jgm56

I've been wondering whether the "arrows-only" definition of natural
transformation, had it appeared in Mac Lane's "Categories Work", would
look something like this:

"Given two functors S,T:C -> B, a natural transformation A from S to T
is a map from {identities of C} to B such that if x and z are
identities in C and y is any arrow in C and xy, yz exist, then
(Tx)(Ax), (Ax)(Sx), (Ty)(Ax), (Az)(Sy) exist and (Ty)(Ax)=(Az)(Sy)"

(xy denotes the composite of x and y, Tx the image of x under T)

Is there perhaps a 'nicer' arrows-only definition possible?

marc olschok

There is an exercise in the MacLane book that gives a description, similar
to the one above, but without restricting A to identities of C:
Given two functors S,T:C -> B, a natural transformation A from S to T
is a map from C to B, satisfying
A(xy) = (Tx)(Ay) = (Ax)(Sy) whenever the composition xy exists.

If you specialise this with either x or y an identity you can see that
for an arrow f: c --> c' the arrow A(f) is just the diagonal Sc ---> Tc' in Ac
Sc ---> Tc
| |
Sf | | Tf v v
Sc' ---> Tc'
Ac'

This definition is exploited further in
D. Pumpluen: "Elemente der Kategorientheorie".

Marc Olschok