Dear Tony Yes, this is true. It is a case of Brian Day's convolution theorem: see [Thesis2] Construction of Biclosed Categories (PhD Thesis, University of New South Wales, 1970) http://www.math.mq.edu.au/~street/DayPhD.pdf. [3] Day, Brian. On closed categories of functors. 1970 Reports of the Midwest Category Seminar, IV pp. 1--38 Lecture Notes in Mathematics, Vol. 137 Springer, Berlin Brian deals with enriched categories. For ordinary categories, your D is a symmetric comonoidal category via the "cotensor product" defined by diagonal D --> D x D; so it becomes a promonoidal category with P (a,b;c) = D(a,c) x D(b,c). The convolution tensor product on Fun(D,C) reduces to the pointwise one. For enriched categories, D would need to be symmetric comonoidal (e.g. if D were a free V-category on an ordinary category). When V = Vect, each "cosymmetric" bialgebra is a one-object such D. However, there are presumably other references for the particular case you have in mind as there are for the bialgebra case. Ross On 19/11/2008, at 8:33 AM, Bockermann Bockermann wrote:
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.