Let $V$ be the monoidal category $[0,\infty)$ (as a poset) with $+$ and $0$. Lawvere shows that $V$-enriched categories are a more natural generalisation of the notion of a metric space (note no symmetry). Where it turns out many classical theorems about metric spaces (and similar structures like ultrametric spaces) are simply special cases of certain theorems in enriched category theory.

Now one of the objects one uses when working with metric spaces are balls. Let $C$ be a $V$-enriched category.

~~Choosing some $v\in V$ we can define the ball centered at some point $x\in C$. As the 'set' of points in $C$ such that there is a morphism $\mathrm{Hom}(x,y)\to v$. This however is not really an ideal definition. It feels very artificial and not very categorical as I would like.~~

What would be a nice categorical way to define an 'open ball' of a Lawvere metric space?

**Edit:**

I have confused the direction of the arrows in $V$, this means my second the last previous paragraph should be rephrased:

Given a $v\in V$, the ball centered at $x \in C$ is the set of points in $C$ such that $v \to \mathrm{Hom}(x,y)$.

This does feel a bit more categorical but it is not quite there yet.