**Definition.** A **\\(\mathbf{Cost}\\)- or \\( \mathbb{R}_+ \\)-category**, say \\(\mathcal{d}\\), consists of:

1. a set of **objects**, \\(\mathrm{Ob}(\mathcal{d})\\), and

2. an element \\(\mathcal{d}(x,y) \in \mathbb{R}_+\\) for any pair of objects \\(x,y\\).

such that:

a) \\( 0\leq\mathcal{d}(x,x) \\) for any object \\(x\\), and

b) \\( \mathcal{d}(x,y) + \mathcal{d}(y,z)\leq\mathcal{d}(x,z) \\) for any objects \\(x,y,z\\).

(Here I'm calling my \\(\mathbf{Cost}\\)- or \\( \mathbb{R}_+ \\)-category, \\(\mathcal{d}\\), to make it easier.)

Edit: Why does my answer feel like it's backwards?