Re: question about monoidal categories

[claudio pisani <pisclau@yahoo.it>]

Dear Steve,

your answer has been very useful to me.

Thus, for a closed complete category V, there are two related doctrines:

1) the V-categories [X,V], indexed by V-Cat;

2) the V-categories V_0^X, indexed by Cat. 

The former is "included" in the latter via the forgetful functor 
V-Cat -> Cat, and the latter has more structure since it is also an indexed  monoidal category.

I would like to know if the second doctrine, and its relationships with the  first one, had been considered explicitly somewhere (perhaps in some paper  by Brian Day?).

Thank you again

Claudio

  

--- Mar 19/4/11, Steve Lack <steve.lack@mq.edu.au> ha scritto:

> Da: Steve Lack <steve.lack@mq.edu.au>
> Oggetto: Re: categories: question about monoidal categories
> A: "claudio pisani" <pisclau@yahoo.it>
> Cc: categories@mta.ca
> Data: Martedì 19 Aprile 2011, 03:40
> Dear Claudio,
> 
> On 18/04/2011, at 8:37 PM, claudio pisani wrote:
> 
>> Dear categorists,
>> 
>> suppose V is a monoidal category, with underlying
> category V_0, and X is an ordinary category. Then (if I am
> not mistaken) the functor category V_0^X has a monoidal
> structure, defined point-wise by that  of V and each
> functor f:X->Y gives a strong monoidal functor.
>> 
>> First question : supposing V closed, under which
> hypothesis is V_0^X closed as well (as in the case V =
> Set)?
>> 
> 
> This will be true if V is complete (as you suppose below
> anyway). Writing, for simplicity, in the case where V is
> symmetric, the internal
> hom [f,g] for functors f,g:X->V_0 is given by the
> formula (which I hope will be legible)
> 
> [f,g]x = int_y [X(x,y).f(y),g(y)]
> 
> Here X(x,y) is the hom-set in X, and X(x,y).f(y) is the
> coproduct of X(x,y) copies of f(y), and int_y denotes the
> end over all object y in X.
> 
> This is a special case of Brian Day's convolution structure
> where the promonoidal structure on X is the ``cartesian''
> one, corresponding
> to the cartesian closed structure on [X,Set].
> 
>> Second question: supposing that V_0^X is  indeed
> closed and that V is suitably complete, so that reindexing
> along f has a right adjoint forall_f, then V_0^X is enriched
> over V by forall_X(A->B). How is this enrichment related
> to the usual one of [X,v]  when X is a V-category?
>> 
> 
> They're the same, essentially because limits commute with
> limits. 
> 
> All the best,
> 
> Steve.
> 
> 
>> Best regards,
>> 
>> Claudio
>> 


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