Subject: categories: Questions on dinatural transformations.

From: "Noson Yanofsky" <[email-masked]>
Date: Tue, 29 Jun 2004 13:19:46 -0400

Hello,

Two quick questions:

a) It is well known that there is no vertical
composition of dinatural transformations.
How about horizontal composition?

i.e. Given
S,S':C^op x C---->B
T,T':C^op x C ----> B^op
U,U':B^op x B --->A
\alpha: S--->S' dinat
\alpha': T--->T' dinat
and
\beta: U--->U' dinat

is there a \beta \circ (\alpha',\alpha) and is it dinat?
It should be. But I can not seem to find the right definition.

How about if we restrict to a nice category of moduals for a nice algebra
over a nice field? Does that help?

I was hoping that the category of small categories, functors and
dinat transformations
should be a graph-category (a category enriched over graphs) but
am having a hard time
finding what the composition is. Did someone write on these
things?

b) Also, I was wondering if anyone ever wrote about
quasi-dinatural transformations. Those
are dinats where the target category is a 2-category and the
hexagon commutes up to a
two cell. They show up in something I am working on. But they are
very painful. Has anyone
worked on such things?

Any thoughts?

All the best,
Noson Yanofsky

---

From: Phil Scott <[email-masked]>
Date: Wed, 30 Jun 2004 21:15:52 -0400 (EDT)

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