Options

Exercise 65 - Chapter 2

edited June 2018 in Exercises
  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, \( \mathcal{X}_f \ne \mathcal{X}_g \) .

Previous Next

Comments

  • 1.
    edited June 2018

    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 \\).
Sign In or Register to comment.