forms

forms

[Al Vilcius]

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

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Re: forms

[Andrej Bauer]

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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Re: forms

[Toby Bartels]

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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Re: forms

[John Baez]

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


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Re: forms

[Al Vilcius]

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



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