From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3832 Path: news.gmane.org!not-for-mail From: Aaron Lauda Newsgroups: gmane.science.mathematics.categories Subject: pivotal adjoints? Date: Thu, 19 Jul 2007 14:05:46 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=ISO-8859-1;DelSp="Yes";format="flowed" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019550 10507 80.91.229.2 (29 Apr 2009 15:39:10 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:39:10 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Jul 20 17:04:37 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 20 Jul 2007 17:04:37 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IByam-0002vb-FY for categories-list@mta.ca; Fri, 20 Jul 2007 16:56:48 -0300 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 26 Original-Lines: 45 Xref: news.gmane.org gmane.science.mathematics.categories:3832 Archived-At: Dear category theorists, Suppose we have chosen left and right adjoints for F:A->B and G:A->B F-| F* -| F and G-| G*-|G i_F: 1_B =3D> FF* i_G: 1_B =3D> GG* e_F: F*F =3D> 1_A e_G: G*G =3D> 1_A j_F: 1_A =3D> F*F j_G: 1_A =3D> G*G k_F: FF* =3D> 1_B k_G: GG* =3D> 1_B Then given any 2-morphism a:F=3D>G there are two obvious duals (mates =20 under adjunction) for the 2-morphsism a a+ :G*=3D>F* :=3D (e_G F*).(G*aF*).(G* i_F) +a :G*=3D>F* :=3D (F* k_G).(F*aG*).(j_F G*) or for those who like pictures: +a a+ __ __ / \ | | / \ | | | | | | | a | | a | | | | | | | | \__/ \__/ | | | In general a+ is not equal to +a because if is was we could always twist one of the units and counits so that it does not hold. Has the condition =20 that a+ =3D +a been investigated in the literature anywhere? In =20 particular, if a 2-category is such that all 1-morphisms F have a =20 simultaneous left and right adjoint then has anyone studied the =20 context where the adjoints are such that a+ =3D +a is always =20 satisfied? Perhaps, this notion has been studied in the language of =20 duals for 1-morphisms? The above condition appears to be related to the notion of pivotal =20 category when we look at Hom(A,A) for any object A. Thanks, Aaron Lauda