Re: Composing modifications

[Tom Leinster <tl@maths.gla.ac.uk>]

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


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