Subject: categories: Pro C

From: Bill Rowan <[email-masked]>
Date: Tue, 29 May 2001 21:38:43 -0700 (PDT)

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

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From: "Dr. P.T. Johnstone" <[email-masked]>
Date: Thu, 31 May 2001 13:57:45 +0100 (BST)

> category is an inversely-directed set.  Does anyone know where this is proved
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

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From: William Boshuck <[email-masked]>
Date: Thu, 31 May 2001 07:42:43 -0400

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