Re: Questions on dinatural transformations.

[Vaughan Pratt <pratt@CS.Stanford.EDU>]

>From: noson@sci.brooklyn.cuny.edu
>I was hoping that the category of small categories, functors and
>dinat transformations...

There's a category problem already at this point.  Dinats don't go between
functors F,G:C->D, they go between sesquifunctors F:C^op x C->D and differ
from n.t.'s of that type by only being defined on the diagonal of C^op x C.
The off-diagonal and non-identity-morphism entries in F,G only participate
in the dinaturality condition, not in the transformation itself.

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

Before you can compose dinats horizontally you have to be able to compose
the sesquifunctors they bridge.  I don't know how others do this, but if I
had to compose G:D^op x D -> E with F:C^op x C -> D, my inclination would
be to restrict the evident composite G(F(a,b),F(c,d)) to a=d, b=c (i.e.
where the variances match up).  That is, GoF:C^op x C -> E is defined by
G(F(c,c'),F(c',c)) on object pairs (c',c) of C^op x C, with the expected
extension to morphism pairs (f',f) where f':c'->d' in C^op (i.e.
f':d'->c' in C) and f:c->d in C, namely

  G(F(f,f'),F(f',f)): G(F(c,c'),F(c',c)) -> G(F(d,d'),F(d',d)).

With that (or some) choice of sesquifunctor composition one can then ask
about horizontal composition tos where s:F->F', t:G->G'.  How would you
whisker a dinatural on the left, i.e. apply the whisker G:D^op x D->E
on the left to the dinat s:F->F' on the right where F,F':C^op x C->D?
For natural transformations, G is just a functor G:D->E, so this is just a
matter of applying G pointwise to each s_c.  For dinaturals however, G is
a sesquifunctor.  What do you want a sesquifunctor to do to a morphism s_c?
Maybe there's some span-like thing one can do here but I don't see it.

For dinaturals, vertical composition may turn out to be easier than
horizontal, in that it at least makes sense provided one solves the
shape-matching problem somehow.  In doing so one also solves another
problem, that dinaturality is too weak a condition, typically admitting
transformations on the internal hom that aren't Church numerals (Pare & Roman,
JPAA 128 33-92 for Set, Pratt, TCS 294:3, bottom of p461, for Chu(Set,K) and
chu(Set,K) which awkwardly seem to need different treatments).  Mike Barr
has a notion of strong dinatural (unpublished?), and the notion of binary
(more generally n-ary) logical transformation also works well here when
definable on the category of interest.

Vaughan Pratt