Re: question about monoidal categories

[Steve Lack <steve.lack@mq.edu.au>]

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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