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From: "Dr. P.T. Johnstone"
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Subject: Re: Pro C
Date: Thu, 31 May 2001 13:57:45 +0100 (BST)
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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