The greast lower bound of a \\(\mathrm{Cost}\\)-category has a name: \\(\mathrm{min}\\).

So to take Jonathan Castello's example above, I would modify it to give,

\\[ \mathcal{X}(x, y) = \mathrm{min}\left\\{\mathcal{d}(p) \mid p \text{ is a path from } x \text{ to } y \right\\} \\]

where \\(\mathcal{d}(p) \\) is a distance function on the path \\(p\\).