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.