Matthew wrote:
> It is perhaps well beyond the scope of this course, but is \\(\mathtt{FinStoch}\\) a \\(\mathcal{V}\\)-enriched profunctor category? It doesn't seem like it is because \\(\sum\\) is not the same as \\(\bigvee\\). Is it some kind of related profunctor category?
Very interesting question. I don't think it's an enriched profunctor category; I think both this category and enriched profunctor categories are special cases of a more general sort of category.
\\(\mathtt{FinStoch}\\) is a subcategory of a simpler category called \\(\mathtt{Mat}(\mathbb{R})\\), where
* objects are finite sets,
* a morphism \\(f \colon X \to Y\\) is a 'matrix', namely a function \\(f \colon Y \times X \to \mathbb{R}\\),
* composition is done via matrix multiplication: we compose morphisms \\(f : X \to Y\\) and \\(g : Y \to Z\\) to get \\(g \circ f : X \to Z\\) as follows:
\[ (g \circ f )\_{zx} = \sum_{y\in Y} g\_{zy} f\_{yz} \]
This category is equivalent (but not isomorphic) to the category of finite-dimensional real vector spaces.
In fact we can define a category \\(\mathtt{Mat}(R)\\) in the exact same way whenever \\(R\\) is any ring, or even any [rig](https://en.wikipedia.org/wiki/Semiring)! A great example is \\( R = [0,\infty) \\) with the usual notions of \\(+\\) and \\(\times\\). \\(\mathtt{FinStoch}\\) is actually a subcategory of \\(\mathtt{Mat}([0,\infty))\\)
You might enjoy this exploration of \\(\mathtt{FinStoch}\\), \\(\mathtt{Mat}([0,\infty))\\) and related categories:
* John Baez, [Relative entropy (part 1)](https://johncarlosbaez.wordpress.com/2013/06/20/relative-entropy-part-1/), _Azimuth_, 20 June 2013.
This is the beginning of a long story, continued in other blog articles in that series.
But notice that any quantale gives a rig with \\(+ = \vee\\) and \\(\times = \otimes\\). So we can also create categories \\(\mathtt{Mat}(\mathcal{V})\\) when \\(\mathcal{V}\\) is a quantale! And in this case there's no need to limit ourselves to finite sets as objects! Infinite joins are well-defined, so we can say
\[ (g \circ f )\_{zx} = \bigvee_{y\in Y} g\_{zy} \otimes f\_{yz}\]
even when \\(Y\\) is infinite!
So we get a category where morphisms are possibly infinite-sized matrices with entries in a quantale \\(\mathcal{V}\\). But these are _not_ the categories of \\(\mathcal{V}\\)-enriched profunctors - do you see why? See how they're related?
So there must be some 'least common generalization' that includes everything I've been talking about, but I'm not sure what it is.
> It is perhaps well beyond the scope of this course, but is \\(\mathtt{FinStoch}\\) a \\(\mathcal{V}\\)-enriched profunctor category? It doesn't seem like it is because \\(\sum\\) is not the same as \\(\bigvee\\). Is it some kind of related profunctor category?
Very interesting question. I don't think it's an enriched profunctor category; I think both this category and enriched profunctor categories are special cases of a more general sort of category.
\\(\mathtt{FinStoch}\\) is a subcategory of a simpler category called \\(\mathtt{Mat}(\mathbb{R})\\), where
* objects are finite sets,
* a morphism \\(f \colon X \to Y\\) is a 'matrix', namely a function \\(f \colon Y \times X \to \mathbb{R}\\),
* composition is done via matrix multiplication: we compose morphisms \\(f : X \to Y\\) and \\(g : Y \to Z\\) to get \\(g \circ f : X \to Z\\) as follows:
\[ (g \circ f )\_{zx} = \sum_{y\in Y} g\_{zy} f\_{yz} \]
This category is equivalent (but not isomorphic) to the category of finite-dimensional real vector spaces.
In fact we can define a category \\(\mathtt{Mat}(R)\\) in the exact same way whenever \\(R\\) is any ring, or even any [rig](https://en.wikipedia.org/wiki/Semiring)! A great example is \\( R = [0,\infty) \\) with the usual notions of \\(+\\) and \\(\times\\). \\(\mathtt{FinStoch}\\) is actually a subcategory of \\(\mathtt{Mat}([0,\infty))\\)
You might enjoy this exploration of \\(\mathtt{FinStoch}\\), \\(\mathtt{Mat}([0,\infty))\\) and related categories:
* John Baez, [Relative entropy (part 1)](https://johncarlosbaez.wordpress.com/2013/06/20/relative-entropy-part-1/), _Azimuth_, 20 June 2013.
This is the beginning of a long story, continued in other blog articles in that series.
But notice that any quantale gives a rig with \\(+ = \vee\\) and \\(\times = \otimes\\). So we can also create categories \\(\mathtt{Mat}(\mathcal{V})\\) when \\(\mathcal{V}\\) is a quantale! And in this case there's no need to limit ourselves to finite sets as objects! Infinite joins are well-defined, so we can say
\[ (g \circ f )\_{zx} = \bigvee_{y\in Y} g\_{zy} \otimes f\_{yz}\]
even when \\(Y\\) is infinite!
So we get a category where morphisms are possibly infinite-sized matrices with entries in a quantale \\(\mathcal{V}\\). But these are _not_ the categories of \\(\mathcal{V}\\)-enriched profunctors - do you see why? See how they're related?
So there must be some 'least common generalization' that includes everything I've been talking about, but I'm not sure what it is.
Version 2.1.8p2
