categories: Re: adjunction of symmetric monoidal closed categories

[Steve Lack <s.lack@uws.edu.au>]

On 20/01/09 6:11 AM, "Bockermann Bockermann" <tonymeman1@googlemail.com>
wrote:

> Dear mathematicians,
> I wonder if the following is true. Has anybody a reference, if this is
> the case?
>
> Let V and W be two complete and cocomplete symmetric monoidal closed
> categories and
> L: V <--> W :R
> an adjunction of (lax) symmetric monoidal functors. Let D be a small V-
> category.
> Is it true that there is a V-isomorphism
> V-Fun(D,RW) = R(W-Fun(LD,W)) ?
>
> (If not, is this at least the case if L is strict symmetric monoidal?)
>
> Thank you for any help.
> Tony
>
>

Dear Tony,

I pointed out this fact in my reply (see below) to one of your earlier
questions. In fact you don't need symmetry, and L is automatically strong
monoidal.

Regards,

Steve Lack.

%%%%%

On 6/12/08 10:21 AM, "Bockermann Bockermann" <tonymeman1@googlemail.com>
wrote:

> Dear mathematicians,
>
> could anybody give me a hint if the following assertion is true?
> Let V be a complete and co-complete symmetric monoidal closed category. The
> category sV of simplicial objects in V is also complete and co-complete
> symmetric monoidal closed with the pointwise tensor. There is a V-adjunction
> D:V<-->sV:Z
> of the V-functor Z which evaluates in 0 and the discrete V-functor D. Does
> this induce a V-Isomorphism of V-categories
> V-Fun(K,ZC)~sV-Fun(DK,C)
> for any small V-category K and any sV-category C?
>
> Please note that a similar statement is true for the non-enriched case [e.g.
> Borceux2, Proposition 6.4.8.].
>
> Thank you for any help.
>
> Tony
>
>
Dear Tony,

Yes, it is true. More generally, let F-|U:W-->V be a monoidal adunction.

This means that V and W are monoidal categories, F and U are monoidal
functors, monoidal natural transformations 1-->UF and FU-->1 satisfying the
triangle equations.

(A monoidal functor F:V-->W involves maps FX\otimes FY-->F(X\otimes Y)
and I_W-->F(I_V), not necessarily invertible, but satisfying coherence
conditions. In a monoidal adjunction, as above, the monoidal functor F is
necessarily strong, so that the comparison maps are invertible. The
comparison maps for U need not be invertible.)

For a small V-category K and a W-category C we do indeed have an isomorphism
    V-Fun(K,UC) = U(W-Fun(FK,C))
of V-categories. I'll do my best to explain this via ascii.


V-functors from K to UC are in bijection with W-functors from FK to C; this
takes care of the object-part. For V-functors M,N:K-->UC,
the hom-object V-Fun(K,UC)(M,N) is the equalizer of the evident maps
               --->
Pi_k UC(Mk,Nk) ---> Pi_{k,l} [K(k,l), UC(Mk,Nl)]
in V, where the products run over all objects k and l of K.

On the other hand, U(W-Fun(FK,C)(M,N)) is given by the equalizer of
                 -->
U(Pi_k C(Mk,Nk)) --> U Pi_{k,l} [FK(k,l),C(Mk,Nl)]
or equivalently, since U is a left adjoint, the equalizer of
               -->
Pi_k UC(Mk,Nk) --> Pi_{k,l} U[FK(k,l),C(Mk,Nl)]

So we are now left to prove

Lemma: U[FX,Y]=[X,UY], for X in V and Y in W.
Proof:

V(Z,U[FX,Y]) = W(FZ,[FX,Y]) = W(FZ\otimes FX,Y) = W(F(Z\otimes X),Y)
             = V(Z\otimes X,UY) = V(Z,[X,UY])

naturally in Z and so U[FX,Y]=[X,UY] as required.

Regards,

Steve Lack.