Products in a compact closed category

Products in a compact closed category

[Robin Houston]

Dear categorists,

I recently proved that products (or coproducts) in a compact closed
category are necessarily biproducts, and I'm wondering whether this
is a known theorem. I can't find any reference to it in the
literature, but the proof is not hugely complicated and it would
not surprise me to learn that someone noticed it before this week!

(More precisely, I can prove that given a monoidal category that has
(finite) sums and products, if the tensor distributes over the sums
on one side and the products on the other -- e.g. for every object A,
-*A preserves sums and A*- preserves products -- then the products and
coproducts are both really biproducts.)

Any references or recollections will be much appreciated.

Yours,
Robin

Re: Products in a compact closed category

[Robin Houston]

On Thu, Mar 23, 2006 at 12:17:05PM +0000, Robin Houston wrote:
> I recently proved that [finite] products (or coproducts) in a compact
> closed category are necessarily biproducts, and I'm wondering whether
> this is a known theorem.

Since a couple of you have been asking about this, I thought
I should post a quick followup message.

I had several private replies, almost all of them asking to see
the proof. None of my respondents knew of an existing proof.

A preliminary paper, describing the proof, in far more detail
than most of the people on this list would need, is available
from

  http://www.arxiv.org/abs/math.CT/0604542

(If any of you have any comments on it, I'd be interested in hearing
them.)

Yours,
Robin