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
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
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
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.
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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
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
< 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
participants (6)
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Bill Rowan -
Dan Isaksen -
Dr. P.T. Johnstone -
Prof. T.Porter -
rosicky＠math.muni.cz -
William Boshuck