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

["jpradines" <jpradines@wanadoo.fr>]

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.