Re: Enriched adjoint functor theorem?

[Ross Street <ross.street@mq.edu.au>]

By Yoneda, what you are asking for is an isomorphism

x@(a@b) =3D~ (x@a)@b

since

V(x,[a@b,c]) =3D~ V(x@(a@b), c)

and

V(x,[a,[b,c]]) =3D~ V((x@a)@b,c).

In general, I see no way around proving a certain associativity constraint i=
nvertible.

Monoidal categories give promonoidal categories via p(a,b;c) =3D V(a@b,c).
On the other hand, closed categories almost give promonoidal categories via p=
(a,b;c) =3D V(a,[b,c]) except that the associativity constraint may not be i=
nvertible.

Ross

Begin forwarded message:

> Subject: Re: categories: Re: Enriched adjoint functor theorem?
>=20

>> ------ Original Message ------
>> Received: Mon, 23 May 2011 02:59:46 AM EDT
>> From:=20
>> To: categories@mta.ca
>> Subject: categories: Enriched adjoint functor theorem?
>>>=20
>>>=20
>>> 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-adjoi=
nt
>>> L^a: V --> V, but in order to obtain a monoidal closed strucutre, one
>>> needs to have a natural isomorphism in V:
>>>=20
>>> [L^a(b),c] -=3D- [b,[a,c]]   (*)
>>>=20
>>> This will also imply associativity and coherence.
>>>=20
>>> 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.
>>>=20
>>>=20

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