Seven Sketches in Compositionality - Exercise 2.68

\section*{Problem}

Find another monoidal monotone $g: \textbf{Cost} \to \textbf{Bool}$ different from the one defined in Eq. 2.66.

\section*{Answer}

Consider $$g(x) = \begin{cases} \mathtt{true} \text{ if } x < \infty,\\ \mathtt{false} \text{ if } x = \infty.\\ \end{cases}$$ We have already shown (Ex. 2.44) that g is a monoidal monotone. Thus, g can turn a Lawvere metric space into a preorder. Intuitively, the preorder relationship will be the one of "reachability": $x \leq y$ if and only if $y$ can be reached from $x$ in a finite time going at a finite velocity.

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