Vaughan Pratt wrote:
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)).
There is a `canonical' choice of composition for such `sesquifunctors' (what follows is presumably folklore and written up somewhere). Consider the category SDCat of *self-dual* categories: objects are categories C, equipped with a duality c: C -> C^op (with c^op c = id), and morphisms F: (C,c) -> (D,d) are functors F:C -> D such that F^op c = d F. The forgetful SDCat -> Cat admits both adjoints (and SDCat is actually both monadic and comonadic over Cat): the right adjoint takes a category A to (A^op x A, s) where s is the switch isomorphism (the second projection \pi' : A^op x A -> A is the counit of the adjunction). We thus get a comonad G on Cat, and sesquifunctors are the morphisms of the resulting Kleisli category Cat_G, which tells us how to compose G:D^op x D -> E with F:C^op x C -> D. The composite is G(F^op s, F)\delta, which agrees indeed with the formula above. (This is of course the composite in SDCat via the adjunction)
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
The counterexamples mentioned in P. Scott's posting concern the lack of a well-defined *vertical* composition of dinaturals. If one persists on endowing them with such a composite, one possible approach is to accept the partiality of this composition and work with *paracategories*. Pushing this simple idea to its logical conclusion leads to a decent enough basic theory, which allows to make sense of the fact that `dinats into a ccc form a cartesian-closed paracat'. See Hermida, C; Mateus, P. Paracategories. II. Adjunctions, fibrations and examples from probabilistic automata theory. Theoret. Comput. Sci. 311 (2004), no. 1-3, 71--103. (also available at my homepage http://maggie.cs.queensu.ca/chermida) Making a 2-dimensional structure with dinats, using their partial vertical composition, leads to consider enrichment over ParCat (the cartesian closed category of paracategories). But whiskering (and therefore *horizontal* composition) is bound to be a partial operation as well, so one has to broaden/weaken ParCat to accommodate this fact. Ultimately, the kind of composite required in Yanosfky's posting: 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 \beta: U--->U' dinat ------------------------------------------------------------------- \beta \circ (\alpha',\alpha) suggests a *partial multicategory* structure (as introduced in the article above), with homs in ParCat. Claudio Hermida