Subject: categories: Re: Pro C

From: "Prof. T.Porter" <[email-masked]>
Date: Fri, 01 Jun 2001 09:48:50 +0100

William Boshuck wrote:
> 
> This is due to Deligne, and can be found towards
> the beginning of SGA4, Expose I, section 8. I would
> like to know of a more recent source that is so (or
> more) thorough on the subject.
> cheers,
> -b
> On Tue, May 29, 2001 at 09:38:43PM -0700, Bill Rowan wrote:
> >
> > I have read that if C is a category, and the axiom of choice is assumed, then
> > Pro C is equivalent to its full subcategory of diagrams where the diagram
> > category is an inversely-directed set.  Does anyone know where this is proved
> > in the literature?
> >
> > Thanks,
> >
> > Bill Rowan

Dear All 
I replied to Bill Rowan directly yesterday but it now seems that others
might be interested in my reply so here it is.

In my book with Cordier, the result you want is Proposition 4 p 42 (The
book is :Categorical Shape Theory, Cordier and Porter, Published by
Ellis Horwood, 1989).  The result is known to some shape theorists as
the Mardesic trick as Sibe Mardesic is thought to have found it, but I
seem to remember seeing a version of it in Grothendieck's work (SGA4 and
earlier) If you can get a copy of our book there is a reasonably
categorical treatment of pro categories.   
   Best wishes,

Tim Porter