Re: Enriched adjoint functor theorem?

Re: Enriched adjoint functor theorem?

[Ross Street]

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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Re: Enriched adjoint functor theorem?

[Gabor Lukacs]

Hi Ross,

On Mon, 23 May 2011, Ross Street wrote:

> By Yoneda, what you are asking for is an isomorphism
>
> x@(a@b) =~ (x@a)@b

You are quite right. The reason that I prefer to seek

[a,[b,c]] =~ [a@b,c]

is because Kelly showed in "Tensor Products in Categories" that the latter
implies associativity, as well as coherence (Theorem 11).

While I understand the structure of [-,-] very well (and for example, I
was able to show that [a,[b,c]] =~ [b,[a,c]]), I do not know much about
the structure of a@b, and I expect its structure to be extremely
complicated.

> In general, I see no way around proving a certain associativity
> constraint invertible.

My question could be rephased as: Is there an extra condition on [-,-],
which can be expressed only using [-,-], that will imply associativity of @ ?

Of course, the alternative way of phrasing this was to seek a
left-V-adjoint for [a,-]. That was the starting point for my question.

Best,
Gabi


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Re: Enriched adjoint functor theorem?

[Fred E.J. Linton]

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
> 




[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Enriched adjoint functor theorem?

[Gabor Lukacs]

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

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]