Completion of a metric space in categorical terms

Is it possible to define the completion of a metric space using categorical terms?

• If you want to work in the category of metric spaces with morphisms continuous functions, I don't think this is possible. You would want the construction to be functorial but the maps that blow up at missing points can't be extended to maps of the completed spaces. Jul 20, 2013 at 18:24
• Can you clarify if by "define" you mean "characterise" or "construct"? (I'd like to know whether I misunderstood and hence should delete my answer.) Jul 20, 2013 at 18:24
• @DanielFischer: I mean definition, that determine completion up to isomorphism. Not matter, with constructive way or not. Don't delete your answer, please; I like it. Jul 20, 2013 at 18:35
• The universal property of the completion "defines"/characterizes it uniquely up to isometry. Jul 20, 2013 at 20:49

The completion of a metric space $X$ is an initial object in the category $\mathcal{U}_X$ whose objects are uniformly continuous maps $\iota_Y \colon X \to Y$, where $Y$ is a complete metric space, and whose morphisms are uniformly continuous maps $f \colon Y \to Z$ such that $f \circ \iota_Y = \iota_Z$.