Re: Function composition of natural transformations?

[Tom LEINSTER <leinster@ihes.fr>]

This may be `mere' pedagogy for ordinary categories, but if you try the
same thing for 2-categories then it becomes a `genuine' issue.  To put it
another way, the two different but equivalent presentations of a concept
(natural transformation) become, on categorification, significantly
different.

First the 1-dimensional situation.  As I understand it, Pat Donaly's
original point was that given functors

    F, G: C ---> D

between categories, you can either define a natural transformation in the
standard way (assigning an arrow of D to each object of C) or in an
alternative way (assigning an arrow of D to each arrow of C).  With the
standard method, vertical composition of transformations is "easy" to
define, and horizontal composition is "hard".  With the alternative
method, horizontal composition is now easy to define (as Pat noted), but
vertical composition is "hard".  So the situation is reversed.

Of course, neither of these "hard"s is really hard, but in both cases you
have two evident ways of defining a composite - one left-handed, one
right-handed - and if you're going to do anything whatsoever with the
definition then you need to show that these two ways give exactly the same
result.

Now suppose that C and D are 2-categories and F and G are 2-functors.  It
doesn't matter whether C, D, F and G are strict or weak for the purposes
of this discussion.  Suppose we're interested in defining weak (=pseudo)
transformations F ---> G.  The usual way is to say that such a
transformation consists of a 1-cell

   alpha_c : Fc ---> Gc

for each c in C, together with an invertible 2-cell inside each naturality
square, satisfying axioms.  With this definition, vertical composition of
transformations is easy to define (and there's only one evident way of
doing it), but horizontal composition can be defined in two different
ways, which are not equal but canonically isomorphic.  An alternative
approach is to say that a transformation consists of a 1-cell

   alpha_f: Fc ---> Gc'

for each 1-cell

   f: c ---> c'

in C, together with certain further 2-cells, satisfying axioms.  You can
guess the rest of this paragraph: with this definition, horizontal
composition is now easily (and canonically) defined, but vertical
composition can be defined in two different ways, which are not equal but
canonically isomorphic.

You might think that this isn't a genuine difference or "problem" so far,
because everything is the same up to isomorphism.  But now suppose that
you're interested in *lax* transformations F ---> G (where F and G are
still 2-functors, as above).  The usual definition is that such a lax
transformation alpha consists of a 1-cell alpha_c as above for each object
c of C, and then a not-necessarily-invertible 2-cell inside each
naturality square (pointing in a direction fixed by convention),
satisfying axioms.  These lax transformations can still be composed
vertically perfectly easily, but horizontal composition is now
*impossible* to define.  (More accurately, you can define two different
horizontal compositions, but they're not isomorphic, only connected by a
non-invertible cell; you could of course choose one over the other, but it
won't have good properties.)  And if you define "lax transformation"
according to the alternative method, then horizontal composition is now
easy and vertical composition impossible.

In summary, if you define transformation of 1- or 2-category in the
standard style then vertical composition is always easy, and regarding
horizontal composition:

- for transformations of categories, it's canonically defined
- for weak transformations of 2-categories, it's not canonically defined
  (you have to choose "left" or "right"), but is canonically defined up to
  isomorphism
- for lax transformations of 2-categories, it's not defined at all.

If you use the alternative style then the situation is similar but with
"vertical" and "horizontal" reversed.

Puzzling.

Tom