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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.

Noah Schweber
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Shaun
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    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. – Simon Rose Mar 26 '14 at 19:45
  • I suppose I should hasten to clarify that I'm not asking others to "do the understanding for me" or anything daft like that; you know what I mean, right? And thank you, @SimonRose. That's quite helpful :) – Shaun Mar 26 '14 at 20:26
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    Possibly related: [Calculus and Category theory](http://math.stackexchange.com/questions/337611/calculus-and-category-theory) (see my answer there). – Dave L. Renfro Mar 26 '14 at 20:46
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    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. :( –  Mar 27 '14 at 06:28
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    That's a big "if", @Hurkyl, but sure, okay. I'd be delighted :) – Shaun Mar 27 '14 at 08:21
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    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. –  Mar 27 '14 at 09:12
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    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. – Gunnar Þór Magnússon Mar 27 '14 at 13:56
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    Reinhard Börger ... mention that here. For searches. This paper has been discussed before, perhaps here, but I cannot find it because nothing matches in my searches. So, at least THIS one will match. (Until this comment dies.) – GEdgar Mar 27 '14 at 17:17
  • Thank you, @Hurkyl. That makes a lot of sense, except that my intuition for your last step is missing. I'm not sure why :/ – Shaun Mar 27 '14 at 22:42
  • @GunnarMagnusson. Fair enough. I respect that. Now you've got me wondering what such a theorem/proof would look like :) – Shaun Mar 27 '14 at 22:47
  • @GEdgar Good idea! I've included the title too :) – Shaun Mar 27 '14 at 22:49
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    @Shaun: Me too. Or more accurately, my intuition is "the authors found something along these lines, so if I travel those lines, I'll probably find something too". –  Mar 28 '14 at 06:41
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    @Shaun: [This nLab article on localizable measure spaces](http://ncatlab.org/nlab/show/localizable+measure) may be of interest; e.g. it gives a direct rationale of why Boolean algebras are a thing to pay attention to. Maybe [measurable locale](http://ncatlab.org/nlab/show/measurable+locale) is interesting too. –  Apr 17 '14 at 07:59
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    @Shaun: I just came across [A SHEAF THEORETIC APPROACH TO MEASURE THEORY](http://www.andrew.cmu.edu/user/awodey/students/jackson.pdf) which you might also find interesting –  Apr 17 '14 at 16:57
  • @Hurkyl Thank you! Yeah, they're all of interest. It'll take me quite a while to digest them though :D – Shaun Apr 17 '14 at 18:17
  • Related, but possibilty not satisfying: https://arxiv.org/pdf/1209.3606v3.pdf mentions how integration operators on a set can be thought as elements of its image under the endofunctor of a codensity monad (and shows how it goes for semirings). – José Siqueira Jan 07 '17 at 20:26
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    I think that Lawvere theory might be a good framework to think about calculus. (Although you would have to restrict to only integrable/differentiable functions on a given set). Basically you think of calculus as a set of rules between functions and then construct a functor to concrete meanings. I'm not explaining it well but you should check out his thesis Functorial Semantics, it's great. – edenstar Feb 13 '17 at 18:10
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    @GunnarÞórMagnússon I found [this](https://ksda.ccny.cuny.edu/PostedPapers/Keigher022015.pdf) on the fundamental theorem of calculus. – Shaun Feb 20 '17 at 10:43
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    @SimonRose An integral is not a functional, because it takes both a function **and** a set for its two arguments. – kakashi10192020 Apr 29 '19 at 05:33
  • Posted in 2014 but active just seven months ago . . . have you considered placing a bounty considering how popular it is? You do have a bit of rep to spare after all – gen-ℤ ready to perish Dec 14 '19 at 00:19
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    @FFcorp Your grammatical edits were incorrect, and I've rolled them back (OK, "a universal" vs. "an universal" is a matter of convention, like "a historical" vs. "an historical," but that's the exception). In particular, "is right adjoint to" is the correct technical phrase (and even if it weren't, since it's part of a quote it would be improper to change it without adding "[sic]"). – Noah Schweber Jun 17 '20 at 12:16
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    @gen-zreadytoperish I'm not OP but that's a great suggestion. I am offering up a sizable bounty from my own rep. – Cameron Williams Jun 17 '20 at 12:19
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    I'd say that the right thing to generalize in order to get to the fundamental calculus theorem is the radon nykodim theorem. This allows to define derivative of a measure wrt another measure using only absolute continuity, which abstract makes sense. With notations from the article, we would need a function $LB \otimes M(B, A) \to M(B, A) $ or equivalently, since $LB$ represents the space of measures $M(B,A)$ in the codomain, $ LB \otimes Hom(LB, A) \to Hom(LB, A) $. This corresponds to multiply a measure by a function. The theorem would then state as: – Andrea Marino Dec 19 '20 at 13:45
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    If a measure $\nu$ is absolute continuous with respect to $\mu$ , there exist an element $f$ of LB such that $ \mu = \int m(f, \nu) $. There could also be a local version of fundamental theorem of calculus, aka Lebesgue differentiation, that does not make use of derivatives. However, all in all, I think that this perspective does not shed any light on integration theory, which is perfectly finely understood by the classical geometric theory of measures. Personally, I am fine with that. – Andrea Marino Dec 19 '20 at 14:01

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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.

Xander Henderson
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