Home Reading Searching Subscribe Sponsors Statistics Posting Contact Spam Lists Links About Hosting Filtering Features Download Marketing Archives FAQ Blog From: Ellis D. Cooper netropolis.net> Subject: Conditions for adjoints -- another variant Newsgroups: gmane.science.mathematics.categories Date: Sunday 1st November 2009 02:20:56 UTC (over 9 years ago) 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\[email protected]{..>}[dd]^{gf}\\a'\ar[dr]_g&\\&a''} Functor prism \xymatrix{ a\[email protected]{..>}[rr]^{gf}\[email protected]{|->}[ddd]\ar[dr]_f&&a''\[email protected]{|->}[ddd]\\ &a'\ar[ur]_g\[email protected]{|->}[d]&\\ &Fa'\ar[dr]_{Fg}&\\ Fa\ar[ur]_{Ff}\[email protected]{..>}[rr]_{Fgf}&&Fa''\\ } Natural Transformation prism \xymatrix{ Fa\ar[rr]^{\eta_a}\[email protected]{..>}[ddd]_{Ff}&&Ga\[email protected]{..>}[ddd]^{Gf}\\ &a\[email protected]{|->}[ul]\[email protected]{|->}[ur]\ar[d]^f&\\ &a'\[email protected]{|->}[dl]\[email protected]{|->}[dr]&\\ Fa'\ar[rr]_{\eta_{a'}}&&Ga'\\ } Semi-adjoint prism \xymatrix{ (Fa\:Gb)\ar[rr]\[email protected]{..>}[ddd]&&(Ka\:Lb)\[email protected]{..>}[ddd]\\ &(a\:b)\[email protected]{|->}[ul]\[email protected]{|->}[ur]\ar[d]&\\ &(a'\:b')\[email protected]{|->}[dl]\[email protected]{|->}[dr]&\\ (Fa'\:Gb')\ar[rr]&&(Ka'\:Lb') } Generalized associativity prism \xymatrix{ (a_1\cdots a_n)\[email protected]{..>}[rr]^{(hg)f}\ar[dr]_f\ar[ddd]_1&&(a'''_1\cdots a'''_n)\ar[ddd]^1\\ &(a'_1 \cdots a'_n)\[email protected]{..>}[ur]_{hg}\ar[d]^g&\\ &(a''_1\cdots a''_n)\ar[dr]_h&\\ (a_1\cdots a_n)\[email protected]{..>}[ur]_{gf}\[email protected]{..>}[rr]_{h(gf)}&&(a'''_1\cdots a'''_n) } Generalized semi-adjoint \xymatrix{ (F_1a_1\cdots F_na_n)\ar[rr]\[email protected]{..>}[ddd]&&(G_1a_1\cdots G_na_n)\[email protected]{..>}[ddd]\\ &(a_1\cdots a_n)\[email protected]{|->}[ul]\[email protected]{|->}[ur]\ar[d]&\\ &(a'_1\cdots a'_n)\[email protected]{|->}[dl]\[email protected]{|->}[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/ ]
CD: 12ms