Date: Tue, 4 Feb 1997 13:29:59 -0400 (AST)
Subject: question on functors adjoint to their dual

Date: Tue, 4 Feb 1997 16:44:39 GMT
From: Hayo Thielecke <ht@dcs.ed.ac.uk>



I am interested in the following situation: a contravariant functor
adjoint to its own dual, with the unit and counit being the same
morphism, but _not_ an iso.

The canonical example is the contravariant internal hom on a cartesian
(or just symmetric monoidal) closed category, [(_) -> A] for some
object A.

My question is: is this typical, or are there (interesting) examples
of such adjunctions that do not come from exponentials?

Thanks,

Hayo Thielecke


Date: Wed, 5 Feb 1997 11:37:26 -0400 (AST)
Subject: Re: question on functors adjoint to their dual

Date: Tue, 4 Feb 1997 15:44:58 -0500 (EST)
From: Fred E.J. Linton <FLinton@wesleyan.edu>

At 01:29 PM 2/4/97 -0400, you wrote:

>I am interested in the following situation: a contravariant functor
>adjoint to its own dual, with the unit and counit being the same
>morphism, but _not_ an iso.
>
>The canonical example is the contravariant internal hom on a cartesian
>(or just symmetric monoidal) closed category, [(_) -> A] for some
>object A.
>
>My question is: is this typical ... ?

        I think it *is* typical:  if we call the functor in question  F ,
and if we write  J  for the unit object, then we should learn easily that
 F  will just be  [(_) -> F(J)] ,  i.e.,  F(J) itself will serve as your  A .

        -- FEJ Linton