Pro C

Pro C

[Bill Rowan]

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

Re: Pro C

[William Boshuck]

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

Re: Pro C

[Dr. P.T. Johnstone]

On Tue, 29 May 2001, 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
>
Choice isn't needed: all you need is the result that, for any filtered
category C, there is a directed poset P and a final functor P --> C.
There is a proof of this somewhere in SGA4 (I don't have the reference
to hand), where it is attributed to Pierre Deligne; but I suspect it
may be older than this.

Peter Johnstone

Re: Pro C

[Prof. T.Porter]

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

Re: Pro C

[Jiri Rosicky]

< Choice isn't needed: all you need is the result that, for any filtered
< category C, there is a directed poset P and a final functor P --> C.
< There is a proof of this somewhere in SGA4 (I don't have the reference
< to hand), where it is attributed to Pierre Deligne; but I suspect it
< may be older than this.

< Peter Johnstone

The proof is also in my book with Adamek, Locally presentable and accessible
categories, Theorem 1.5,
Jiri Rosicky

Re: Pro C

[Dan Isaksen]

A slightly more recent exposition is given in 

D. A. Edwards and H. M. Hastings, Cech and Steenrod homotopy theories with
applications to geometric topology, Lecture Notes in Mathematics 542,
Springer, 1976

on pages 6--7.  It is a bit strange that Edwards and Hastings credit
Mardesic and do not mention Deligne.  Understandably, they must have been
more familiar with the literature on shape theory than on algebraic
geometry.

Dan Isaksen
University of Notre Dame
isaksen.1@nd.edu

> Date: Thu, 31 May 2001 07:42:43 -0400
> From: William Boshuck <boshuk@triples.math.mcgill.ca>
> To: categories@mta.ca
> Subject: categories: Re: Pro C
> 
> 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