Re: Matrices Category

Re: Matrices Category

[Mike Stay]

On Mon, Oct 13, 2008 at 9:34 AM, Hugo Macedo <hugodsmacedo@gmail.com> wrote:
> Hello
>
> I'm trying to study the Category of Matrices but I found almost nothing. Do
> you know where
> I can find information about them?

I think you may have more success searching for information about the
category Vect_K of vector spaces and K-linear transformations between
them, where K is the field of interest (usually the reals K=R or the
complex numbers K=C).

> More specifically can we consider the tensor product as the product
> bi-functor?

In Set, the cartesian product is different from the coproduct, and the
product satisfies
   hom(A x B, C) is isomorphic to hom(A, C^B)
making Set into a cartesian closed category, a special kind of
symmetric monoidal closed category; but this is not true in Vect.

The product and coproduct are the same in Vect, namely the "direct
sum", while the tensor product is what makes Vect into a symmetric
monoidal closed category:
   hom(A tensor B, C) = hom(A, B -o C)
where -o is linear implication.  Vect also happens to be a compact
closed category, which means that B -o C is isomorphic to B* tensor C.
-- 
Mike Stay - metaweta@gmail.com
http://math.ucr.edu/~mike
http://reperiendi.wordpress.com

Re: Matrices Category

[Fred E.J. Linton]

Hi, Hugo,

Given the spec "as its morphisms," there are at least three perspectives.

1) (narrowest) you're focussed on some field F and on the very 
particular vector spaces over F that are of form F^n for natural 
numbers n, along with the matrices that "are" the F-linear
transformations among them. Any linear algebra text should help.

2) (intermediate) you're focussed on some semiring R (equipped with
a multiplication and an addition, each associative and unital, 
and together satisfying a distributive law) and are after the
category with objects the natural numbers and maps from k to n
the n by k matrices with coefficients from R.

This can be construed also as the full subcategory of R-semimodules
whose objects are the finitely generated free ones, which is the
Lawvere-style "algebraic theory" of R-semimodules.

"Functorial semantics" (google it?) can offer broad general insights.

Perspective 1) arises from the special case of 2) with R = F.

3) (broadest) what Ronnie Brown pointed out -- from almost any 
category A one can form a new category whose objects are the finite
sequences of objects of A and whose maps, from say A1 ... An to
B1 ... Bk are the n by k matrices whose various ij'th entries are
A-morphisms from Ai to Bj [or, if you prefer, the exact opposite].

Some Russians in the '60s or '70s in the Kurosh school (i.e. [students 
of ...]* students of Kurosh) exploited that construction to explain 
how to embed a less-than-additive category in an additive one. The names 
Kurosh, Lifshutz and Shulgeifer come to mind (perhaps inappropriately?),
but the publications I'd like to cite for you, or my copies of them,
anyway, are in storage, and inaccessible to me at the moment, sorry.

Perspective 2) arises from the special case of 3) with 
A = the one-object pre-additive category R.

May these pointers help get you started. Cheers,

-- Fred

------ Original Message ------
Received: Thu, 16 Oct 2008 09:41:42 AM EDT
From: "Hugo Macedo" <hugodsmacedo@gmail.com>
To: "Fred E.J. Linton" <fejlinton@usa.net>
Subject: Re: categories: Matrices Category

> Hello Fred E.J. Linton
> 
> Thanks for the answer, I meant Matrices as its morphisms.
> 
> Best regards,
> Hugo
> 
> On Wed, Oct 15, 2008 at 4:38 AM, Fred E.J. Linton <fejlinton@usa.net>
wrote:
> 
> > "Hugo Macedo" <hugodsmacedo@gmail.com> wrote:
> >
> > > I'm trying to study the Category of Matrices but I found almost
nothing.
> > Do
> > > you know where
> > > I can find information about them?
> > > --- [snip] ---
> >
> > Do you mean matrices as the objects of this category?
> > or as its morphisms (in which case, what objects do you see?)?
> > Might make a difference in the references you get pointed to.
> >
> >
> >
> 

Re: Matrices Category

[vs27]

I am surprised that nobody suggested to Hugo
to have a look at modules/profunctors.
??


On Oct 16 2008, R Brown wrote:

>On any additive category A you can define a category of matrices Mat(A).
> The objects of Mat(A) are n-tuples A_*= (A_1, ...,A_n) for all n >= 1, of
>objects of A. A morphism a_**: A_* \to B_* consists of arrays of morphisms
>a_{ij}: A_i \to B_j (or is it the other way round? I leave you to look for
>the conventions). If you get the conventions right, then composition in
>Mat(A) is just matrix composition.
>
>The nice point about this is that Mat(A) is again an additive category, so
>the process can be iterated.  Actually only semi-additive is needed (matrix
>composition does not use negatives.)
>
>The above idea essentially yields  partitioned matrices (see old books on
>matrices).
>
>This passage A \mapsto Mat(A) ought to be available on computer software!
>
>I expect the above is in a reference somewhere!
>
>Ronnie Brown
>www.bangor.ac.uk/r.brown
>
>----- Original Message -----
>From: "Hugo Macedo" <hugodsmacedo@gmail.com>
>To: <categories@mta.ca>
>Sent: Monday, October 13, 2008 5:34 PM
>Subject: categories: Matrices Category
>
>
>> Hello
>>
>> I'm trying to study the Category of Matrices but I found almost nothing.
>> Do
>> you know where
>> I can find information about them?
>>
>> More specifically can we consider the tensor product as the product
>> bi-functor?
>>
>> --
>> Hugo
>>
>>
>
>
>
> --------------------------------------------------------------------------------
>
>
>
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>Checked by AVG - http://www.avg.com
>Version: 8.0.173 / Virus Database: 270.8.0/1724 - Release Date: 14/10/2008
>02:02
>
>
>
>

Re: Matrices Category

[Ross Street]

"Hugo Macedo" <hugodsmacedo@gmail.com> wrote:

> I'm trying to study the Category of Matrices but I found almost
> nothing. Do
> you know where I can find information about them?

Once upon a time I handwrote some incomplete notes on "skeletal"
representation theory
which started out looking at Mat (a skeleton of Vect) as a monoidal
category with direct sums
(the tensor product is Kronecker product of matrices). I tried to
strictify as much as possible
all the associativity, distributivity, . . constraints. I typed it at
some point. The pdf file has some
funny things (notably copyright signs!) in it but it has been on the
web (linked from my
Publications page) for quite awhile. Maybe it is the kind of thing
Hugo has in mind. But
remember, it is sort of "in progress"; no jokes please about "droup"
which is meant as
short for "dual group"!

<http://www.math.mq.edu.au/~street/Droup.pdf>

Ross

Re: Matrices Category

[R Brown]

On any additive category A you can define a category of matrices Mat(A).
 The objects of Mat(A) are n-tuples A_*= (A_1, ...,A_n) for all n >= 1, of
objects of A. A morphism a_**: A_* \to B_* consists of arrays of morphisms
a_{ij}: A_i \to B_j (or is it the other way round? I leave you to look for
the conventions). If you get the conventions right, then composition in
Mat(A) is just matrix composition.

The nice point about this is that Mat(A) is again an additive category, so
the process can be iterated.  Actually only semi-additive is needed (matrix
composition does not use negatives.)

The above idea essentially yields  partitioned matrices (see old books on
matrices).

This passage A \mapsto Mat(A) ought to be available on computer software!

I expect the above is in a reference somewhere!

Ronnie Brown
www.bangor.ac.uk/r.brown

----- Original Message -----
From: "Hugo Macedo" <hugodsmacedo@gmail.com>
To: <categories@mta.ca>
Sent: Monday, October 13, 2008 5:34 PM
Subject: categories: Matrices Category


> Hello
>
> I'm trying to study the Category of Matrices but I found almost nothing.
> Do
> you know where
> I can find information about them?
>
> More specifically can we consider the tensor product as the product
> bi-functor?
>
> --
> Hugo
>
>


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02:02

Re: Matrices Category

[Fred E.J. Linton]

"Hugo Macedo" <hugodsmacedo@gmail.com> wrote:

> I'm trying to study the Category of Matrices but I found almost nothing. Do
> you know where
> I can find information about them? 
> --- [snip] ---

Do you mean matrices as the objects of this category?
or as its morphisms (in which case, what objects do you see?)?
Might make a difference in the references you get pointed to.

Re: Matrices Category

[Mike Stay]

On Tue, Oct 14, 2008 at 10:43 AM, Mike Stay <metaweta@gmail.com> wrote:
> The product and coproduct are the same in Vect, namely the "direct
> sum", while the tensor product is what makes Vect into a symmetric
> monoidal closed category:
>   hom(A tensor B, C) = hom(A, B -o C)

Sorry, that should read "is isomorphic to", not strict equality.
-- 
Mike Stay - metaweta@gmail.com
http://math.ucr.edu/~mike
http://reperiendi.wordpress.com

Matrices Category

[Hugo Macedo]

Hello

I'm trying to study the Category of Matrices but I found almost nothing. Do
you know where
I can find information about them?

More specifically can we consider the tensor product as the product
bi-functor?

-- 
Hugo