Dear Claudio, One place with a spelled out proof that you could refer to is Theorem 2.1 of http://dx.doi.org/10.1016/j.entcs.2008.10.012: Let (C,+,0) be a symmetric monoidal category. Then (+,0) provides finite coproducts if and only if the forgetful functor cMon(C)->C is an isomorphism of categories. At that point I, like you, couldn't find any references, but I'm sure it is a well-known piece of folklore, and would be interested if you can trace its earliest appearance in the literature. Best wishes, Chris On Tue, Mar 12, 2013 at 1:58 PM, claudio pisani <pisclau@yahoo.it> wrote:
Dear categorists,
in several places I have seen variants of the following statement (or its dual):
If C is a symmetrical monoidal category and every object has a natural monoid structure (that is any map is a monoid morphism) then C is cocartesian monoidal (tensor = sums).
I have not found any proof of the general statement. In fact the tensor product of two objects has obvious maps A -> AxB <-B and for any A -> X <- B it's easy to see that there is at least one factorization AxB -> X. The problem is unicity. If the monoid structure on objects is unique, then that on AxB (and on I) has to be the usual one and unicity follows.
Question: is the statement true in the above general form or one has to assume that the monoid structure on objects is unique? Or that it is commutative?
I would be grateful for any suggestion or reference.
Claudio
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]