Preprint: A direct proof that the category of 3-computads is not cartesian closed

Preprint: A direct proof that the category of 3-computads is not cartesian closed

[Eugenia Cheng]

Dear All,

I've just posted a note on the arXiv, giving a short proof by counterexample 
that the category of 3-computads is not cartesian closed.

http://arxiv.org/abs/1209.0414

This result was first proved by Makkai and Zawadowski:

3-computads do not form a presheaf category. Journal of Pure and Applied 
Algebra, 212(11):2543--3546, 2008.

Their original proof depended on some results on Artin glueing of Carboni and 
Johnstone, and in turn some results of Day.  Recently François Métayer asked me 
if I knew of a direct counterexample instead, that is, a 3-computad B such that 
the functor _ x B does not have a right adjoint.

I did not know of anywhere that such a counterexample had been written up.   So 
I constructed one, and after showing it to François and others at Paris 7, 
decided it might be useful to make the notes available.

The counterexample uses the same ideas as the proof of Makkai and Zawadowski 
(essentially coming down to an Eckmann-Hilton argument) but the proof is 
self-contained in this 8 page note, and the counterexample itself is given in 
two pages in the middle.  I hope that this will help people further understand 
the very interesting and crucial result of Makkai and Zawadowski.

Comments are welcome.
Regards,
Eugenia


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Re: Preprint: A direct proof that the category of 3-computads is not cartesian closed

[Ross Street]

Dear Eugenia

On 04/09/2012, at 10:48 AM, Eugenia Cheng wrote:

> I've just posted a note on the arXiv, giving a short proof by counterexample that the category of 3-computads is not cartesian closed.
> 
> http://arxiv.org/abs/1209.0414
> 
> This result was first proved by Makkai and Zawadowski:
> 
> 3-computads do not form a presheaf category. Journal of Pure and Applied Algebra, 212(11):2543--3546, 2008.

A bit of shock hit me to see that the Makkai and Zawadowski paper was over 1000 pages!
But I see from MathSci that the correct page numbers are 2543–2546. 

I look forward to reading your approach.

Regards,
Ross

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