This is a continuation of this question about the paper Categories of Space and of Quantity by W. Lawvere.
As intuitively clear by the very broad (and tentative) definitions suggested by W. Lawvere, intensive quantities act on extensive quantities by a sort of "multiplication".
Here's the last paragraph of page 22 (boldface was added by me).
How can systems of extensive and intensive quantities, with action of the latter on the former, be realized on various distributive categories which mathematically arise? As mentioned above, the intensive quantities are often representable (indeed more often than commonly noticed, for example differential forms can be represented via the "fractional exponentiation" which exists in certain gros toposes). An important class of extensive quantities can be identified with the (smooth linear) functionals (with codomain a fixed linear space such as that of constant scalars) on the given intensive quantities, i.e. a distribution may sometimes be determined by the ensemble of all definite integrals (with respect to it) of all appropriate intensive quantities. This identification, supported in a particular context by the classical Riesz representation theorem (and in the homotopical context of section 2 below, by the universal coefficient theorem), contributed to the flourishing of functional analysis, but perhaps also distracted attention from the fact that extensive quantities are at least as basic as the intensive ones. At any rate, the fundamental projection formula/canonical commutation relation is automatic for those extensive quantities which can be identified as functionals on the intensive ones; here the action is defined in terms of the integral of the multiplication of intensive quantities.
How to understand this analogy between the Riesz representation theorem and the universal coefficient theorem? This seems very interesting conceptually but I can't make any sense of it.
In the case of the Riesz theorem, I think the extensive quantity is 'measure' (volume), and it is identified with the integrals of its density, which makes perfect sense. How and what is the universal coefficient identifying?
