categories: Re: Enriched category theory, again

[maxk@maths.usyd.edu.au (Max Kelly)]

This is in reply to the queries in Mark Hovey's letter of 12 Jan, of which I
shall try to include a copy below; I am limited by the inadequacies of this 
home computer.

The first isomorphism he asks about, namely (using o for tensor product)

           [A,D](Y o A(X,-),F] =~ D(Y,FX),

may be seen as a simple consequence of the fact that Y o A(X,-) is the left
Kan extension of Y : I --> D along X : I --> A, where I is the unit V-category;
see formula (4.18) or (4.25) of my book.

The second question concerns an easy extension of Brian Day's convolution 
monoidal structure on a presheaf category. An easy way of seeing the truth of 
Hovey's observation is to recall that a tensored V-category structure on a 
(mere) category A corresponds to an _action_ of V on A having an appropriate
right adjoint; see [Janelidze and Kelly, TAC 9 (2001), 61 - 91], foot of page
66; but the idea itself is quite old. Now V --> [D,D] easily gives 
[A,V] --> [[A,D],[A,D]]. Of course there is checking to do.

Max Kelly.