Al Vilcius wrote:
to which category do multilinear maps belong?
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?
The slickest answer is perhaps that the middle arrows live in the *multicategory* k-mod: X, Y ---> Z bilinear in k-mod The problem with this is that multicategories are little bit more complicated than categories. http://ncatlab.org/nlab/show/multicategory http://en.wikipedia.org/wiki/Multicategory Because k-mod is a monoidal category (aka tensor category), that is it has a well-behaved operation tensor, we can use mere linear maps X tensor Y ---> Z instead. http://ncatlab.org/nlab/show/monoidal+category http://unapologetic.wordpress.com/2007/06/28/monoidal-categories/ http://en.wikipedia.org/wiki/Monoidal_category And because k-mod is a closed category, that is it has a well-behaved operation hom, we can use mere linear maps X ---> Y hom Z instead. http://en.wikipedia.org/wiki/Closed_category Since these two operations tensor and hom are compatible, in that they correspond to the same multicategory structure, k-mod is in fact a *closed monoidal category*. http://ncatlab.org/nlab/show/closed+monoidal+category http://en.wikipedia.org/wiki/Closed_monoidal_category Category theorists usually turn a bilinear map into a linear map in one or the other of these ways, to avoid multicategories.
From a structural perpsective, all three perspectives are equivalent, so it really doesn't matter which way you look at them. But multicategories are there if you want them.
--Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]