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/ ] |
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