Re:Fundamental Theorem of Category Theory?

Re:Fundamental Theorem of Category Theory?

[Miles Gould]

On Fri, Jun 05, 2009 at 04:36:23PM -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?

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.

I've occasionally referred to this theorem as the Fundamental Theorem of
Category Theory by way of emphasizing its importance while teaching, but
I've always immediately clarified that it's only me who uses this term :-)

Miles

-- 
Sometimes it's best to do nothing, if it's the right sort of nothing.
  -- The Doctor


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Re:Fundamental Theorem of Category Theory?

[Ronnie Brown]

Limits, colimits, and adjoints:

I go along with this: it is the result of general category theory that I
have used most in studying colimits of forms of multiple groupoids, for
homotopical applications. It really does come under `categories for the
working mathematician'.

I have also been attracted in the same vein by fibrations and cofibrations
of categories: see a recent paper in TAC.

I well remember a remark of Henry Whitehead in response to a visiting
lecturer saying: `The proof is trivial.' JHCW: `It is the snobbishness of
the young to suppose that a theorem  is trivial because the proof is
trivial.'  (There  was and is no answer to that!) (His example was
Schroder-Bernstein.)

The leads to the interesting question of what makes a theorem nontrivial?
Good discussion topic for the young (at heart).

Ronnie Brown




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