1 Jul 03:15 2004

### Re: Questions on dinatural transformations.

Phil Scott <phil <at> site.uottawa.ca>

2004-07-01 01:15:52 GMT

2004-07-01 01:15:52 GMT

In general, the naive horizontal merging of dinaturals fails to be dinatural. This is discussed in the article "Functorial Polymorphism" by Bainbridge, Freyd, Scedrov and me (Theoretical Computer Science 1990, pp. 35-64). Several counterexamples are given there. For example, in a cartesian closed category of domains or CPO's, consider a dinatural family Y_A: A^A --> A (e.g. in domains, let Y_A = the least fixed point operator). If you were able to compose this with the "polymorphic identity" dinat id_A: 1---> A^A (i.e. a dinat from constant functor 1 to (-) ==> (-) where id_A = the transpose of the identity on A), then the category would be degenerate (proved in BFSS, Appendix A.4). Of course, if the middle diamond (of an attempted merging of two dinat families) is a pullback or pushout, then merging works. (see BFSS, Fact 1.2). Re vertical merging, some things can be said quite generally: e.g BFSS, Propn. 1.3. For various generalizations, see Peter Freyd's paper "Structural Polymorphism" (in TCS, 1993, pp.107-129). Soloviev has also discussed compositionality of dinats in several articles in JPAA. Philip Scott On Tue, 29 Jun 2004, Noson Yanofsky wrote:(Continue reading)