**Puzzle 89:** Recall that \\(\mathbf{Cost} = \langle [0, \infty], \le, +, 0 \rangle\\) is a symmetric monoidal poset (a total order, even). Consider a \\(\mathbf{Cost}\\)-category \\(\mathcal{X}\\). This is effectively a function \\(\mathcal{X} : \mathrm{Ob}(\mathcal{X}) \times \mathrm{Ob}(\mathcal{X}) \to \mathbf{Cost}\\) for some choice of object set \\(\mathrm{Ob}(\mathcal{X})\\) obeying the following properties:

1. \\(0 \le \mathcal{X}(x, x)\\) for all \\(x \in \mathrm{Ob}(\mathcal{X})\\)

2. \\(\mathcal{X}(x, y) + \mathcal{X}(y, z) \le \mathcal{X}(x, z)\\) for all \\(x, y, z \in \mathrm{Ob}(\mathcal{X})\\).

Taking John's hint into consideration, property #2 is suggestive of the [triangle inequality](https://en.wikipedia.org/wiki/Triangle_inequality) -- but the inequality goes the wrong way! If we started with \\(\mathbf{Cost}^{op}\\), we would get the triangle inequality, and \\(\mathcal{X}(x, x) = 0\\) -- but we didn't, so we don't. (No Lawvere metric spaces for us!)

So what does this \\(\mathbf{Cost}\\)-enriched category tell us? First, going nowhere might actually cost you. Maybe you're renting. Second, it pays to take the scenic route! Even if you end up back where you started, you won't be worse off than if you'd stayed put. Shop around!

(I feel like there's a [contraction mapping](https://en.wikipedia.org/wiki/Contraction_mapping) lurking here, but I'm not sure I can tease it out.)

1. \\(0 \le \mathcal{X}(x, x)\\) for all \\(x \in \mathrm{Ob}(\mathcal{X})\\)

2. \\(\mathcal{X}(x, y) + \mathcal{X}(y, z) \le \mathcal{X}(x, z)\\) for all \\(x, y, z \in \mathrm{Ob}(\mathcal{X})\\).

Taking John's hint into consideration, property #2 is suggestive of the [triangle inequality](https://en.wikipedia.org/wiki/Triangle_inequality) -- but the inequality goes the wrong way! If we started with \\(\mathbf{Cost}^{op}\\), we would get the triangle inequality, and \\(\mathcal{X}(x, x) = 0\\) -- but we didn't, so we don't. (No Lawvere metric spaces for us!)

So what does this \\(\mathbf{Cost}\\)-enriched category tell us? First, going nowhere might actually cost you. Maybe you're renting. Second, it pays to take the scenic route! Even if you end up back where you started, you won't be worse off than if you'd stayed put. Shop around!

(I feel like there's a [contraction mapping](https://en.wikipedia.org/wiki/Contraction_mapping) lurking here, but I'm not sure I can tease it out.)