Matthew wrote:

> Are we allowed to relax the definition of a \\(\mathcal{V}\\)-functor?

I know what you're thinking, but one important lesson from category theory is that most systematic definition of a map between mathematical gadgets is usually a map that preserves all their structure. Other definitions of map can be useful, and even tremendously important - but they're not "systematic", so it's harder to apply category theory to study them.

So, the most systematic concept of a map between metric spaces is _not_ a continuous map! Since a metric space \\(X\\) is equipped with a distance function \\(d: X \times X \to \mathbb{R}\\), the most natural concept of a map between metric spaces is an **isometry**: a map that preserves distances.

The concept of continuous map is natural for _topological spaces_. When we're studying continuous maps between metric spaces, we're secretly applying a "forgetful functor" to turn these metric spaces into topological spaces, and then looking at maps between _those_.

This whole story has an analogue for Lawvere metric spaces, which I'm trying to probe with Puzzle 92.

The concept of \\(\mathcal{V}\\)-functor, introduced above, cleverly slips an inequality into the game. One might have tried demanding

\[ \mathcal{X}(x,x') = \mathcal{X'}(F(x),F(x')) \]

and this would change the answers to "What is a \\(\mathbf{Bool}\\)-functor?" and "What is a \\(\mathbf{Cost}\\)-functor?" in interesting and instructive ways. It turns out that

\[ \mathcal{X}(x,x') \le \mathcal{X'}(F(x),F(x')) \]

is, in general, a wiser choice... as becomes apparent when we proceed deeper into category theory.

> Are we allowed to relax the definition of a \\(\mathcal{V}\\)-functor?

I know what you're thinking, but one important lesson from category theory is that most systematic definition of a map between mathematical gadgets is usually a map that preserves all their structure. Other definitions of map can be useful, and even tremendously important - but they're not "systematic", so it's harder to apply category theory to study them.

So, the most systematic concept of a map between metric spaces is _not_ a continuous map! Since a metric space \\(X\\) is equipped with a distance function \\(d: X \times X \to \mathbb{R}\\), the most natural concept of a map between metric spaces is an **isometry**: a map that preserves distances.

The concept of continuous map is natural for _topological spaces_. When we're studying continuous maps between metric spaces, we're secretly applying a "forgetful functor" to turn these metric spaces into topological spaces, and then looking at maps between _those_.

This whole story has an analogue for Lawvere metric spaces, which I'm trying to probe with Puzzle 92.

The concept of \\(\mathcal{V}\\)-functor, introduced above, cleverly slips an inequality into the game. One might have tried demanding

\[ \mathcal{X}(x,x') = \mathcal{X'}(F(x),F(x')) \]

and this would change the answers to "What is a \\(\mathbf{Bool}\\)-functor?" and "What is a \\(\mathbf{Cost}\\)-functor?" in interesting and instructive ways. It turns out that

\[ \mathcal{X}(x,x') \le \mathcal{X'}(F(x),F(x')) \]

is, in general, a wiser choice... as becomes apparent when we proceed deeper into category theory.