Re: no fundamental theorems please

[Vaughan Pratt <pratt@cs.stanford.edu>]

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


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