Date: Fri, 21 Oct 1994 12:38:06 +0500 (GMT+4:00)
From: categories <cat-dist@mta.ca>
Subject: colimits of Hilbert spaces

Date: Thu, 20 Oct 94 21:45:50 PDT
From: john baez <baez@ucrmath.UCR.EDU>

I'm interested in colimits in the category of Hilbert spaces
with isometries as morphisms.  Is there a good reference on
when these exist?

John Baez

Date: Sat, 22 Oct 1994 11:29:25 +0500 (GMT+4:00)
From: categories <cat-dist@mta.ca>
Subject: Re: colimits of Hilbert spaces

Date: Fri, 21 Oct 94 12:23:33 EDT
From: Michael Barr <barr@triples.math.mcgill.ca>

>
> Date: Thu, 20 Oct 94 21:45:50 PDT
> From: john baez <baez@ucrmath.UCR.EDU>
>
> I'm interested in colimits in the category of Hilbert spaces
> with isometries as morphisms.  Is there a good reference on
> when these exist?
>
> John Baez
>

I don't know a reference, but I know that very few colimits exist.
For example, the sum of two non-zero spaces cannot exist for you can
give the direct sum of the normed spaces either the sup or euclidean
norm and these would both have to embed isometrically in the sum,
which is clearly impossible.  Clearly coequalizers are likewise
precluded.  It seems possible that filtered colimits might make it;
you would take the ordinary colimit as inner product spaces and
complete.  I don't offhand see what can go wrong with that.

Michael