2006-05-30 § 2 Comments
Update: One correction and one addition; changes in red.
Second update: another correction.
I had a couple of very interesting responses to my speculation about n-categories. As well as John Baez’s comment, I had an email from Tom Leinster pointing out his Structures in higher-dimensional category theory, whose section III.7 contains a similar conjecture.
Thinking about what John wrote, I realise that I made a mistake. I was wrong to claim that the full sub-tricategory of Bicat determined by the 2-categories is a Gray-category: the problem is that post-whiskering with a pseudo-functor is not 2-functorial, even when the objects are 2-categories. (It looks as though Tom’s section III.3 contains the same error? Note that the 1-cells of his 2-Cat are pseudo-functors rather than 2-functors.)
So, if we’re looking at what kind of semi-strictness result can be obtained purely from Yoneda, we get something weaker than the Gordon-Power-Street theorem, but still useful. John’s comment seems to be hinting at a way to do better, and I’m starting to see what he’s getting at. The point is that Yoneda is not the only tool available, as the two-dimensional case illustrates. The Yoneda argument shows that every bicategory is biequivalent to a 2-category, but in fact more is true. Given a bicategory B, one can construct a 2-category st(B) in which a 1-cell is a composable sequence of 1-cells from B, and a 2-cell is a 2-cell in B from “the” composite of the source to “the” composite of the target. Then given a pseudo-functor F: B → C, one can construct a 2-functor st(B) → st(C), and this process preserves identities and composition. Presumably this is the construction that John thinks ought to define a triequivalence between Bicat and the tricategory he calls Gray.
I think I agree. In fact, I think section 5.6 of [GPS] proves that Bicat is triequivalent to Gray, precisely along these lines.
I’m not sure how this argument would generalise to higher dimensions. Defining st more or less seems to require having an “all diagrams commute” sort of coherence theorem. But if we are working with n-categories in which k-ary composition is a primitive operation, the difficulties are obviously lessened, so probably there is a sensible generalisation. I just haven’t thought about what it is. All that said, proving a weak semi-strictness theorem based purely on Yoneda would be an interesting way to get the ball rolling.
I’m still a bit surprised that proving Yoneda for tricategories is said to be so hard, but I’m sure that’s just because I haven’t actually tried to do it! (I recently wrote out a proof of the bicategory version, and while it requires a clear head to keep track of what needs to be done, each individual step is simple.) I’ll have a stab at the tricategory version, which should give me a better idea of what’s hard about it.