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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 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.

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    $\begingroup$ Reading the first sentence of the paper certainly suggests that it is what you're looking for. An integral can be considered as a functional, i.e. a function which takes (a certain class of) functions as input and spits out numbers. $\endgroup$
    – Simon Rose
    Commented Mar 26, 2014 at 19:45
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    $\begingroup$ If you work out the sketch of the details of what they're actually doing, you should post it as an answer. I feel like I'd have an easier time trying to rederive their work myself than trying to understand it from the paper. :( $\endgroup$
    – user14972
    Commented Mar 27, 2014 at 6:28
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    $\begingroup$ Best guess starting point: think of the forgetful functor from $\mathbf{R-Vect}$ to $\mathbf{Set}$. This has a left adjoint, the free functor $F$ which assigns to a set a vector space whose basis is that set. Now let $B$ be the underlying set of a boolean algebra. Functions $B \to UV$ (I think the paper calls these to be $V$-valued "measures") are in one-to-one correspondence with linear maps $FB \to V$. Now, think about whether $F$ transfers the boolean algebra structure on $B$ to be a boolean algebra structure on $FB$. Hopefully, the space of integrable functions appears naturally now. $\endgroup$
    – user14972
    Commented Mar 27, 2014 at 9:12
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    $\begingroup$ Let's suppose that paper does what it says and defines integration in a category-theoretic manner. That's fun, but to do any work with that definition we need a version of the fundamental theorem of calculus. Until I see a category-theoretic proof of that I'll remain only politely but distantly interested. $\endgroup$
    – Gunnar Þór Magnússon
    Commented Mar 27, 2014 at 13:56
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    $\begingroup$ @Shaun: I just came across A SHEAF THEORETIC APPROACH TO MEASURE THEORY which you might also find interesting $\endgroup$
    – user14972
    Commented Apr 17, 2014 at 16:57

1 Answer 1

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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.

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