Re: Symmetric monoidal closed categories

Re: Symmetric monoidal closed categories

[Ross Street]

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.

Re: Symmetric monoidal closed categories

[vs27]

[From moderator: apologies to Vincent Schmitt. He posted the answer below
and the first item recently reposted. The correct From: field was
inadvertently omitted.]


A bit more.
You may be interested by the V-category
of V-functors [A,B] for V-categories
A and B -- Take care of the sizes though.
V= SSet, Ab etc...
References for this: Day and Kelly certainly.
Kelly's "Basic concepts of enriched category theory"
or Day's thesis and early papers.

Best,
V.


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.
>
>Tony
>
>
>

Re: Symmetric monoidal closed categories

[vs27]

[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
>
>
>

Symmetric monoidal closed categories

[Bockermann Bockermann]

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.

Tony