From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1627 Path: news.gmane.org!not-for-mail From: "Dr. P.T. Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: question on "model functor" Date: Fri, 15 Sep 2000 18:08:14 +0100 (BST) Message-ID: References: <39C22090.5802016D@informatik.uni-bremen.de> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241017977 32140 80.91.229.2 (29 Apr 2009 15:12:57 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:12:57 +0000 (UTC) To: categories@mta.ca (Categories) Original-X-From: rrosebru@mta.ca Sun Sep 17 12:15:29 2000 -0300 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id LAA30488 for categories-list; Sun, 17 Sep 2000 11:14:52 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f In-Reply-To: <39C22090.5802016D@informatik.uni-bremen.de> from "Lutz Schroeder" at Sep 15, 2000 03:13:52 PM X-Mailer: ELM [version 2.5 PL2] Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 13 Original-Lines: 37 Xref: news.gmane.org gmane.science.mathematics.categories:1627 Archived-At: > The following question looks so natural that somebody's > bound to have looked into it: > > Does the functor > > Cat^op ---> CAT > > A |--> [A,Set] > > reflect isomorphisms (more generally: limits)? The question is not very well posed: since Cat and CAT are 2-categories, one ought to be asking about whether it reflects equivalences. For this, the answer is negative (and well known): a functor A --> B induces an equivalence [B,Set] --> [A,Set] iff it induces an equivalence between the idempotent-completions of A and B. So the inclusion of any non-idempotent- complete category in its idempotent-completion provides a counterexample. However, if you insist on asking about isomorphisms rather than equivalences, the answer is yes. It's easy to see that if F: A --> B fails to be surjective (resp. injective) on objects then the induced functor [F,Set] fails to be injective (resp. surjective); so if [F,Set] is an isomorphism then F must be bijective on objects, and this combined with inducing an equivalence of idempotent-completions is enough to make it an isomorphism. But this is not a very meaningful result. Provided you assume a sufficiently powerful form of the axiom of choice, [A,Set] and {B,Set] will be isomorphic whenever they are equivalent (since each has a proper class of objects in each isomorphism class, except for the initial object which is unique in its isomorphism class). The isomorphism will not, of course, be induced by a functor from A to B; but it will be naturally isomorphic to a functor that is (at least provided B is idempotent-complete). Peter Johnstone