Dinatural transformations

Dinatural transformations

[Max Kelly]

Nason Yanofsky asked a question about composition of dinatural
transformations, and there have been a number of replies - but none
giving the reference I should have expected, namely [S.Eilenberg and
G.M.Kelly, A generalization of the functorial calculus,, J.Algebra 3
(1966), 366 - 375] .

What Sammy and I  were concerned with were such families as  the
evaluationn e_A,B : [A,B] o A --> B , where o is a tensor product and [
, ] is an internal hom. Here e_A,B is natural in B in the usual sense;
it is also natural, in our extended sense, in the variable A, which
appears twice on one side of the arrow, but with two opposite variances.
Similar for d_A,B : A --> [B, AoB]. In certain circumstances one can
compose such "natural transformations" to form new ones:one example is
the composite

                  AoB ---------> [B, AoB] o B ----------> AoB
                             d_A,B o B                     e_B, AoB

which is in fact the identity natural transformation.

Sammy and I gave a general treatment in the article above. Later, others
generalised our "extended naturals" to get the notion of dinatural
transformation. Since these do not compose except in some very special
cases, I find them to be of limited interest.   In contrast, I find that
I still use the Eilenberg-Kellycalculus from time to time.

Max Kelly.