Here‘s an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it’s called, “A Categorical Approach to Integration“.
“We present a general treatment of measures and integrals in certain (monoidal
closed) categories. Under appropriate conditions, the integral can be defined by a universal property, and the universal measure is at the same time a universal multiplicative measure. In the multiplicative case, this assignment is right adjoint to the formation of the Boolean algebra of idempotents. Now coproduct preservation yields an approach to product measures.”
I’d like to find a way to use category theory to define or think about integration, at least over Rk, ideally in some pragmatic fashion, without borrowing too heavily from some other theory of integration. So before I invest lots more time & effort than usual trying to understand the thing . . .
Please excuse my ignorance. I am trying.
NB: Yeah, it does seem to be talking about integration, but let’s go a little deeper there if possible. My first question is now highlighted. It’s still open. I’ve thrown in the soft-question tag for good measure.
Another user suggested that Tom Leinster’s The categorical origins of Lebesgue integration is relevant to this question (the link is to the arXiv). The abstract reads:
We identify simple universal properties that uniquely characterize the Lebesgue Lp spaces. There are two main theorems. The first states that the Banach space Lp[0,1], equipped with a small amount of extra structure, is initial as such. The second states that the Lp functor on finite measure spaces, again with some extra structure, is also initial as such. In both cases, the universal characterization of the integrable functions produces a unique characterization of integration. Using the universal properties, we develop some of the basic elements of integration theory. We also state universal properties characterizing the sequence spaces ℓp and c0, as well as the functor L2 taking values in Hilbert spaces.
I am not an expert on category theory by any means, but the abstract is clearly referencing the Lebesgue theory, which is a broad framework for integration (in the sense of “finding the area under a curve”). Thus it appears to me that this paper is highly relevant to the question asked.