One of my favourite recent books is Tom Leinster’s Higher Operads, Higher Categories. Among the things I like about it is that he’s not afraid to dwell on issues that are conventionally dismissed as trivial, such as the distinction between classical and unbiased monoidal categories and bicategories. The more I think about it, the more I think that the distinction deserves attention: perhaps I’ll return to that in a future post.
There’s a brief remark on p. 123, where he mentions one of the benefits of unbiased bicategories as compared to classical ones. Any bicategory B has a hom pseudo-functor Bop×B→Cat; but defining its action on 1-cells requires a non-canonical choice, when working with classical bicategories. The pseudofunctor should take a pair (f,g) of arrows to the function p ↦ gpf, but classically there is no ternary composition, so we are forced to make an arbitrary choice to bracket this as either g(pf) or (gp)f.
Recently I needed to define the curried version of this pseudo-functor, which goes Bop→[B,Cat]. I’m working with classical bicategories, but I didn’t have to make any arbitrary choices in the definition. (Try it!) That didn’t surprise me at the time, but in the context of Leinster’s remark it seems distinctly strange. It seems that the mere act of uncurrying has introduced some arbitrariness! And sure enough, if you try to define a biequivalence
Bicat(A×B,C) ≈ Bicat(A, [B,C]),
some arbitrary choice does seem to be needed to go from right to left. Suppose we have a pseudofunctor F: A→[B,C], and denote its uncurried version F’: A×B→C. How should we define F'(f,g), for arrows f: W→X and g: Y→Z? Clearly it must be either F(X)(g)⋅F(f)Y or F(f)Z⋅F(W)(g), but neither choice is canonical. There is an invertible 2-cell F(f)g between them, so in a sense it doesn’t matter which we choose, but still one of them must be chosen.
And it matters not a jot whether the bicategories are classical or unbiased; the problem stems from the way pseudo-natural transformations are defined. In the special case of the hom pseudofunctor, ternary composition provides a way out, but in general the difficulty remains.