Re: Fundamental Theorem of Category Theory?

[Makoto Hamana <hamana@cs.gunma-u.ac.jp>]

Dear Ellis,

On Fri,  5 June 2009 16:36:23 -0400, Ellis D. Cooper wrote:
| There are Fundamental Theorems of Arithmetic, Algebra, Calculus, and
| indeed, many more.
| My question is, What would be candidates for the Fundamental Theorem
| of Category Theory?
| Yoneda Lemma comes to my mind. What do you think?

I have asked Prof. Yoneda many years ago why Yoneda Lemma is
called "Lemma", not "Theorem". He said that perhaps it was a
bit about internal of category theory rather than insisting
on applications to other mathematics. Doesn't Yoneda Lemma
satisfy (c) in Mile Gould's post? I don't know how much
Yoneda Lemma is useful in other areas of mathematics, and
I have wanted to know it.

On Sat,  6 June 2009 23:22:52 +0100, Miles Gould wrote:
| My suggestion would be the theorem that left adjoints preserve colimits,
| and right adjoints preserve limits.
| This may not be the deepest theorem in category theory, but
| (a) it's pretty darn deep,
| (b) it describes a beautiful connection between two fundamental notions
| in the subject,
| (c) it admits a huge variety of applications in "ordinary" mathematics.

Best Regards,
Makoto Hamana



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