Re: Composing modifications

[David Leduc <david.leduc6@googlemail.com>]

Dear Nick and Tom,

Thank you very much for your replies. It is very helpful.

I had in mind to form a tricategory of bicategories, therefore I guess
I was talking of what Tom calls strong transformations. They are also
called weak??? I am a bit confused by the lax, weak, pseudo, strict
and so on terminology in higher category theory. Nick, could you
confirm that you were talking of strong transformations in your mail?
Now I am not sure anymore what are natural transformations in category
theory. They are strict transformations, right?

Unfortunately, I do not have a copy of the paper "Coherence for
tricategories" by Gordon, Power and Street. I guess such reference
would help me a lot with such questions.

I have another question. For strong and strict (and maybe lax?)
transformations, we have the interchange law relating vertical and
horizontal composition.  What is the equivalent of interchange law for
compositions of modifications?

Thank you,

David


On 3/3/10, Tom Leinster <tl@maths.gla.ac.uk> wrote:
> 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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