John Baez wrote:

>Keith: _don't forget the fiendish reversal of inequality in the definition of \$$\mathbf{Cost}\$$._

Luckily, such a fix is easy.

>**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

>$\mathcal{d}(x,x') \geq \mathcal{d'}(F(x),F(x'))$

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

>I would call this is a ***non-expanding*** map, since the generalized distance or cost metric may only stay the same or ***contract***.