Keith wrote:

> Isn't puzzle 185 asking for an algorithm that computes shortest possible paths over the profunctor \\(\Phi\\)?

No, I'm a mathematician: if I ever want an algorithm, I'll ask for it loud and clear!

> **Puzzle 185.** In general, a \\(\mathbf{Cost}\\)-enriched profunctor \\(\Phi : C \nrightarrow D\\) is defined to be a [\\(\mathbf{Cost}\\)-enriched functor](

> \[ \Phi : C^{\text{op}} \times D \to \mathbf{Cost} \]

> This is a function that assigns to any \\(c \in C\\) and \\(d \in D\\) a cost \\(\Phi(c,d)\\). However, to be a \\(\mathbf{Cost}\\)-enriched functor it needs to obey some properties! What are these properties? What do they mean in terms of trips between cities?

I'm asking:

A) What properties must a function \\(\Phi : C^{\text{op}} \times D \to \mathbf{Cost}\\) obey for it to be a \\(\mathbf{Cost}\\)-enriched functor?

Simon has pointed out a problem here, which is that I may never have come out and said how \\(\mathbf{Cost}\\) becomes a \\(\mathbf{Cost}\\)-enriched category! Without knowing this my question is not really well-posed.

So, how does \\(\mathbf{Cost}\\) becomes a \\(\mathbf{Cost}\\)-enriched category?

A \\(\mathbf{Cost}\\)-enriched category \\( \mathcal{C}\\) has a cost \\(\mathcal{C}(x,y)\\) for any two objects \\(x,y\\), so here we need a cost for any two costs \\(x,y\\). And here's what it is: it's \\(y-x\\), or zero if this is less than zero.

Simon has explained how we get this from general principles. I will say more about this later - I was always planning to explain it; I just forget I needed to explain it for this puzzle!

And given the answer to part A), I'm asking

B) What does this answer mean in terms of our favorite example, where

\[ \Phi : C^{\text{op}} \times D \to \mathbf{Cost} \]

describes the cost of travelling from a city \\(c \in C\\) to a city \\(d \in D\\)?