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