Balls in Lawvere metric spaces

Question

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?

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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.

Answer 1

Luckily for you, Lawvere has already considered this in Taking categories seriously.

On page 18, he defines the family: $$\mathcal V^{\mathrm op}\times A\xrightarrow{B} \mathcal{V}^{A^{\mathrm{op}}}$$ defined by $$B(r,c)(a)=\mathcal V(r,A(a,c))$$ Where $B(r,c)$ reads the closed ball of given radius and center, since $$0 \ge B(r,c)(a)\iff r\ge A(a,c)$$ This is quite a cool construction because you can consider closed balls in any $\mathcal V$-enriched category.

Answer 2

I would probably define balls as follows (maybe with left and right transposed):

• The left ball of radius $r$ centered at $x$ is the enriched presheaf $d(-,x)-r: X^{op} \to V$

• The right ball of radius $r$ centered at $x$ is the enriched copresheaf $d(x,-) - r: X \to V$

Here $a-b = \mathrm{max}(a-b,0)$ is the internal hom in $V$.

The idea is that subsets typically categorify to (co)presheaves. The set of $y$ such that $d(y,x)-r = 0$ is exactly the set of $y$ such that $d(y,x) \leq r$, and dually.

As a presheaf, we can ask when the left ball is representable by an object $r \ast x$. In fact, there is a standard name in enriched category theory for such a representing object: $r\ast x$ is the tensor of $x$ by $r$. Dually, a corepresenting object for the right ball is the cotensor $\{r,x\}$.

Another way of saying this is that the left ball is the cotensor $\{r,d(-,x)\}$ in the presheaf space $Fun^V(X^{op},V)$, and right ball is the cotensor $\{r,d(x,-)\}$ in the copresheaf space $Fun^V(X,V)$.