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? Putting Z = k , what are the categories of forms? I can't seem to find these discussed (explicitly) anywhere? My questions arose in thinking about a "probe and measure" schema for investigating systems. Thank you in advance for any kind responses. Greatly appreciated. ....... Al Al Vilcius Campbellville, ON, Canada [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
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/ ]
Al Vilcius wrote:
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
Andrej Bauer replied:
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".
Or, you can reverse your attitude and say "don't talk about linear maps out of tensor products--talk about arrows in the *multicategory* of multilinear maps'. You may not like multicategories - but as Toby said, they're there when you want them, waiting with the kind of patience that only mathematical objects can muster: http://ncatlab.org/nlab/show/multicategory http://en.wikipedia.org/wiki/Multicategory Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Thank you for your kind responses. Yes indeed, now I remember Jim Lambek talking about multicategories (Boulder'87 and before). I attribute my brain-freeze to being a senior citizen probably time for me to give up cats for dogs. But maybe before I do, I would like to understand: what are the correct dimensions for the Grassman algebras? either remembering or forgetting some distinguished origin. The idea of "forget" (structure) seemed unreasonably hard for me to explain to a philosopher friend of mine, and I wound up getting confused myself because it is not always clear to me when "forgetting" is trivial or non-trivial. Best to you all. ........ Al Al Vilcius Campbellville, ON, Canada [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
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Al Vilcius -
Andrej Bauer -
John Baez -
Toby Bartels