According to the stats I’ve already had one reader, so perhaps it’s time for me to stick my neck out: this speculation tag is intended for things that might be utter nonsense, a fair proportion of which I shall no doubt be embarrassed to have written once I’ve realised what rubbish they are. The hope, of course, is that one or two might turn out to contain a useful idea. We shall see.
The theory of weak n-categories is very much in its infancy. (This recent-ish project proposal paints a helpful picture of what has been, and what remains to be done.) A mature theory of n-categories ought at least to have definitions of ‘n-functor’ and the n different kinds of ‘transformation’ needed to define an (n+1)-category of n-categories. I’m not aware that any of the proposed definitions has such a thing, though I must confess that I have read only a small fraction of the relevant literature, so I could be mistaken in that. Once that is done, there is an obvious desideratum that I haven’t seen mentioned: a Yoneda theorem, which at the very least should show how, for any (n+1)-category C and (n+1)-functor F: C → n-Cat, to exhibit an appropriate equivalence
FA ≈ (n+1)-Cat(C, n-Cat)(C(A, -), F).
I would imagine that such a theorem, together with a few supporting lemmas, would fairly readily yield a coherence theorem of the form “every weak n-category is equivalent to a semi-strict one”, along with a specification of what it means for an n-category to be semi-strict. It certainly works in dimensions two and three, as follows: the bicategorical Yoneda lemma shows that every bicategory B is biequivalent to its image in [Bop, Cat], which is a 2-category (since Cat is). In dimension three, the tricategorical Yoneda lemma (I should really say conjecture, since I don’t think there is a full proof yet) would show that every tricategory is triequivalent to one that is as strict as the full sub-tricategory of Bicat determined by the 2-categories. How strict is that? Well, I can easily convince myself that it’s a Gray-category. It is surely not much of a stretch to conjecture that every weak 4-category is equivalent to one that’s as strict as the full sub-4-category of Tricat determined by the Gray-categories. It can’t be hard to figure out just how strict that is, can it? I think it’s locally a Gray-category, and horizontal composition is strictly associative with a strict unit, and is also strict in each argument separately.
[See here for an update.]