Dear Jamie, I agree that there is no canonical choice of horizontal composition of pseudonatural transformations, but that the various possible choices are, in a suitable sense, equivalent. Whether or not you should be bothered by this I can't really say. But perhaps it's worth pointing out that there are various different possible descriptions of the structure of weak 3-category, and not all of them include a chosen horizontal composition of pseudonatural transformations. Some, particularly, the simplicial approaches, include *no* choices of compositions. Others include some choices of composition, but not this particular one. For example, the notion of Gray-category does include chosen composition of 1-cells, and vertical composition of 2-cells, but does not include a chosen horizontal composition of 2-cells. Best wishes, Steve Lack. On 21/10/2011, at 9:17 PM, Jamie Vicary wrote:
Dear categorists,
Suppose you have categories A, B, C, and functors S,S': A-->B, T,T': B-->C, and natural transformations alpha: S==>S', beta: T==>T'. Suppose we want to see these as part of a 2-category of categories; then we had better know the horizontal composite of alpha and beta. There are two possible ways to evaluate this composite: as the natural transformation having components beta_{S'X}.T(alpha_X), and as the natural transformation having components T'(alpha_X).beta_{S(X)}. But these are equal, since beta is a natural transformation. So we have no difficulty uniquely defining our horizontal composite, and obtaining a canonical 2-category of categories.
But now suppose that A, B, C are bicategories, S,S',T,T' are pseudofunctors, and alpha and beta are pseudonatural transformations. Then the two possible definitions for the horizontal composite of alpha and beta will not necessarily be equal, although of course they will be related by an invertible modification. But then we have a problem forming the tricategory of bicategories, pseudofunctors, pseudonatural transformations and modifications: there is no longer a canonical choice available for horizontal composition of pseudonatural transformations.
Presumably this choice can be made, and a tricategory is the result, and different choices yield equivalent tricategories. But it bothers me that there seems to be no canonical tricategory of bicategories. Should it? Or is my reasoning flawed?
Jamie.
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