What's the real name for these things? Categories whose morphisms have "length." A fairly obvious "categorification" of metric spaces is as follows.
First, let us agree to view $\mathbb{R}_+$ as an ordered Abelian monoid, where by "Abelian monoid" we really mean a category whose law of composition is denoted $+$ and required to be commutative, having a single object whose identity arrow is denoted $0.$
Then we can define that a categorial pseudometric space is a small, strict category $C$ whose objects are called "points" and whose morphisms are called "unparametrized paths" or simply "paths," equipped with a functor $l$ (called "length") into the Abelian monoid $\mathbb{R}_+$, such that for any two points $x,y$ there exists a path $x \rightarrow y$ of minimum length among all other such paths.
To get a genuine categorial metric space, we should also require that if $x$ and $y$ are distinct objects of $C$ and $f : x \rightarrow y$ is a path, then $l(f) \neq 0.$
Now I haven't actually proved it, but it seems obvious that we get a (classical) metric space from any categorial metric space (defined as above). Just define a distance function $d : (\mathrm{Ob} C)^2 \rightarrow \mathbb{R}_+$ by asserting that that for all objects $x$ and $y$ of $C$, we have that $d(x,y)$ equals the minimum of the expression $l(f),$ where $f$ ranges over all $f : x \rightarrow y$. Such a minimum necessarily exists, by hypothesis.
What's the real name for these so called "categorial metric spaces"?

A while back I asked a similar question on MO. There did not seem to be a consensus regarding the name of such a gadget, but perhaps you will find the discussion interesting.
