>**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.