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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?

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).