Re: Function composition of natural transformations?

[Marc Olschok <sa796ol@uni-duisburg.de>]

> 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.[...]

At least in the book "Elemente der Kategorientheorie" by D. Pumpl\"un
the above characterization of natural maps is used explicitely; there is
also a short discussion on obtaining simpler proofs this way.

For the above reason \circ is used for the "horizontal composition";
\cdot or \ast (I do not remember which one) is used for the
"vertical composition", which after all looks more "point-wise".

Unfortunately some authors use these symbols just the other way round.

Marc