative structure within the intensive quantities and along any inducing spatial map, an extremely important strong homogeneity condition which so far has carried different names in different fields: in algebraic topology this homogeneity is called the "projection formula", in group representation theory it lies at the base of "Frobenius reciprocity", in quantum mechanics it is called "covariance" or the "canonical commutation relation", while in subjective logic it is often submerged into a side condition on variables for the validity of the rule of inference for existential quantification when applied to a conjunction.

In particular this brings to mind these two questions on (the origins of) projection formulas.

My question is simple - can anyone explain, illustrate, simplify, and/or give details about this? (The concept of "total" is discussed on page 19.) In particular:

1. In what sense is the projection formula a "homogeneity" condition?
2. What is the intuitive geometric meaning of the projection formula at this great and conceptual generality?

Of course all additional stories, references, details, examples, anecdotes, warnings, etc are welcome!

Answer 1

One way to interpret the projection formula as a "homogeneity" condition is by thinking of it as saying that the push-forward is a module map, i.e. it preserves an action. I don't know if this is what is meant by Lawvere, however.

I will be vague and general. We start with some category of things (maybe spaces or groups or whatever). To each object $X$ we associate a thing $M(X)$ which might be a category or an abelian group or a vector space (maybe a category of sheaves, or the representation ring, or cohomology) and $M(X)$ has a product $\times$ making it into a monoid in a suitable sense (a monoidal category or a ring or an algebra).

Given a morphism $f:X\to Y$ we get a pull-back $f^\ast\colon M(Y) \to M(X)$ and this respects the products in $M(X)$ and $M(Y)$, so $f^\ast$ is a morphism of monoids: $$f^\ast(y_1\times y_2) = f^\ast y_1 \times f^\ast y_2.$$

Using this fact we can make $M(X)$ into an $M(Y)$-module: $y\cdot x:= (f^\ast y) \times x$. On the other hand, being a monoid $M(Y)$ is also an $M(Y)$ module using the regular action: $y\cdot y' := y\times y'$.

Associated to $f\colon X\to Y$ we also have the push-forward $f_\ast\colon M(X)\to M(Y)$. The projection formula says $$f_\ast(f^\ast y \times x)= y \times f_\ast x$$ in other words the projection formula says that $f_\ast$ is a morphism of $M(Y)$-modules: $$f_\ast( y \cdot x)= y \cdot f_\ast x.$$