Re: Conditions for adjoints -- another variant

Re: Conditions for adjoints -- another variant

[Paul Levy]

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
> 
> 
> 
> 
> 
> 


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Conditions for adjoints -- another variant

[Ellis D. Cooper]

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



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Conditions for adjoints -- another variant

[Ellis D. Cooper]

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



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Re: Conditions for adjoints -- another variant

[Dusko Pavlovic]

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/ ]
>


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Conditions for adjoints -- another variant

[robin]

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






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