[From moderator: apologies to Vincent Schmitt. He posted the answer below and the second item which will be reposted. The correct From: field was inadvertently omitted.] On Nov 19 2008, Bockermann Bockermann wrote:
Dear mathematicians,
I wonder if the following is true. Has anybody a reference, if this is the case?
For a complete and co-complete symmetric monoidal closed category C and a small category D the functor category Fun(D,C) is pointwise a symmetric monoidal category. Is this a closed symmetric monoidal structure? This is true for simplicial sets and simplicial abelian groups for example.
Thank you for any help.
What is true is the following: if D,C are symmetric monoidal then the category of symmetric monoidal functors D->C admits a symmetric monoidal structure given pointwise by that of C. The crucial point is the symmetry.
Tony