Dear David,
I am reading Basic Bicategories by Tom Leinster, and I have basic questions about modifications.
1) Suppose that n, n', m and m' are transformations such that m * n and m' * n' are well defined, where * denotes horizontal (= Godement) composition of transformations. From given modifications a:m-->m' and b:n-->n' is there a way to derive a modification from m * n to m' * n' ?
The first thing to be careful about is horizontal composition of transformations. In that paper, "transformation" was used to mean what might more systematically be called "lax transformation". The paper also refers to "strong" transformations (Gray's terminology?, also called pseudo or weak), and strict transformations. For horizontal composition of transformations, the situation is this: i. Lax: can't be done ii. Strong: can be done, after making a fairly harmless non-canonical choice of "left" or "right" iii. Strict: can be done, canonically. So in order for your question to make sense, I think you need to assume that the transformations are strong, at least. And in that case, yes, there is a canonical way to horizontally compose modifications in the way that you describe.
2) There are two ways to compose transformations: vertical and horizontal. What are the ways to compose modifications?
Provided that you're using strong or strict transformations (so that horizontal composition makes sense), there are three ways. You could call them vertical, horizontal and... transversal? But it's probably better to adopt a more systematic terminology and talk about "i-composition" for i = 0, 1, 2. Here i is the dimension of the cell that your two composable things have in common. For example, suppose that we were talking about composing 2-cells x and y inside a 2-category. Then: * vertical composition would be "1-composition", because you can do it when the 1-dimensional domain dom(x) of x is equal to the 1-dimensional codomain cod(y) of y * horizontal composition would be "0-composition", because you can do it when the 0-dimensional domain dom(dom(x)) of x is equal to the 0-dimensional codomain cod(cod(y)) of y. Best wishes, Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]