Re: Enriched adjoint functor theorem?

[Gabor Lukacs <dr.gabor.lukacs@gmail.com>]

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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