Re: Question on "On Closed Categories of Functors"

[street@ics.mq.edu.au]

> Let V be a symmetric monoidal closed category and C a small V-category.
> Brian Day constructs a symmetric monoidal closed structure ([C,V],@,E) on
> the V-category of V-functors [C,V] for some cases [3.3, 3.6], e.g. if
> (C,*,e) is a symmetric monoidal V-category [4.1]. The underlying
> *category*
> [C,V] of V-functors admits a closed symmetric monoidal structure from the
> enriched one by taking the underlying functor of each V-functor, the
> underlying natural transformation of each V-natural transformation.
>
> Because a closed symmetric monoidal category is canonically enriched over
> itself, the category [C,V] gets a [C,V] enrichment in this way.
>
> My question is: What does this [C,V]-enrichment of [C,V] have to do with
> the V-enrichment of [C,V]?
> Suppose C have a terminal object t. One gets a evaluation functor
> Ev_t:[C,V]-CAT-->V-CAT. Is this the connection between the two enrichments?

I think what you want here is the following observation.
Every closed monoidal V-category E is also an E-category.
The unit object j for tensor in E  is a monoid and so E(j,-) : E --> V
is a monoidal V-functor. Therefore by applying it on hom objects,
it induces a 2-functor E-Cat --> V-Cat. In particular, you can apply
the 2-functor to E itself to see it as a V-category.

Your example is for E = [C,V].

Ross

PS I have ordinary- (not enriched-) mailed your message to Brian himself.
He may want to add something when he gets it. But I hope I have the story
you need!