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 $\mathbb{R}^k$, 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 meanintegration(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.