gn.general topology - What is a metric space?

A classic category-theoretic view of metric spaces says that the "correct" maps are the distance-decreasing ones:
$$
d(f(a), f(a')) leq d(a, a')
$$
where $A$ and $B$ are metric spaces, $f: A to B$, and $a, a' in A$. Then all maps are continuous, and the isomorphisms are the isometries onto.

This comes from viewing metric spaces as enriched categories, as proposed by Lawvere. The enriched functors are then exactly the distance-decreasing maps.

Edit Let me add some detail. Consider the set $V=[0, infty]$ of non-negative reals. (The inclusion of $infty$ isn't important here.) It's ordered by $geq$, and so can be regarded as a category: there's one map $x to y$ if $x geq y$, and there are no maps $x to y$ otherwise. It becomes a monoidal category under $+$ and $0$.

A $V$-enriched category is then a set $A$ of objects (or points) together with, for each pair $(a, b)$ of points, an object $A(a, b)$ of $V$ --- that is, a non-negative real, which you might prefer to call $d(a, b)$. Composition then becomes the triangle inequality, and identities the assertion that the distance from a point to itself is $0$. So, a $V$-enriched category is a "generalized metric space": there's no requirement of symmetry (so you could take distance to be the work done in moving between points of a mountainous region) or that points distance $0$ apart are equal (which is just like not asking for isomorphic objects of a category to be equal).

You should then be able to see that $V$-enriched functors are what I said they were.

Edit re Lipschitz maps I don't want to evangelize this point
of view too much. But it's a matter of fact that Lipschitz maps do
arise naturally in this framework.

To explain this I first need to explain a little about 'change of
base' for enriched categories. Any lax monoidal functor $Phi:
mathcal{V} to mathcal{W}$ induces a functor $Phi_*:
mathcal{V}-mathbf{Cat} to mathcal{W}-mathbf{Cat}$, in an obvious
way. For example, if $Phi: mathbf{Vect} to mathbf{Set}$ is the
forgetful functor, then $Phi_*$ sends a linear category to its
underlying ordinary category.

This means that given a lax monoidal $Phi: mathcal{V} to mathcal{W}$,
a $mathbf{V}$-enriched category $mathbf{A}$, and a
$mathbf{W}$-enriched category $mathbf{B}$, we can define a
$Phi$-enriched functor $mathbf{A} to mathbf{B}$ to be a
$mathcal{W}$-enriched functor $Phi_*(mathbf{A}) to mathbf{B}$. One
might also call this a 'functor over $Phi$'.

That's completely general enriched category theory. Now let's
apply it to $mathcal{V} = mathcal{W} = [0, infty]$. For any $M
geq 0$, multiplication by $M$ defines a (strict) monoidal functor
$Mcdot -: [0, infty] to [0, infty]$. Let $A$ and $B$ be metric
spaces. Then an $(Mcdot -)$-enriched functor from $A$ to $B$ is
precisely a function $f: A to B$ such that
$$
d(f(a), f(a')) leq Mcdot d(a, a')
$$
for all $a, a' in A$. In other words, it's a Lipschitz map.

A bit more can be squeezed out of this. The maps $Mcdot -$ are the
strict monoidal endofunctors of $[0, infty]$. But we can talk
about $phi$-enriched maps of metric spaces for any lax
monoidal endofunctor of $[0, infty]$. `Lax monoidal' means that
$$
phi(0) = 0,

phi(x + y) leq phi(x) + phi(y),
$$
which is a kind of concavity property (satisfied by $phi(x) = sqrt{x}$,
for instance). Then a $phi$-enriched map from $A$ to $B$ is a
function $f: A to B$ such that
$$
d(f(a), f(a')) leq phi(d(a, a'))
$$
for all $a, a' in A$. Is that kind of map found useful?