>**Puzzle 89.** Figure out exactly what a \\(\mathbf{Cost}\\)-category is, starting with the definition above. Again, what matters is not your final answer so much as your process of deducing it!

>Hint: given a \\(\mathbf{Cost}\\)-category \\(\mathcal{X}\\), write \\(\mathcal{X}(x,y)\\) as \\(d(x,y)\\) and call it the **distance** between the objects \\(x,y \in \mathrm{Ob}(\mathcal{X})\\). The axioms of an enriched category then say interesting things about "distance".

Both the book (on page 53) and [Simon Willerton, on his Catester videos on enriched categories](https://www.youtube.com/watch?v=be7rx29eMr4&list=PL398472E8DD6B73C8), spill the beans that such a \\(\mathbf{Cost}\\)- or \\( \mathbb{R}_+ \\)-category is a metric space.

Now I'm not sure if *proof by book* or *proof by Catster* will count, but you did want us to get creative with the proving process.

Anyways, joking aside... For those wanting to give a proof, I would recommend reading *'Seven Sketches'* and watching that video playlist.

>Hint: given a \\(\mathbf{Cost}\\)-category \\(\mathcal{X}\\), write \\(\mathcal{X}(x,y)\\) as \\(d(x,y)\\) and call it the **distance** between the objects \\(x,y \in \mathrm{Ob}(\mathcal{X})\\). The axioms of an enriched category then say interesting things about "distance".

Both the book (on page 53) and [Simon Willerton, on his Catester videos on enriched categories](https://www.youtube.com/watch?v=be7rx29eMr4&list=PL398472E8DD6B73C8), spill the beans that such a \\(\mathbf{Cost}\\)- or \\( \mathbb{R}_+ \\)-category is a metric space.

Now I'm not sure if *proof by book* or *proof by Catster* will count, but you did want us to get creative with the proving process.

Anyways, joking aside... For those wanting to give a proof, I would recommend reading *'Seven Sketches'* and watching that video playlist.