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Exercise 52 - Chapter 4

What are identity elements in Lawvere metric spaces (that is, Cost-categories)? How do we interpret this in terms of distances?

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  • 1.

    In a Lawvere metric space, \(I=0\). So, an identity element is a map \(\mathrm{id}_x:0\to\mathcal{X}(x,x)\). In terms of distances, this means that \(0\geq d(x,x)\) (since \(0\) is the smallest element of \([0,\infty]\), this means \(d(x,x)=0\) (this was already observed in the text between the definition of a Lawvere metric space as a Cost-category and the example of \(\mathbb{R}\) as a Lawvere metric space, in chapter 2), which is the statement of definiteness for a Lawvere metric).

    Comment Source:In a Lawvere metric space, \\(I=0\\). So, an identity element is a map \\(\mathrm{id}_x:0\to\mathcal{X}(x,x)\\). In terms of distances, this means that \\(0\geq d(x,x)\\) (since \\(0\\) is the smallest element of \\([0,\infty]\\), this means \\(d(x,x)=0\\) (this was already observed in the text between the definition of a Lawvere metric space as a **Cost**-category and the example of \\(\mathbb{R}\\) as a Lawvere metric space, in chapter 2), which is the statement of definiteness for a Lawvere metric).
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