categories: Re: Pro C

["Dr. P.T. Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

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