# All metric spaces are Lawvere metric spaces

18 Apr 2020 ⇐ Notes archive(This is an entry in my technical diary. There will likely be typos, mistakes, or wider logical leaps—the intent here is to “[let] others look over my shoulder while I figure things out.”)

I’m definitely the rookie in my research group, so the diary will be a bit math-heavy as I try to catch up.

To start, here’s an entry on a topic—amongst many—Jade walked us through during our first meeting, Lawvere metric spaces.

nLab’s definiton is a bit impenetrable. At a glance, it seems like tacking on Lawvere’s name, to an already general concept, means added axioms.

It’s…surprisingly the opposite.

All metric spaces *are* Lawvere metric spaces—that is, we lift some of the constraints on plain ol’ metrics.

Recapping, a metric space is a set $X$ equipped with a distance function $d: X \times X \rightarrow [0, \infty)$ under the following coherences:

Assuming $x, y, z \in X$,

- $d(x, y) = 0 \iff x = y$ (zero-distance coincides with equality).
- $d(x, y) = d(y, x)$ (symmetry).
- $d(x, y) + d(y, z) \geq d(x, z)$ (the triangle inequality).

And Lawvere relaxed a few bits. A Lawvere metric space has a distance function

- that respects the triangle inequality,
- whose codomain includes $\infty$ (which is helpful when we want a “disconnectedness” between points),
- and $d(x, x) = 0$ (points are zero-distance from themselves).

We’re dropping the symmetry requirement and allowing for possibly zero distances between *distinct* points.

The former lets us represent, e.g. in a distance as cost situation, non-symmetric costs. Borrowing from Baez, imagine the commute from $x$ to $y$ being cheaper than from $y$ to $x$.

The easing of zero-distance being necessary and sufficient for equality to only one side of the implication adds the ability to reach points “for free” (continuing with the transportation theme).

I need to read up on more applications this freedom affords us. In the meantime, here’s some links I’ve come across:

- Jeremy Kun’s metric spaces primer.
- Lecture 31’s notes from MIT’s ’19 ACT course.
- Section 2.3.3 of Seven Sketches.