categories: Re: Limits

[Dusko Pavlovic <dusko@kestrel.edu>]

> Tobias Schroeder writes:
>  > - Can the limit of a sequence of real numbers be expressed
>  >   as a categorical limit (of course it can if the sequence is
>  >   monotone, but what if it is not)?
>
> I think I have an answer to this question (without cheating). It may
> be well known or wrong (I haven't carefully checked the details, but I
> believe that they are correct).

the view of (quasi)metric spaces as R+-categories with the hom-objects
d(x,y) goes back to lawvere's "metric spaces, generalized logic and
closed categories" from 1973. the cauchy completeness of a space was
identified with what came to be known as the cauchy completeness of the
corresponding category (see kelly's book on enriched categories, or
francis borceux handbook). and cauchy completeness of a category amounts
to the existence of certain absolute (co)limits: eg over Set, Ab, Cat...
to splitting idempotents (which can be done by taking (co)equalizers with
id).

-- dusko