From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2400 Path: news.gmane.org!not-for-mail From: Tom Leinster Newsgroups: gmane.science.mathematics.categories Subject: Re: Compatibility of functors with limits Date: Sun, 20 Jul 2003 19:50:35 +0200 (CEST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset="US-ASCII" X-Trace: ger.gmane.org 1241018634 3961 80.91.229.2 (29 Apr 2009 15:23:54 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:23:54 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Jul 21 10:54:31 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 21 Jul 2003 10:54:31 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19eb7Q-0006FO-00 for categories-list@mta.ca; Mon, 21 Jul 2003 10:54:24 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 28 Original-Lines: 34 Xref: news.gmane.org gmane.science.mathematics.categories:2400 Archived-At: Mamuka Jibladze wrote: > A related question: does anybody know any analogs of the > Freyd's Adjoint Functor Theorems for functors between > in(co)complete categories? Borceux states the 'More General Adjoint Functor Theorem' in Vol 1, 6.6.1 of his Handbook. This requires only that the codomain of the hoped-for left adjoint is Cauchy-complete (and of course that the known functor has some properties: it is 'absolutely flat' and satisfies some solution set conditions). Here's a representability theorem, presumably related. Let C be a small, Cauchy-complete category and let X: C ---> Set. Then X is representable <=> X respects small limits. The same goes for familial representability and connected limits. Proofs are at http://www.ihes.fr/~leinster/rr.ps Tom