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