# Is this really a categorical approach to integration?

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“.

The Abstract:

“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.”

The Problem:

I’d like to find a way to use category theory to define or think about integration, at least over $$Rk\mathbb{R}^k$$, 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 . . .

Does the pdf (or whatever) achieve anything like this? Does its “integration” really mean integration (like “area under the curve” and so on) or is it a false friend, as in “integral domain”?

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.

We identify simple universal properties that uniquely characterize the Lebesgue $$LpL^p$$ spaces. There are two main theorems. The first states that the Banach space $$Lp[0,1]L^p[0,1]$$, equipped with a small amount of extra structure, is initial as such. The second states that the $$LpL^p$$ 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 $$ℓp\ell^p$$ and $$c0c_0$$, as well as the functor $$L2L^2$$ taking values in Hilbert spaces.