Conditions for adjoints

Conditions for adjoints

[robin]

(Apologies to those who received the earlier type mashed version ...)

Jeremy Dawson and I were discusing whether one can express the conditions
for an adjoint without requiring functors ... this is what we came up
with:


There is an adjoint between two categories if and only if
there are object functions F and G (not functors) and
for each X in \X and Y in \Y there are functions:

#: \X(X,G(Y)) -> \Y(F(X),Y)  ---- sharp
@: \Y(F(X),Y) -> \X(X,G(Y))  ---- flat

between the homsets such that
(1) @(#(1)) = 1 and dually #(@(1)) = 1 (inverse on identities)
(2) @(1) @(#(1) #(f)) = f  and dually  #(@(g) @(1)) #(1) = g
(3) @(#(f @(1)) h k) = f @(h) @(#(1) k)
            and dually
        #(x y @(#(1) z)) = #(x @(1)) #(y) z.

I find it hard to believe that such conditions have not been recorded. 
Does anyone have a reference or similar conditions which do not require
functors?

-robin






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Re: Conditions for adjoints

[laurent.mehats]

robin@ucalgary.ca a écrit :
> (Apologies to those who received the earlier type mashed version ...)
> 
> Jeremy Dawson and I were discusing whether one can express the conditions
> for an adjoint without requiring functors ... this is what we came up
> with:
> 
> 
> There is an adjoint between two categories if and only if
> there are object functions F and G (not functors) and
> for each X in \X and Y in \Y there are functions:
> 
> #: \X(X,G(Y)) -> \Y(F(X),Y)  ---- sharp
> @: \Y(F(X),Y) -> \X(X,G(Y))  ---- flat
> 
> between the homsets such that
> (1) @(#(1)) = 1 and dually #(@(1)) = 1 (inverse on identities)
> (2) @(1) @(#(1) #(f)) = f  and dually  #(@(g) @(1)) #(1) = g
> (3) @(#(f @(1)) h k) = f @(h) @(#(1) k)
>             and dually
>         #(x y @(#(1) z)) = #(x @(1)) #(y) z.
> 
> I find it hard to believe that such conditions have not been recorded. 
> Does anyone have a reference or similar conditions which do not require
> functors?
> 
> -robin

Hello,

Such conditions are discussed in detail in:
Kosta Došen, Cut Elimination in Categories, Trends in Logic 6, Kluwer, 1999.

Those you mention already appear on p. 258 of:
Kosta Došen, Deductive Completeness, Bull. Symbolic Logic Volume 2, Number
3 (1996), 243-283.
(http://www.math.ucla.edu/~asl/bsl/0203/0203-001.ps).

Regards,
Laurent Méhats



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