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# Exercise 65 - Chapter 2

edited June 2018
1. Find another monoidal functor $$g : \textbf{Cost} \rightarrow \textbf{Bool}$$ different from the one defined in Eq. (2.63).

2. Using Construction 2.61, both your monoidal functor $$g$$ and the functor $$f$$ in Eq. (2.63) can be used to convert a Lawvere metric space into a preorder. Find a Lawvere metric space $$\mathcal{X}$$ on which they give different answers, $$\mathcal{X}_f \ne \mathcal{X}_g$$ .

This is the second quasi-inverse function from the exercise 2.41: $$g(x)= \begin{cases} \mathtt{true}, & \text{if x < \infty}.\\\\ \mathtt{false}, & \text{if  x = \infty}. \end{cases}$$ Both of them map $$0$$ to $$\mathtt{true}$$ and $$\infty$$ to $$\mathtt{false}$$, but everything else to a different value. So to find a metric space where they give different answers, we need one of the distances $$d(a, b)$$ to be $$\neq 0$$ and $$\neq \infty$$. Take a Lawvere metric space $$\mathcal{X}$$ with two points $$a$$ and $$b$$ and define $$d(a, b) = 4$$.
This gives $$\mathcal{X}_f \neq \mathcal{X}_g$$.
Comment Source:This is the second quasi-inverse function from the exercise 2.41: $g(x)= \begin{cases} \mathtt{true}, & \text{if x < \infty}.\\\\ \mathtt{false}, & \text{if  x = \infty}. \end{cases}$ Both of them map \$$0 \$$ to \$$\mathtt{true} \$$ and \$$\infty \$$ to \$$\mathtt{false} \$$, but everything else to a different value. So to find a metric space where they give different answers, we need one of the distances \$$d(a, b) \$$ to be \$$\neq 0 \$$ and \$$\neq \infty \$$. Take a Lawvere metric space \$$\mathcal{X} \$$ with two points \$$a \$$ and \$$b \$$ and define \$$d(a, b) = 4 \$$. This gives \$$\mathcal{X}_f \neq \mathcal{X}_g \$$.