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