Re: Pro C

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