Re: forms

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

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


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