Proofs from THE BOOK

Proofs from THE BOOK

[Ellis D. Cooper]

Dedicated to the memory of Paul Erdos, Martin Aigner and Gunter M.
Ziegler's book (with the Subject title) offers examples of proofs
with "brilliant ideas, clever insights and wonderful observations."
They include chapters on Number Theory, Geometry, Analysis,
Combinatorics, and Graph Theory. My question is, what would likely be
included if there were a chapter on Category Theory?

Ellis D. Cooper



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Re: Proofs from THE BOOK

[Wojtowicz, Ralph]

The proof of Proposition 2.7.1 on page 59 in Volume I of "Handbook of Categorical Algebra" by Francis Borceux may be a candidate.  John Gray once told me that the proof is due to Peter Freyd.  When I was studying category theory in graduate school, proofs of this proposition, the Adjoint Functor Theorem, and some other results in "Categories, Allegories" struck me as demonstrating a creative, original, and insightful use of limits and colimits.

Best wishes,
Ralph Wojtowicz
Metron, Inc.
1818 Library Street, Suite 600
Reston, VA  20190


On Wed, 24 Jun 2009, Ellis D. Cooper wrote:

> Dedicated to the memory of Paul Erdos, Martin Aigner and Gunter M.
> Ziegler's book (with the Subject title) offers examples of proofs
> with "brilliant ideas, clever insights and wonderful observations."
> They include chapters on Number Theory, Geometry, Analysis,
> Combinatorics, and Graph Theory. My question is, what would likely be
> included if there were a chapter on Category Theory?
>
> Ellis D. Cooper
>




[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Proofs from THE BOOK

[Michael Barr]

Well, you might want to look at this:

\item  (With. M.-A. Knus), Extensions of Derivations. Proc. Amer.
      Math. Soc. {\bf 28} (1971), 313--314.

This is more homological algebra than category theory. The story is that
someone had done this in a special case (where the center was local I
think) in a dozen pages and someone else had extended it to semilocal and
Knus was lecturing at the ETH on his extension to arbitrary centers (but
the ambient ring was always semi-simple = Hochschild dimension 0).  As
Knus was lecturing I said to myself that there had to be a better way.
And there was.  It took only a paragraph and use only Hochschild dimension
1 besides.  The rest of the two pages was intro and bibliography.

Although it isn't category theory it exemplifies the categorical way of
thinking, dealing with generic properties and the like.

Incidentally, the paper was originally rejected.  The referee's report
said, "The only possible reason for publishing this is that it has been so
badly handled in the literature"!  I would have thought that an excellent
reason to publish it.  Only the fact that the editor was a personal friend
who said he would publish it if I insisted, allowed it to see the light of
day.  But I have long thought it a "proof from the book".

Michael

On Wed, 24 Jun 2009, Ellis D. Cooper wrote:

> Dedicated to the memory of Paul Erdos, Martin Aigner and Gunter M.
> Ziegler's book (with the Subject title) offers examples of proofs
> with "brilliant ideas, clever insights and wonderful observations."
> They include chapters on Number Theory, Geometry, Analysis,
> Combinatorics, and Graph Theory. My question is, what would likely be
> included if there were a chapter on Category Theory?
>
> Ellis D. Cooper
>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]