I think another thing which it suffices to assume is that the monoid structure maps AxA -> A and I -> A are not merely natural transformations (which is what it means for every map to be a monoid morphism) but *monoidal* natural transformations. This should also make the monoid structures on AxB and I into the usual ones. But I don't know whether one can get away without any such condition. On Tue, Mar 12, 2013 at 6:58 AM, 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
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