Dear Jamie, Let me add to Steve's remarks that: If one wishes to work in a purely algebraic setting, then it is possible to give a notion of tricategory which is such that the bicategories together with their homomorphisms, strong natural transformations and modifications comprise a canonical instance of this structure. This can be done by using this latter requirement to inform the very definition of tricategory that one gives. More specifically, I mean to suggest that one works in the setting of Michael Batanin's globular operads. Such a globular operad encodes a notion of higher category by describing the forms of composition that each instance of this notion must come equipped with. There are a SET of such composition operations for each reasonable shape of diagram which one might wish to compose; in particular, for the diagram whose shape is that of a pair of horizontally composable 2-cells. We might derive such a set of operations as follows. Consider the category M whose objects are given by a triple of finitely presentable bicategories A, B, C, homomorphisms f, g : A -> B and h, k : B -> C, and a pair of strong natural transformations a : f => g and b : h => k; and whose morphisms (A,B,C,f,g,h,k,a,b) -> (A',B',C',f',g',h',a',b') are given by a triple of STRICT homomorphisms u:A -> A', v:B -> B', w:C -> C' commuting with all the other data. There is a corresponding category N whose objects are given by a single strong transformation a : f => g : A -> B between homomorphisms between finitely presentable bicategories, and whose morphisms are given by compatible strict homomorphisms A->A' and B->B'. We now consider the set of all functors M -> N which on objects send (A,B,C,f,g,h,k,a,b) to a strong natural transformation of the form fh => gk and which on morphisms send (u,v,w) to (u,w). This defines a set of composition operations for diagrams whose shape is that of a pair of horizontally composable 2-cells. We may in a similar manner define sets of composition operations for other shapes of diagrams of 2- and 3-cells, in this way obtaining a globular operad. (At the 0- and 1-cell level, we take it that there is a unique composition operation of each shape; we do this because we have a canonical, strictly unital and associative, way of composing homomorphisms of bicategories). This construction does indeed yield a 3-dimensional globular operad as it is really just the construction of the endomorphism operad of a globular object in a monoidal globular category, which you may find described in Michael's paper on the subject. (In fact this is not quite true as we are doing something slightly different at the 0- and 1-cell level but that is easily accounted for). The point is that this globular operad is, unless something is terribly wrong with the world, contractible, meaning that it provides a fully sensible notion of weak 3-category. And almost by definition, the totality of bicategories and their weak cells comprise a canonical instance of this structure. Richard On 24 October 2011 09:59, Steve Lack <steve.lack@mq.edu.au> wrote:
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.
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