Re: Matrices Category

["Fred E.J. Linton" <fejlinton@usa.net>]

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.
> >
> >
> >
>