Question on "On Closed Categories of Functors"

Question on "On Closed Categories of Functors"

[Tony Meman]

Dear category theorists,

I have a question concerning the paper "On Closed Categories of Functors"
from Brian Day (By the way, this is an excellent paper).

Let V be a symmetric monoidal closed category and C a small V-category.

The (ordinary) category [C,V] of V-functors admits the sturcure of a
V-category in a canonical way.

A symmetric monoidal V-category is the enriched analogue of a
symmetric-monoidal structure on an ordinary category, i.e. all the structure
morphisms are V-morphisms and the coherence conditions are fullfilled. The
underlaying category of a symmetric monoidal V-category admits the structure
of an ordinary symmetric monoidal 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 underlaying *category*
[C,V] of V-functors admits a closed symmetric monoidal structure from the
enriched one by taking the underlaying functor of each V-functor, the
underlaying 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?

Thank you in advance for any help.
Tony

Re: Question on "On Closed Categories of Functors"

[street]

> 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!