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.