Is this really a categorical approach to integration?

Solution 1:

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 $L^p$ spaces. There are two main theorems. The first states that the Banach space $L^p[0,1]$, equipped with a small amount of extra structure, is initial as such. The second states that the $L^p$ 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 $\ell^p$ and $c_0$, as well as the functor $L^2$ 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.