Function composition of natural transformations? (Pat Donaly)

Function composition of natural transformations? (Pat Donaly)

[Jpdonaly]

Here is a technical/pedagogical question which has been bothering me for
about twelve years.

In problem 5 on page 19 of "Categories for the Working Mathematician" (CWM),
Saunders Mac Lane points out that a natural transformation may be fully
extended to an intertwining function from one category to another, intertwining
meaning (except in the void case), that the function transforms on one side
according to its domain functor and on the other according to its codomain functor.
Then on page 42 Mac Lane introduces what he calls "horizontal" composition
diagramatically and without reference to the fully extended intertwining
functions. But the function composite of such a pair of functions trivially
intertwines the function composite of the domain functors with that of the codomain
functors, and this function composition operation is very quickly verified to be
"horizontal" composition when written in terms of restrictions to sets of
objects. Thus Mac Lane and everyone else I have read leaves the impression that an
honest verification of, say, the associativity of "horizontal" composition
would require a somewhat involved diagrammatic demonstration which, in fact,
would be nothing other than the hard way to prove the associativity of function
composition. Presumably this has been noticed for a long, long time, but the
1998 edition of CWM did not mention it, and I can't help but be struck by the
fact that other authors' terminologies leave the impression that they don't know
or don't care that "horizontal", star or Godement composition is function
composition. Notationally, I am bemused to see the standard symbol for function
composition (the small circle) degraded into a generic symbol for the
composition of just about any category, while a star or an asterisk is frequently used
to denote what amounts to function composition done awkwardly.

This worries me that I am somehow overlooking something fairly blatant. Can
someone tell me what it is?

Jpdonaly@aol.com

Re: Function composition of natural transformations? (Pat Donaly)

[jpradines]

Just a few naive remarks about natural transformations and their two compositions.
I think comfortable to define natural transformations as functors, which is not the case, neither for the classical "object to arrow" nor for the "arrow to arrow" definition.
Now this can be achieved in two obviously equivalent ways.

In the following the letter I (though or because it is frequently used also to denote the unit interval) will picture the category denoted by 2 in CWM (just one arrow e going from the object 0 to the object 1) while QA will denote the category of commutative squares in the category A, endowed with the "horizontal" composition. (Note that the arrows of A may be described as functors from I to A and the arrows of QA as functors from the product category I x I to A).

Then the first definition is : a natural transformation between the categories A and B is nothing else than a functor from A to QB. 
With this first definition, the vertical composition of natural transformations comes immediately from the "vertical" composition in QB (which is indeed a double category).

The secund definition (which is just a more symmetric reformulation of the "arrow to arrow" definition) seems to be less popular, though very useful in my opinion : a natural transformation between the categories A and B is also nothing else than a functor from the product category I x A to B. 
This can also may be viewed as a functor from I x A to I x B, letting the first component be just the canonical projection from I x A to I.
Then the horizontal composition is just the composition of functors.

The latter definition is especially convenient when working with categories (or groupoids) in a category. For instance one knows immediately what is a smooth natural transformation (of course I is endowed with the discrete structure). It also allows to define immediately the horizontal composition for "higher order natural transformations", replacing I by its n'th power.

                                                Jean PRADINES




----- Original Message ----- 
From: <Jpdonaly@aol.com>
To: <categories@mta.ca>
Sent: Saturday, May 31, 2003 5:19 AM
Subject: categories: Function composition of natural transformations? (Pat Donaly)


> Here is a technical/pedagogical question which has been bothering me for
> about twelve years.
> 
> In problem 5 on page 19 of "Categories for the Working Mathematician" (CWM),
> Saunders Mac Lane points out that a natural transformation may be fully
> extended to an intertwining function from one category to another, intertwining
> meaning (except in the void case), that the function transforms on one side
> according to its domain functor and on the other according to its codomain functor.