RE: Set as a monoidal category

["Stephen Lack" <S.Lack@uws.edu.au>]

Dear Peter,

There's a paper 

Algebraic categories with few monoidal biclosed structures or none

of Foltz, Lair, and Kelly which studies monoidal closed structures on various
categories, and shows that the cartesian closed one is the only possibility for Set.
More generally, it shows that for many categories we know well, the only possible
monoidal closed structures are the ones we know well.

But this depends heavily on the closedness. Without that, as you say, one can use
the cocartesian monoidal structure (the coproduct). 

Here's a further infinite family of monoidal structures on Set. Let A be any set. Then 
define the tensor product * by X*Y=AXY+X+Y.

Steve.

-----Original Message-----
From: cat-dist@mta.ca on behalf of Peter Selinger
Sent: Wed 8/13/2008 10:23 AM
To: Categories List
Subject: categories: Set as a monoidal category
 
Dear Categoreans,

I know three monoidal structures on the category of sets, all of them
symmetric. Two are the product and coproduct, and I'll leave it to
your imagination to figure out the third one.

My question is: are these the only three? Proofs, counterexamples, or
references appreciated.

Thanks, -- Peter