Matthew wrote:

> I have been wondering how this might generalize to Lawvere metric spaces.

Great! That's coming soon!

This chapter is secretly about profunctors between \\(\mathcal{V}\\)-enriched categories. When \\(\mathcal{V}\\) is \\(\mathbf{Bool}\\), we get feasibility relations between preorders... and those are an easy way to get some intuition for the general case, so I want to talk about those first. The next simplest example is \\(\mathcal{V} = \mathbf{Cost}\\), and then we get _some sort_ of maps going between [Lawvere metric spaces](https://forum.azimuthproject.org/discussion/2128/lecture-31-chapter-2-lawvere-metric-spaces/p1). It's fun to guess what those might be.

But before doing the case \\(\mathcal{V} = \mathbf{Cost}\\), I should probably explain the general theory. At least in the way Fong and Spivak do it, they assume \\(\mathcal{V}\\) is more than a symmetric monoidal preorder: they take it to be a commutative unital quantale. So I need to explain those. Maybe I should have done it sooner, because they showed up in an earlier chapter! But now is a good time.

> I have been wondering how this might generalize to Lawvere metric spaces.

Great! That's coming soon!

This chapter is secretly about profunctors between \\(\mathcal{V}\\)-enriched categories. When \\(\mathcal{V}\\) is \\(\mathbf{Bool}\\), we get feasibility relations between preorders... and those are an easy way to get some intuition for the general case, so I want to talk about those first. The next simplest example is \\(\mathcal{V} = \mathbf{Cost}\\), and then we get _some sort_ of maps going between [Lawvere metric spaces](https://forum.azimuthproject.org/discussion/2128/lecture-31-chapter-2-lawvere-metric-spaces/p1). It's fun to guess what those might be.

But before doing the case \\(\mathcal{V} = \mathbf{Cost}\\), I should probably explain the general theory. At least in the way Fong and Spivak do it, they assume \\(\mathcal{V}\\) is more than a symmetric monoidal preorder: they take it to be a commutative unital quantale. So I need to explain those. Maybe I should have done it sooner, because they showed up in an earlier chapter! But now is a good time.