Is this *really* a categorical approach to *integration*?
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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.
integration measure-theory soft-question category-theory boolean-algebra
edited Nov 17 at 9:55
Viktor Glombik
489321
asked Mar 26 '14 at 19:20
Shaun
8,006113577
|
show 13 more comments
up vote
102
down vote
favorite
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.
integration measure-theory soft-question category-theory boolean-algebra
edited Nov 17 at 9:55
Viktor Glombik
489321
asked Mar 26 '14 at 19:20
Shaun
8,006113577
7
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
4
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. :(
– Hurkyl
Mar 27 '14 at 6:28
3
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.
– Hurkyl
Mar 27 '14 at 9:12
11
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
3
@Shaun: I just came across A SHEAF THEORETIC APPROACH TO MEASURE THEORY which you might also find interesting
– Hurkyl
Apr 17 '14 at 16:57
|
show 13 more comments
up vote
102
down vote
favorite
up vote
102
down vote
favorite
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.
integration measure-theory soft-question category-theory boolean-algebra
edited Nov 17 at 9:55
Viktor Glombik
489321
asked Mar 26 '14 at 19:20
Shaun
8,006113577
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.
integration measure-theory soft-question category-theory boolean-algebra
integration measure-theory soft-question category-theory boolean-algebra
edited Nov 17 at 9:55
Viktor Glombik
489321
asked Mar 26 '14 at 19:20
Shaun
8,006113577
edited Nov 17 at 9:55
Viktor Glombik
489321
asked Mar 26 '14 at 19:20
Shaun
8,006113577
edited Nov 17 at 9:55
Viktor Glombik
489321
edited Nov 17 at 9:55
Viktor Glombik
489321
edited Nov 17 at 9:55
Viktor Glombik
489321
489321
asked Mar 26 '14 at 19:20
Shaun
8,006113577
asked Mar 26 '14 at 19:20
Shaun
8,006113577
asked Mar 26 '14 at 19:20
Shaun
8,006113577
8,006113577
7
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
4
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. :(
– Hurkyl
Mar 27 '14 at 6:28
3
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.
– Hurkyl
Mar 27 '14 at 9:12
11
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
3
@Shaun: I just came across A SHEAF THEORETIC APPROACH TO MEASURE THEORY which you might also find interesting
– Hurkyl
Apr 17 '14 at 16:57
|
show 13 more comments
7
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
4
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. :(
– Hurkyl
Mar 27 '14 at 6:28
3
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.
– Hurkyl
Mar 27 '14 at 9:12
11
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
3
@Shaun: I just came across A SHEAF THEORETIC APPROACH TO MEASURE THEORY which you might also find interesting
– Hurkyl
Apr 17 '14 at 16:57
7
7
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
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
4
4
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. :(
– Hurkyl
Mar 27 '14 at 6:28
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. :(
– Hurkyl
Mar 27 '14 at 6:28
3
3
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.
– Hurkyl
Mar 27 '14 at 9:12
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.
– Hurkyl
Mar 27 '14 at 9:12
11
11
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
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
3
3
@Shaun: I just came across A SHEAF THEORETIC APPROACH TO MEASURE THEORY which you might also find interesting
– Hurkyl
Apr 17 '14 at 16:57
@Shaun: I just came across A SHEAF THEORETIC APPROACH TO MEASURE THEORY which you might also find interesting
– Hurkyl
Apr 17 '14 at 16:57
|
show 13 more comments
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7
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
4
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. :(
– Hurkyl
Mar 27 '14 at 6:28
3
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.
– Hurkyl
Mar 27 '14 at 9:12
11
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
3
@Shaun: I just came across A SHEAF THEORETIC APPROACH TO MEASURE THEORY which you might also find interesting
– Hurkyl
Apr 17 '14 at 16:57