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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