* Re: forms
@ 2010-01-19 19:27 Al Vilcius
0 siblings, 0 replies; 5+ messages in thread
From: Al Vilcius @ 2010-01-19 19:27 UTC (permalink / raw)
To: categories
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/ ]
^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: forms
@ 2010-01-15 2:51 John Baez
0 siblings, 0 replies; 5+ messages in thread
From: John Baez @ 2010-01-15 2:51 UTC (permalink / raw)
To: categories
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/ ]
^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: forms
2010-01-13 17:18 forms Al Vilcius
2010-01-14 15:19 ` forms Andrej Bauer
@ 2010-01-14 17:15 ` Toby Bartels
1 sibling, 0 replies; 5+ messages in thread
From: Toby Bartels @ 2010-01-14 17:15 UTC (permalink / raw)
To: Al Vilcius, categories
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/ ]
^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: forms
2010-01-13 17:18 forms Al Vilcius
@ 2010-01-14 15:19 ` Andrej Bauer
2010-01-14 17:15 ` forms Toby Bartels
1 sibling, 0 replies; 5+ messages in thread
From: Andrej Bauer @ 2010-01-14 15:19 UTC (permalink / raw)
To: Al Vilcius, categories
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/ ]
^ permalink raw reply [flat|nested] 5+ messages in thread
* forms
@ 2010-01-13 17:18 Al Vilcius
2010-01-14 15:19 ` forms Andrej Bauer
2010-01-14 17:15 ` forms Toby Bartels
0 siblings, 2 replies; 5+ messages in thread
From: Al Vilcius @ 2010-01-13 17:18 UTC (permalink / raw)
To: categories
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/ ]
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