Re: Enriched adjoint functor theorem?

["Fred E.J. Linton" <fejlinton@usa.net>]

As I recall, tho' it's a fallible memory by now 4 decades or more old,
the enrichedness of the left adjoint in an adjunction that starts with 
an enriched functor in the first place follows automatically in the event
that V's underlying-set functor is faithful, i.e., that the unit object 
for the internal hom-functor [-,-] is a generator.

Seems to me I probably have that in an old SLNM proceedings volume -- not
# 80, I'd think, but maybe # 99, or not much later.

HTH, and that I'm not mistaken, -- Fred 

------ Original Message ------
Received: Mon, 23 May 2011 02:59:46 AM EDT
From: Gabor Lukacs <dr.gabor.lukacs@gmail.com>
To: categories@mta.ca
Subject: categories: Enriched adjoint functor theorem?

> Dear All,
> 
> I was wondering if there is a known generalization of the adjoint functor
> theorem to enriched categories. This is what I am trying to figure out:
> 
> I have a closed category V with internal hom-functpr [-,-], and I am
> trying to show that it is *monoidal* closed. I was able to prove (using
> the adjoint functor theorem) that the hom-functor [a,-] has a left-adjoint
> L^a: V --> V, but in order to obtain a monoidal closed strucutre, one
> needs to have a natural isomorphism in V:
> 
> [L^a(b),c] -=- [b,[a,c]]   (*)
> 
> This will also imply associativity and coherence.
> 
> So, I am asking if there is a way to prove (*) based on some form of
> enriched adjoint functor theorem, without figuring out the structure
> of L^a(b) explicitly.
> 
> Best,
> Gabi
> 




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