9
$\begingroup$

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

$\endgroup$
4
  • $\begingroup$ 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. $\endgroup$
    – Marek
    Jul 20, 2013 at 18:24
  • $\begingroup$ 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.) $\endgroup$ Jul 20, 2013 at 18:24
  • $\begingroup$ @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. $\endgroup$ Jul 20, 2013 at 18:35
  • 2
    $\begingroup$ The universal property of the completion "defines"/characterizes it uniquely up to isometry. $\endgroup$
    – Julien
    Jul 20, 2013 at 20:49

2 Answers 2

9
$\begingroup$

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 $f \circ \iota_Y = \iota_Z$.

$\endgroup$
6
  • $\begingroup$ We wanted to define complete metric space. You assume its defined a merely build on that. -1 $\endgroup$
    – Marek
    Jul 20, 2013 at 18:21
  • 3
    $\begingroup$ @Marek I understood "define" as "characterise" here. If it was meant as "construct", that would of course be an entirely different beast, and I'd have no idea whether that'd be possible. Thanks for explaining the downvote, much appreciated. $\endgroup$ Jul 20, 2013 at 18:23
  • 1
    $\begingroup$ Thanks for answer. Can you recommend books when I can read about such a definition? $\endgroup$ Jul 20, 2013 at 19:06
  • 1
    $\begingroup$ @user14284 Terribly sorry, but no. I hardly know any category theory, can't name any books that might include something like that. $\endgroup$ Jul 20, 2013 at 19:08
  • 3
    $\begingroup$ @user14284 Almost the same definitins, theorems and explanations are given for the case of Normed spaces in the book Lectures And Exercises on Functional Analysis (Translations of Mathematical Monographs) by A. Ya. Helemskii $\endgroup$
    – Norbert
    Jul 20, 2013 at 23:25
2
$\begingroup$

At least for Lawvere metric spaces it is a special case of the Cauchy completion.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .