TITLE: What is a metric space?
QUESTION [42 upvotes]: According to categorical lore, objects in a category are just a way of separating morphisms.  The objects themselves are considered slightly disparagingly.  In particular, if I can't distinguish between two objects by using morphisms, then I should consider them equivalent (not equal, that would be evil).
In this view, then, metric spaces with continuous functions are just plain wrong.  The category of metric spaces is equivalent to the full subcategory of topological spaces consisting of metrisable spaces.  Choosing a metric is evil.
So what's the right view of metric spaces, that makes the metric both worth having and not (so much of) an arbitrary choice?
I put the "so much of" there because the obvious answer is having isometries as morphisms, but then the category becomes too rigid to be of any conceivable use.  So what's the best middle ground?

REPLY [2 votes]: An interesting class of morphisms that sits between isometric embeddings and bi-Lipschitz maps (both the map and its inverse are Lipschitz) is the class of almost isometric embeddings.  There are two natural definitions of this.  Let $X$ and $Y$ be metric spaces.
(1) An almost isometric embedding from $X$ into $Y$ is a sequence $(f_n)_{n\in\mathbb N}$ of maps $f_n:X\to Y$ such that for some sequence $(\varepsilon_n)_{n\in\mathbb N}$ of positive reals converging to zero, for every $n$, $f_n$ is bi-Lipschitz of constant 
$\varepsilon_n$.
The composition of two almost isometric embeddings $(f_n)_{n\in\mathbb N}$ and $(g_n)_{n\in\mathbb N}$ (of appropriate domains and ranges) is $(f_n\circ g_n)_{n\in\mathbb N}$.
(2) If you don't like the fact that an almost isometric embedding as defined above does not have a fixed range in the classical sense, change the definition by requiring that
each $f_n$ has the same range.  
$X$ and $Y$ are almost isometric if there is an almost isometric embedding as in (2) from $X$ onto $Y$ (all $f_n$ are onto $Y$).
For example, any two countable dense subsets of $\mathbb R$ are almost isometric (back and forth argument), but not necessarily isometric, simply because there are $|\mathbb R|$ many possible distances but only countably many of these are realized in a fixed countable set.