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Exercise 52 - Chapter 2

Consider the symmetric monoidal preorder \( ( \mathbb{R} \ge_0 , \ge, 0, +) \), which is almost the same as \( \textbf{Cost} \), except it does not include \( \infty \). How would you characterize the difference between a Lawvere metric space and a category enriched in \( ( \mathbb{R} \ge_0 , \ge, 0, +) \)?

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  • 1.
    edited June 2018

    A Lawvere metric space is a set of objects between which we have some sort of notion of a distance. It can be infinite which intuitively means that no matter the effort, sometimes you can't get from a point \( a \) to the point \( b \). On the other hand, a category \( \mathcal{X} \) enriched in \( (\mathbb{R_{\geq_0}}, \geq, 0, +) \) defines a finite distance between every two of its objects. This means that we can get from point \( a \) to point \( b \), for any two points \( a, b \in \mathcal{X} \).

    Comment Source:A Lawvere metric space is a set of objects between which we have some sort of notion of a distance. It can be infinite which intuitively means that no matter the effort, sometimes you can't get from a point \\( a \\) to the point \\( b \\). On the other hand, a category \\( \mathcal{X} \\) enriched in \\( (\mathbb{R_{\geq_0}}, \geq, 0, +) \\) defines a finite distance between every two of its objects. This means that we _can_ get from point \\( a \\) to point \\( b \\), for any two points \\( a, b \in \mathcal{X} \\).
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