Re: forms

[Andrej Bauer <andrej.bauer@andrej.com>]

On Wed, Jan 13, 2010 at 6:18 PM, Al Vilcius <al.r@vilcius.com> wrote:
> Dear Categorists,
> Please help me relieve my confusion on a simple question:
>
> to which category do multilinear maps belong?
>
> For example, in vector spaces (k-mod)
> there are the usual bijective correspondences:
>
> X tensor Y  --->  Z     linear in k-mod
> ____________________________________
>
> X cartesian Y  --->   Z     bilinear
> ____________________________________
>
> X  --->  Y hom Z     linear in k-mod
>
> where do the middle arrows live?

I always understood the correspondence between the first and the
second line as saying "don't talk about bilinear maps on
products--talk about linear maps on tensor products instead". But if
you twisted my arm (and I did not exectute a proper defense) I would
cook up the following:

Take the category whose objects are tuples of vector spaces and a morphism

(X_1, ..., X_n) -> (Y_1, ..., Y_m)

is an m-tuple (f_1, f_2, ..., f_m) of multi-linear maps

f_i : X_1 \times ... \times X_n -> Y_i

and composition is composition. Doesn't that work? So the answer to
your question is: the cartesian sign in the second row is a mirage.
But that's something we can easily figure out: it makes no sense to
talk about a bilinear map unless we are told how its domain is
decomposed into a product (there can be many ways).

With kind regards,

Andrej


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