Home
Reading
Searching
Subscribe
Sponsors
Statistics
Posting
Contact
Spam
Lists
Links
About
Hosting
Filtering
Features Download
Marketing
Archives
FAQ
Blog
 
Gmane
From: <laurent.mehats <at> gmail.com>
Subject: Re: Conditions for adjoints
Newsgroups: gmane.science.mathematics.categories
Date: Sunday 25th October 2009 09:48:39 UTC (over 8 years ago)
[email protected] 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



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
 
CD: 3ms