In a message with [Subject: A diagram lemma] Mike Barr asks:
Has anyone seen the following diagrammatic lemma?
Sure -- in the general case. Remember the business about "lim-pacing" functors in Lawvere's Columbia PhD thesis? A functor C ---> D is lim-pacing iff: for every diagram, anywhere, of shape D , if the "restricted" diagram of shape C has a limit, then so does the original one, and the canonical "comparison" between those two limits is an isomorphism. There are n.a.s.c.'s internal to C and D for the lim-pacing condition (some comma category being non-empty and connected); and your "diagrammatic lemma" is just the instance of all these obtained with C = NNO^op and D = (NNO x NNO)^op (and C ---> D the diagonal). [No one would probably want to bother promoting this particular case to the status of a "lemma".] Hope this helps. -- Fred ==================================
Yes. Please excuse me if I think of (filtered) colimits instead, but this says that the co/limit over N, then over another N is that over NxN, in which the diagonal is co/final. =========================
There are many arguments of this kind, one of them (as I recall) in my paper with Pultr in JPAA 12(1978), 207 - 244. Isn't the point merely that the diagonal is initial in the product? Max Kelly, 19 Sept. =========================
Mike and Fred, The modern term is "initial functor" (dual to "final" which Mac Lane preferred to "cofinal" which is classical for posets). Ross ==========================
Please make it clear to us which is final and which initial. ==========================
From categories@mta.ca Tue Sep 24 19:11:01 1991 Received: from macc1.mta.ca by adder.maths.su.oz.au with SMTP (5.61++/10.0) id AA10186; Tue, 24 Sep 91 19:10:44 +1000 Date: Tue, 24 Sep 1991 00:26:58 EDT From: categories@mta.ca To: sydcat@maths.su.oz.au Message-Id: <0094f18d.df2677a0.4226@mta.ca> Subject: Re: A diagram lemma Date: Sat, 21 Sep 91 22:33+0000 From: Paul Taylor <pt@doc.ic.ac.UK> Please make it clear to us which is final and which initial. K: A ---> C is initial if lim TK = lim T for all T: C ---> B, either existing if the other does. See Section 4.5 of my book "Basic Concepts of Enriched Category Theory", C.U.P. 1982, for a complete treatment of both the classical and enriched cases. Max Kelly, 25 Sept.91. =========================
Date: Sat, 21 Sep 91 22:33+0000 From: Paul Taylor <pt@doc.ic.ac.UK>
Please make it clear to us which is final and which initial.
Restriction along a final functor induces an isomorphism between colimits. { Of interest may be the fact that every functor factors into a final functor followed by a discrete fibration (Street-Walters Bull. AMS 79 (1973)); a discussion of such functors is contained there. Unfortunately, the proof of the factorisation contains a misleading step about composites of Kan extensions; we were over-zealous in trying to keep the account short for the Bulletin (we had proved it other ways).} Regards, Ross ===============================
participants (5)
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Fred E.J. Linton -
kelly_m＠maths.su.oz.au -
Paul Taylor -
Paul Taylor -
street＠macadam.mpce.mq.edu.au