@John - I was looking at your old [Network Theory III](http://math.ucr.edu/home/baez/networks_oxford/networks_entropy.pdf) notes and I saw a familiar pattern:

> Given finite sets \\(X\\) and \\(Y\\) , a stochastic map \\(f : X \rightsquigarrow Y\\) assigns a

> real number \\(f\_{yx}\\) to each pair \\(x \in X\\), \\(y \in Y\\) in such a way that for

> any \\(x\\), the numbers \\(f\_{yx}\\) form a probability distribution on Y .

> We call \\(f\_{yx}\\) the probability of \\(y\\) given \\(x\\).

>

> So, we demand:

>

> - \\(f\_{yx} \geq 0\\) for all \\(x ∈ X\\) \\(y ∈ Y\\)

> - \\(\displaystyle{\sum\_{y\in Y} f\_{yx} = 1} \\) for all \\(x \in X\\)

>

> We can compose stochastic maps \\(f : X \rightsquigarrow Y\\) and \\(g : Y \rightsquigarrow Z\\) by

matrix multiplication:

>

> \[ (g \circ f )_{zx} = \sum_{y\in Y} g_{zy} f_{yz}\]

>

> and get a stochastic map \\(g ◦ f : X \rightsquigarrow Z\\).

>

> We let \\(\mathtt{FinStoch}\\) be the category with

>

> - finite sets as objects,

> - stochastic maps \\(f : X \rightsquigarrow Y\\) as morphisms.

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?

> Given finite sets \\(X\\) and \\(Y\\) , a stochastic map \\(f : X \rightsquigarrow Y\\) assigns a

> real number \\(f\_{yx}\\) to each pair \\(x \in X\\), \\(y \in Y\\) in such a way that for

> any \\(x\\), the numbers \\(f\_{yx}\\) form a probability distribution on Y .

> We call \\(f\_{yx}\\) the probability of \\(y\\) given \\(x\\).

>

> So, we demand:

>

> - \\(f\_{yx} \geq 0\\) for all \\(x ∈ X\\) \\(y ∈ Y\\)

> - \\(\displaystyle{\sum\_{y\in Y} f\_{yx} = 1} \\) for all \\(x \in X\\)

>

> We can compose stochastic maps \\(f : X \rightsquigarrow Y\\) and \\(g : Y \rightsquigarrow Z\\) by

matrix multiplication:

>

> \[ (g \circ f )_{zx} = \sum_{y\in Y} g_{zy} f_{yz}\]

>

> and get a stochastic map \\(g ◦ f : X \rightsquigarrow Z\\).

>

> We let \\(\mathtt{FinStoch}\\) be the category with

>

> - finite sets as objects,

> - stochastic maps \\(f : X \rightsquigarrow Y\\) as morphisms.

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?