From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2557 Path: news.gmane.org!not-for-mail From: Topos8@aol.com Newsgroups: gmane.science.mathematics.categories Subject: Equivalences and psuedo-equivalences (two items) Date: Sat, 21 Feb 2004 08:46:40 EST Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 X-Trace: ger.gmane.org 1241018744 4718 80.91.229.2 (29 Apr 2009 15:25:44 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:25:44 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sun Feb 22 16:06:53 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 22 Feb 2004 16:06:53 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1Auzka-0001jl-00 for categories-list@mta.ca; Sun, 22 Feb 2004 15:58:52 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 18 Original-Lines: 46 Xref: news.gmane.org gmane.science.mathematics.categories:2557 Archived-At: The answer to this question must be in the literature somewhere but I haven't been able to track it down or figure it out for myself. Let A and B be bicategories. By a functor from A to B I shall mean a morphism of underlying reflexive globular sets that preserves all operations and identities on the nose and satisfies the standard coherence conditions. A psuedo-functor is a morphism that preserves operations and identities only up to 1 cells that are equivalences in B ( a 1-cell f is an equivalence if there is a 1-cell g such that fg and gf are both defined and isomorphic to the respective identitity 1-cells). (These 1-cells must of course satisfy additional, standard coherence conditions.) An equivalence from A to B is a functor that is essentially surjective on 0-cells (every 0-cell of B is equivalent in B to a 0-cell in the image of F) and which induces an equivalence of 1-categories between A( x,y ) and B ( Fx, Fy ) for all 0-cells x, y in A. A psuedo-equivalence from A to B is a psuedo-functor that has these same properties. Question 1 : If F is a psuedo-equivalence from A to B does there exist an equivalence G from A to B? (references/counterexamples?) Question 2: Same as question 1 but with A and B strict 2-categories. Qestion 3: If the answer to question 1 is yes, then can G be chosen to be equivalent to F in the bicategory whose 0-cells are psuedo-functors from A to B? I shall be very grateful for any guidance the list can provide on these questions. Carl Futia ------------------------------------------------------ SECOND ITEM: Subject: Correction to my last query I have to apologize for a silly error in my last request for help. The problem lies with the definition of psuedo functor. I should have said that a psuedo functor preserves operations up to a 2-cell isomorphism, not a 1-cell equivalence. Sorry! Carl Futia