Conditions for adjoints -- another variant
BTW. Here are some even cleaner conditions .... There is an adjoint between two categories iff there are two object functions F and G (not required to be functors) and For each X \in \X and Y \in \Y there are two functions: #: \X(X,G(Y)) -> \Y(F(X),Y) ---- sharp @: \Y(F(X),Y) -> \X(X,G(Y)) ---- flat (i)' @ and # are inverse @(#(f)) = f and #(@(g)) = g (ii)' @(h k) = @(h) @(#(1) k) and dually #(xy) = #(x @(1)) #(y) Still hoping to find where these all are recorded!! -robin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
cca 1991, i mentioned to mike barr that the functoriality of adjoints was derivable, and he knew it already. i think he said that this was used by isbell. (it's embarassing that i remember things from so long ago; and also that i don't remember them clearly enough to be confident. but does it really matter? when we optimize, we often find it more efficient not to store some data, but to regenerate when needed...) -- dusko On Sat, 24 Oct 2009, robin@ucalgary.ca wrote:
BTW. Here are some even cleaner conditions ....
There is an adjoint between two categories iff there are two object functions F and G (not required to be functors) and For each X \in \X and Y \in \Y there are two functions:
#: \X(X,G(Y)) -> \Y(F(X),Y) ---- sharp @: \Y(F(X),Y) -> \X(X,G(Y)) ---- flat
(i)' @ and # are inverse @(#(f)) = f and #(@(g)) = g (ii)' @(h k) = @(h) @(#(1) k) and dually #(xy) = #(x @(1)) #(y)
Still hoping to find where these all are recorded!!
-robin
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Hi Robin, In my thesis http://www.cs.bham.ac.uk/~pbl/papers/thesisqmwphd.pdf Def. 109 - 110, pages 220-222 I listed six (equivalent) definitions of adjunction, one of which (Def. 110(4)) resembles yours, and two of which (Def. 110(4)-(5)) don't mention any functors. (Some of these definitions - though not the one that resembles yours - use the notion of "representing object", which itself can be defined in either element style or naturality style.) The list also appears in my "Call-by-push-value" book, Def. 9.33 (page 235) and Def. 11.17 (pages 278-280). Also see the discussion in Sect. 1.2 of my TAC paper "Adjunction models for call-by-push-value with stacks" http://www.tac.mta.ca/tac/volumes/14/5/14-05abs.html regards, Paul On Sat, 24 Oct 2009 17:11:26 -0600 (MDT), robin@ucalgary.ca wrote:
BTW. Here are some even cleaner conditions ....
There is an adjoint between two categories iff there are two object functions F and G (not required to be functors) and For each X \in \X and Y \in \Y there are two functions:
#: \X(X,G(Y)) -> \Y(F(X),Y) ---- sharp @: \Y(F(X),Y) -> \X(X,G(Y)) ---- flat
(i)' @ and # are inverse @(#(f)) = f and #(@(g)) = g (ii)' @(h k) = @(h) @(#(1) k) and dually #(xy) = #(x @(1)) #(y)
Still hoping to find where these all are recorded!!
-robin
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
Dear categorists, My Composition prism, Functor prism, Natural Transformation prism, Semi-adjoint prism, Generalized associativity prism, and Generalized semi-adjoint prism previously transmitted as XY-pic code may not have been the most direct way to convey my idea for subsuming these basic concepts under the prism umbrella. Hence, both source code and PDF generated from it are available for download at http://www.distancedrawing.com/Prism091102.tex and http://distancedrawing.com/Prism091102.pdf . An Identity prism has subsequently been drawn, but I bet anybody can do that for themselves. Ellis D. Cooper [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
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Dusko Pavlovic -
Ellis D. Cooper -
Paul Levy -
robin＠ucalgary.ca