Let me second Paul's observation. Going from the Yoneda lemma to a more sophisticated version would be analogous to saying that the Fundamental Theorem of calculus is "really" Stokes's theorem in its most general form (that the integral of form over the boundary of an n-dimensional compact region is the integral of the differential of the form over the region). The point of fundamental theorems is that they are easy to state and somehow capture something essential about the subject. The original Yoneda lemma states that all the properties of an object are present in its homfunctor. Thus objects are really captured by the morphisms. That it then leads to very sophisticated extensions is the point, but it, not they, are the basis for it all. Michael On Mon, 22 Jun 2009, Paul Taylor wrote:
I see that Takuo and I have still not overcome the "royalist" forces.
Claudio, Vaughan, Ross and others have mentioned various more "sophisticated" versions of the Yoneda Lemma.
However, I contend that the more sophisticated the result is, the LESS it deserves to be called "the fundamental theorem of category theory".
...
And the Yoneda Lemma will survive as long as Category Theory does in a recognisable form.
But it still shouldn't be called the "fundamental theorem"!
Paul
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]