On 6/22/2009 2:07 PM, Paul Taylor wrote:
Claudio, Vaughan, Ross and others have mentioned various more "sophisticated" versions of the Yoneda Lemma.
I don't see how supplying the missing half of an if-and-only-if is "more sophisticated." The Yoneda Lemma usually states that J embeds (meaning fully) in [J^op,Set], but it could just as well state this for any factor C between J and [J^op,Set]. In such situations it is natural to ask whether the converse holds; it doesn't, but if one adds to full-and-faithful the (very natural) requirement of density then it does: the dense extensions of J are precisely the categories of presheaves on J, by which I mean the full subcategories of [J^op,Set] that retain J as a full subcategory (the sense of "on"). I don't call that sophisticated.
However, I contend that the more sophisticated the result is, the LESS it deserves to be called "the fundamental theorem of category theory".
Indeed. Any branch of mathematics that does so only contributes to the image of mathematics as a difficult subject.
I particularly like Euclid's algorithm for the highest common factor as a historical and methodological example, Without meaning to dictate to number theorists how their subject should be organised, let me suggest for the sake of argument that it is a pretty good candidate for being called the "fundamental theorem of number theory".
That's an algorithm. The relevant theorem is also called the fundamental theorem of arithmetic. One could state the essential idea in sophisticated language as "Z is a principal ideal domain" but the more usual statement about uniqueness of factorization of positive integers makes number theory a more accessible subject.
As I say, I like Euclid's algorithm because the idea has survived many many revolutions in the "official" foundations of mathematics.
[...] Cantor's "theorem" about a powerset being
strictly bigger than a set. This belongs entirely to the dogma of set theory. When set theory is overturned, this miserable and wholely misguided "theorem" will go in the dustbin of mathematical history with it.
I like Cantor's theorem because, like Euclid's algorithm, it will survive the revolution Paul is trying to foment here.
But Euclid's algorithm will live forever.
As will diagonalization arguments like Cantor's.
But [Yoneda's Lemma] still shouldn't be called the "fundamental theorem"!
Unlike the Fundamental Theorems of Arithmetic and of Algebra, the Yoneda Lemma has not yet established itself as *the* fundamental theorem of category theory. Nor will it unless a reasonable consensus to that effect emerges. I suggest waiting a few years before coming to any conclusion about whether CT has an FT, and if so what it is. Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]