Re: Function composition of natural transformations? (Pat Don...

Re: Function composition of natural transformations? (Pat Don...

[Jpdonaly]

Thanks, Jean. As an assiduous student of CWM, I was aware of this and will
always wonder why Mac Lane didn't just make the point explicit in his first
edition in 1971. The only thing left to realize is that the category of
commutative squares which you mention is a subcategory of a product category and thus
has a couple of projection functors on it which can be used to follow a functor
to get the domain and codomain functors of the natural transformation, so that
this version of naturality is much more neatly packaged than the usual
diagram. I believe that there is a worker named John Baez (deep apologies for any
naive and unforgivable errors here) who says that Mac Lane claimed to be
interested not in functoriality so much as naturality when he was coinventing
category theory; I wonder when and if he realized that naturality is a brand of
functoriality. It would seem that this realization would come very early. In
general, if one fixes an argument in a bifunctor, the resulting function is a fully
extended intertwining function, and I believe that your point is that every
natural transformation arises in this way. So already naturality is an artifact
of functoriality. Mitchell notices much of this in his 1965 book.

Re: Function composition of natural transformations? (Pat Don...

[jpradines]

Thank you for your quick answer.
I am not sure to understand it quite well. I must confess that I consider myself much more as a differential geometer than as a "categoricist", so that I am not at all well informed about categorical literature. In fact I am often considered as a dissident in both fields, since my general philosophy is :
-first that there is a lot of interesting things to do in Geometry using categories, an idea with which unfortunately most geometers, save Ehresmann, disagree ;
- secund that most of the classical tools developed by category theorists, including Ehresmann, are not the best adapted for that purpose, and I tried to develop some alternative tools (which indeed most people ignore).
It is not the place here to say more on that subject. 
About the precise point in discussion the main object of my letter was to emphasize the secund aspect of the definition (functor from I x A to B or more symmetrically to I x B), which seems less familiar (is it written somewhere ?) than the first one, and in my opinion the simplest and the most useful (at least for my personal use). For each arrow f of A from a to a', the image of  I x {a, f, a'}, which may also be regarded as an image of I x I, gives immediately a commutative square in A and its diagonal (3 x 3 = 9 arrows including the unit ones), where one reads immediately all the useful aspects of the data, displayed in a perfectly geometric and symmetric way, in contrast with the usual definitions, in which the two functors and the naturality arrow seem to play quite unsymmetrical roles. (Of course all of this is so perfectly obvious that I am convinced that plenty of people have noticed this, but I must confess that I have never read (nor heared) it, though in my opinion it is the best way of describing a natural transformation.) 
I think also than it would be often useful to consider more frequently the obvious canonical structure of double category on a product category and especially on I x A, and I x I.
                Jean PRADINES
  ----- Original Message ----- 
  From: Jpdonaly@aol.com 
  To: jpradines@wanadoo.fr 
  Cc: categories@mta.ca 
  Sent: Tuesday, June 10, 2003 2:17 AM
  Subject: Re: categories: Re: Function composition of natural transformations? (Pat Don...


  Thanks, Jean. As an assiduous student of CWM, I was aware of this and will always wonder why Mac Lane didn't just make the point explicit in his first edition in 1971. The only thing left to realize is that the category of commutative squares which you mention is a subcategory of a product category and thus has a couple of projection functors on it which can be used to follow a functor to get the domain and codomain functors of the natural transformation, so that this version of naturality is much more neatly packaged than the usual diagram. I believe that there is a worker named John Baez (deep apologies for any naive and unforgivable errors here) who says that Mac Lane claimed to be interested not in functoriality so much as naturality when he was coinventing category theory; I wonder when and if he realized that naturality is a brand of functoriality. It would seem that this realization would come very early. In general, if one fixes an argument in a bifunctor, the resulting function is a fully extended intertwining function, and I believe that your point is that every natural transformation arises in this way. So already naturality is an artifact of functoriality. Mitchell notices much of this in his 1965 book.