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\$$.