From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5211 Path: news.gmane.org!not-for-mail From: "Ellis D. Cooper" Newsgroups: gmane.science.mathematics.categories Subject: Conditions for adjoints -- another variant Date: Sat, 31 Oct 2009 22:20:56 -0400 Message-ID: Reply-To: "Ellis D. Cooper" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Trace: ger.gmane.org 1257118745 29878 80.91.229.12 (1 Nov 2009 23:39:05 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 1 Nov 2009 23:39:05 +0000 (UTC) To: P.B.Levy@cs.bham.ac.uk,categories@mta.ca,Dusko.Pavlovic@comlab.ox.ac.uk Original-X-From: categories@mta.ca Mon Nov 02 00:38:57 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1N4k0f-0005Ch-Gp for gsmc-categories@m.gmane.org; Mon, 02 Nov 2009 00:38:57 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N4ja0-0001eB-Fz for categories-list@mta.ca; Sun, 01 Nov 2009 19:11:24 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5211 Archived-At: Dear categorists, I cannot resist wondering whether it has been observed that (almost) everything is a prism, including the Subject, and there is a generalization of category theory beyond homsets with merely two parameters. I have omitted a lot of labels because anyone on this list can fill them in, and I have omitted prisms for identity diagrams for the same reason. Dotted arrows are induced, outer squares commute. (By "semi-adjoint" I mean the family of set maps in either of the two directions in the usual bifunctor definition of adjoint.) Composition \xymatrix{ &a\ar[dl]_f\ar@{..>}[dd]^{gf}\\a'\ar[dr]_g&\\&a''} Functor prism \xymatrix{ a\ar@{..>}[rr]^{gf}\ar@{|->}[ddd]\ar[dr]_f&&a''\ar@{|->}[ddd]\\ &a'\ar[ur]_g\ar@{|->}[d]&\\ &Fa'\ar[dr]_{Fg}&\\ Fa\ar[ur]_{Ff}\ar@{..>}[rr]_{Fgf}&&Fa''\\ } Natural Transformation prism \xymatrix{ Fa\ar[rr]^{\eta_a}\ar@{..>}[ddd]_{Ff}&&Ga\ar@{..>}[ddd]^{Gf}\\ &a\ar@{|->}[ul]\ar@{|->}[ur]\ar[d]^f&\\ &a'\ar@{|->}[dl]\ar@{|->}[dr]&\\ Fa'\ar[rr]_{\eta_{a'}}&&Ga'\\ } Semi-adjoint prism \xymatrix{ (Fa\:Gb)\ar[rr]\ar@{..>}[ddd]&&(Ka\:Lb)\ar@{..>}[ddd]\\ &(a\:b)\ar@{|->}[ul]\ar@{|->}[ur]\ar[d]&\\ &(a'\:b')\ar@{|->}[dl]\ar@{|->}[dr]&\\ (Fa'\:Gb')\ar[rr]&&(Ka'\:Lb') } Generalized associativity prism \xymatrix{ (a_1\cdots a_n)\ar@{..>}[rr]^{(hg)f}\ar[dr]_f\ar[ddd]_1&&(a'''_1\cdots a'''_n)\ar[ddd]^1\\ &(a'_1 \cdots a'_n)\ar@{..>}[ur]_{hg}\ar[d]^g&\\ &(a''_1\cdots a''_n)\ar[dr]_h&\\ (a_1\cdots a_n)\ar@{..>}[ur]_{gf}\ar@{..>}[rr]_{h(gf)}&&(a'''_1\cdots a'''_n) } Generalized semi-adjoint \xymatrix{ (F_1a_1\cdots F_na_n)\ar[rr]\ar@{..>}[ddd]&&(G_1a_1\cdots G_na_n)\ar@{..>}[ddd]\\ &(a_1\cdots a_n)\ar@{|->}[ul]\ar@{|->}[ur]\ar[d]&\\ &(a'_1\cdots a'_n)\ar@{|->}[dl]\ar@{|->}[dr]\\ (F_1a'_1\cdots F_na'_n)\ar[rr]&&(G_1a'_1\cdots G_na'_n) } Ellis D. Cooper [For admin and other information see: http://www.mta.ca/~cat-dist/ ]