>**Puzzle 92.** What's a \\(\mathbf{Cost}\\)-functor?

**Definition.** Let \\(\mathcal{X}\\) and \\(\mathcal{Y}\\) be \\(\mathbf{Cost}\\)-categories. A \\(\mathbf{Cost}\\)-functor from \\(\mathcal{X}\\) to \\(\mathcal{Y}\\), denoted \\(F\colon\mathcal{X}\to\mathcal{Y}\\), is a function

\[ F\colon\mathrm{Ob}(\mathcal{X})\to \mathrm{Ob}(\mathcal{Y}) \]

such that

\[ \mathcal{d}(x,x') \leq \mathcal{d'}(F(x),F(x')) \]

for all \\(x,x' \in\mathrm{Ob}(\mathcal{X})\\).

I would call this is a non-contracting map, since the generalized distance or cost metric may only stay the same or expand.

**Definition.** Let \\(\mathcal{X}\\) and \\(\mathcal{Y}\\) be \\(\mathbf{Cost}\\)-categories. A \\(\mathbf{Cost}\\)-functor from \\(\mathcal{X}\\) to \\(\mathcal{Y}\\), denoted \\(F\colon\mathcal{X}\to\mathcal{Y}\\), is a function

\[ F\colon\mathrm{Ob}(\mathcal{X})\to \mathrm{Ob}(\mathcal{Y}) \]

such that

\[ \mathcal{d}(x,x') \leq \mathcal{d'}(F(x),F(x')) \]

for all \\(x,x' \in\mathrm{Ob}(\mathcal{X})\\).

I would call this is a non-contracting map, since the generalized distance or cost metric may only stay the same or expand.