Jonathan wrote:

> should that be changed as well to \$$x \ge \infty\$$?

No, if we did that we'd also have to say \$$2 \ge 1\$$, etc. What Fong and Spivak are doing instead is saying that the ordering on \$$\textbf{Cost} \$$ is the usual \$$\ge \$$ on the set \$$[0,\infty]\$$, not the usual \$$\le \$$.

Yeah, I know it's confusing! It would be probably less confusing if Fong and Spivak had defined \$$\textbf{Cost} \$$ to be the monoidal poset \$$([0,\infty],\le,0) \$$ and then studied \$$\textbf{Cost}^{\text{op}}\$$-categories. But they are instead defining \$$\textbf{Cost} \$$ to be \$$([0,\infty],\ge,0) \$$.

As I've said before:

> _Category theory reduces all of mathematics to the study of arrows. The only mistake you can make with an arrow is to get it pointing the wrong way. Thus, in category theory, this is the mistake you will always make._