As a followup to the discussion on Mal'cev relations these days I thought it might be a good idea to describe some ideas I had recently concerning these. They seem quite close to Peter Freyd's structors, so I don't claim originality. Let BB be a ccc with finite limits. A pullback square (a:X->A,b:X->B,p:A->Y,q:B->Y) is called a Mal'cev relation between A and B, (by a slight distortion of Michael Barr's definition). Are there any good terms for X and Y? One might call them kernel and image, perhaps. The Mal'cev relations in BB form a ccc with finite limits (morphisms being given by hexagons.). Now given a class of -+ bifunctors on BB we can axiomatise an action on Mal'cev relations as follows: If R:=(a:X->A,b:X->B,p:A->Y,q:B->Y) is a M. rel between A and B and F:BB\op x BB -> BB is in the class of bifunctors, then we assign to this situation a Mal'cev relation F*R between FAA and FBB such that the following axioms are satisfied: 1) (F x G)*R =~= F*R x G*R 2) (F => G)*R =~= F*R => G*R 3) If F_R is the relation between FAA and FBB defined by the span (Fpa:FYX->FAA, Fqb:FYX->FBB) and F^R is the relation defined by the cospan (Fap:FAA->FXY, Fqb: FBB->FXY) then F_R \subset F*R \subset F^R. In 3) the term \subset and "relations are used in an informal way. What 3) precisely means is that FAA FAA /\ \ /\ \ / \ / \ / \/ / \/ W FXY FYX Z \ /\ \ /\ \ / \ / \/ / \/ / FBB FBB commute for W the kernel and Z the image of F*R. => and x denote twisted exponential and pointwise product for bifunctors and exponential and cartesian product for Mal'cev relations. The clauses 1)-3) can easily be generalised to -^n+^n-functors. One can show by induction that the class of bifunctors definable by constants +(if it exists), x and => admit an action on Mal'cev relations, so the definition is consistent. Now given such a class of functors and an action on Mal'cev relations we can define a notion of dinaturality as follows: Def: Let F and G be two -+ bifunctors. A "very strong dinatural transformation" t is then given by a family of maps t_A:FAA to GAA for each object A in C, such that for each Mal'cev relation R between A and B the pair (t_A:FAA->GAA, t_B:FBB->GBB) is a morphism from F*R to G*R in the category of Mal'cev relations. I.e. the "famous hexagon" t_A FAA -------> GAA /\ \ / \ / \ / \/ W Z \ /\ \ / \ / \/ t_B / FBB -------> GBB for W the kernel of F*R and Z the image of G*R. (end of definition) These transformations compose and, in effect, form a ccc (with bifunctors as objects). We may call a polymorphic function parametric if it comes from such a transformation. Remark: By letting R := the graph of some morphism it follows that every "very strong dinatural transfomation" is dinatural in the Yoneda-MacLane-Bainbridge-Freyd-Scedrov-Scott sense. Now we are ready to define the corresponding notion of an "end" as a universal transformation from some constant functor to a bifunctor and prove that in the case of FXY == (TY=>X)=>Y (for T covariant) such an end gives the initial T-algebra for T. Moreover one can show that the initial T-algebra _is_ an end, so that e.g. in the category of sets an end for FXY=(Y=>X)=>(X=>Y) exists (namely the NNO). In this sense the category provides a "bad model for polymorphism" looked for by E.Robinson in that it distinguishes different notions of parametricity. At least as far as ML polymorphism is concerned. As a side effect this provides a characterisation of a strong NNO in a ccc with finite limits by pure equations. Martin Hofmann ==========================================================================