Re: enriched Kan-extensions

Re: enriched Kan-extensions

[Steve Lack]

Dear Tony,

As you say, this depends on how the monoidal structures on the functor
categories are defined. The only sensible way I know of which uses the given
structures on A and B is by Day convolution, and then your result will hold
if F itself is strong monoidal.

See the Day-Street paper "Kan extensions along promonoidal functors" in
volume 1 of TAC.

Regards,

Steve Lack.


On 14/02/09 12:47 AM, "Tony Meman" <tonymeman1@googlemail.com> wrote:

> Dear category theorists,
> I have a question concerning enriched left Kan-extensions.
>
> My situation is the following:
> V is a complete and cocomplete symmetric monoidal closed category, A and B
> two small V categories and F:A-->B a V-functor. Via the V-left-Kan extension
> one gets a V-adjunction
> Lan_F: V-Fun(A,V)<-->V-Fun(B,V):F*
> where F* denotes the precomposition with F.
>
> Moreover, the V-category V-Fun(A,V) and the V-category V-Fun(B,V) are
> equipped with a symmetric V-monoidal structure respectively. Is it known,
> under which conditions the adjunction (Lan_F,F*) is actually a monoidal
> adjunction? Surely, it must have something to do with F: I suppose that F
> have to be a symmetric monoidal functor with respect to a symmetric
> V-monoidal structure on and A and B. The V-monoidal structures on A and B
> also should have something to do with the V-monoidal structure on V-Fun(A,V)
> and V-Fun(B,V).
>
> Does anyone know a reference for this situation?
>
> Thank you in advance for any help.
> Tony
>
>

enriched Kan-extensions

[Tony Meman]

Dear category theorists,
I have a question concerning enriched left Kan-extensions.

My situation is the following:
V is a complete and cocomplete symmetric monoidal closed category, A and B
two small V categories and F:A-->B a V-functor. Via the V-left-Kan extension
one gets a V-adjunction
Lan_F: V-Fun(A,V)<-->V-Fun(B,V):F*
where F* denotes the precomposition with F.

Moreover, the V-category V-Fun(A,V) and the V-category V-Fun(B,V) are
equipped with a symmetric V-monoidal structure respectively. Is it known,
under which conditions the adjunction (Lan_F,F*) is actually a monoidal
adjunction? Surely, it must have something to do with F: I suppose that F
have to be a symmetric monoidal functor with respect to a symmetric
V-monoidal structure on and A and B. The V-monoidal structures on A and B
also should have something to do with the V-monoidal structure on V-Fun(A,V)
and V-Fun(B,V).

Does anyone know a reference for this situation?

Thank you in advance for any help.
Tony