Metric Spaces are Categories!
A guest post by Tom Leinster over at The n-Category Café reminded me of an interesting fact I haven’t mentioned yet: a metric space is actually an example of an enriched category!
First we’ll need to pick out our base category 
, in which we’ll find our hom-objects. Consider the set of nonnegative real numbers with their real-number order, and add in a point called 
that’s above all the other points. This is a totally ordered set, and orders are categories. Let’s take the opposite of this category. That is, the objects of our category 
are the points in the “interval” ![\left[0,\infty\right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B0%2C%5Cinfty%5Cright%5D&bg=e6e6e6&fg=333333&s=0&c=20201002)
, and we have an arrow 
exactly when 
.
This turns out to be a monoidal category, and the monoidal structure is just addition. Clearly this gives a monoid on the set of objects, but we need to check it on morphisms to see it’s functorial. But if 
and 
then 
, and so we can see addition as a functor.
So we’ve got a monoidal category, and we can now use it to form enriched categories. Let’s keep out lives simple by considering a small -category 
. Here’s how the definition looks.
We have a set of objects 
that we’ll call “points” in a set 
. Between any two points 
and 
we need a hom-object 
. That is, we have a function ![d:X\times X\rightarrow\left[0,\infty\right]](https://s0.wp.com/latex.php?latex=d%3AX%5Ctimes+X%5Crightarrow%5Cleft%5B0%2C%5Cinfty%5Cright%5D&bg=e6e6e6&fg=333333&s=0&c=20201002)
.
For a triple 
of objects we need an arrow 
. In more quotidian terms, this means that 
.
Also, for each point 
there is an arrow from the identity object of to the hom-object 
. That is, 
, so 
.
These conditions are the first, fourth, and half of the second conditions in the definition of a metric space! In fact, there’s a weaker notion of a “pseudometric” space, wherein the second condition is simply that , and so we’re almost exactly giving the definition of a pseudometric space.
The only thing we’re missing is the requirement that 
. The case can be made (and has been, by Lawvere) that this requirement is actually extraneous, and that it’s in some sense more natural to work with “asymmetric” (pseudo)metric spaces that are exactly those given by this enriched categorical framework.
February 11, 2008 -
Posted by John Armstrong |
Algebra, Category theory, Point-Set Topology, Topology
The Unapologetic Mathematician 
Is there a standard term for something that satisfies the axioms of a pseudometric space except for symmetry?
The Wikipedia page on pseudometric spaces doesn’t list any more general concept.
I really don’t know whether there is or not. Tom certainly didn’t mention one in the post that reminded me about this fact.
Aha! It’s a quasimetric space:
http://en.wikipedia.org/wiki/Quasimetric_space
http://planetmath.org/?op=getobj&from=objects&name=QuasimetricSpace
Sorry, Jeremy, the links triggered the spam demon. Fixed now.
Anyhow, that’s brilliant. Thanks for turning up that term.
Incidentally, on a closer look it seems that quasimetric spaces keep the condition that
if and only if 
. That is, maybe what we’re looking at are pseudoquasimetric spaces. Or vice versa.
Or almost pseudoquasimetric, since the Lawvere definition you’re using includes
as a possible distance.
You might want to check out Dictionary of Distances by Deza & Deza (Elsevier, 2006) for names for various almost-metrics. They aren’t quite consistent with those listed in Wikipedia (which I liked, because they had a name for every subset of the metric axioms, including hemimetric and prametric, for example). It also seems that the names pseudo-, quasi- and (especially) semi-metric aren’t used 100% consistently in the literature.