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# Exercise 52 - Chapter 2

edited June 2018

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 $$( \mathbb{R} \ge_0 , \ge, 0, +)$$?

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} \$$.