From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7649 Path: news.gmane.org!not-for-mail From: Paul Levy Newsgroups: gmane.science.mathematics.categories Subject: Re: Reference: equivalences can be made into adjunctions Date: Mon, 8 Apr 2013 08:57:37 +0100 Message-ID: References: Reply-To: Paul Levy NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v936) Content-Type: text/plain; charset=UTF-8; format=flowed; delsp=yes Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1365438251 25117 80.91.229.3 (8 Apr 2013 16:24:11 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 8 Apr 2013 16:24:11 +0000 (UTC) Cc: categories list To: =?UTF-8?Q?Jonathan_CHICHE_=E9=BD=8A=E6=AD=A3=E8=88=AA?= Original-X-From: majordomo@mlist.mta.ca Mon Apr 08 18:24:12 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1UPErO-0002AI-Aj for gsmc-categories@m.gmane.org; Mon, 08 Apr 2013 18:23:58 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:37547) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UPBlE-0006m1-BZ; Mon, 08 Apr 2013 10:05:24 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UPBlC-0007jq-VJ for categories-list@mlist.mta.ca; Mon, 08 Apr 2013 10:05:22 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7649 Archived-At: On 6 Apr 2013, at 21:22, Jonathan CHICHE =E9=BD=8A=E6=AD=A3=E8=88=AA = wrote: > Dear all, > > This is a standard fact, proved for instance in details in =20 > "Categories for the Working Mathematician" (Chapter 4, Section 4, =20 > Theorem 1 in the second edition), that a functor is an equivalence =20 > of categories if and only if it is part of an adjoint equivalence. I =20= > would like to use the fact that this is true in an arbitrary 2-=20 > category, i.e. that given an equivalence in a 2-category the =20 > invertible 2-cells can be required to satisfy the triangle =20 > identities. Is there a standard reference for this fact? Dear Jonathan, A reference is "Two-dimensional monad theory" by Blackwell, Kelly and =20= Power. Journal of Pure and Applied Algebra 59, 1989. I asked a similar question in 2001: http://facultypages.ecc.edu/alsani/ct01%289-12%29/msg00071.html which led to some very interesting responses. Paul -- Paul Blain Levy School of Computer Science, University of Birmingham +44 121 414 4792 http://www.cs.bham.ac.uk/~pbl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]