Re: no fundamental theorems please

[Michael Barr <barr@math.mcgill.ca>]

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/ ]