reference on enriched monoidal categories

reference on enriched monoidal categories

[Fernando Muro]

Dear colleagues,

I'm looking for a reference where the following fact (that I believe to
be clearly true) is discussed:

Let V and C be biclosed monoidal categories. Suppose that V is symmetric
and that we have a strong braided monoidal functor z : V --> Z(C) to the
center of C in the sense of Joyal-Street. Assume further that the
functor z(-) \otimes Y : V --> C has a right adjoint Hom(Y,-) : C --> V
for any object Y in C. Then C is a monoidal V-category with Hom objects
in V given by this right adjoint.

You may assume that V and C are (co)complete if you wish.

It is easy to construct compositions morphisms, etc. in an elementary
way, but verifying all laws is a pain. This is why I'm willing to find a
reference.

All the best for the new year,

Fernando Muro


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Re: reference on enriched monoidal categories

[Richard Garner]

Dear Fernando,

I do not know of a discussion of the exact result you state, but much
of what you need to prove it is in the appendix to:

A Note on Actions of a Monoidal Category
G. Janelidze and G.M. Kelly
Theory and Applications of Categories, Vol. 9, 2001, No. 4, pp 61-91.

Best wishes,

Richard

On 12 January 2011 18:50, Fernando Muro <fmuro@us.es> wrote:
> Dear colleagues,
>
> I'm looking for a reference where the following fact (that I believe to
> be clearly true) is discussed:
>
> Let V and C be biclosed monoidal categories. Suppose that V is symmetric
> and that we have a strong braided monoidal functor z : V --> Z(C) to the
> center of C in the sense of Joyal-Street. Assume further that the
> functor z(-) \otimes Y : V --> C has a right adjoint Hom(Y,-) : C --> V
> for any object Y in C. Then C is a monoidal V-category with Hom objects
> in V given by this right adjoint.
>
> You may assume that V and C are (co)complete if you wish.
>
> It is easy to construct compositions morphisms, etc. in an elementary
> way, but verifying all laws is a pain. This is why I'm willing to find a
> reference.
>
> All the best for the new year,
>
> Fernando Muro

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]