Re: Conditions for adjoints -- another variant

[Paul Levy <P.B.Levy@cs.bham.ac.uk>]

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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