Re: Matrices Category

["Mike Stay" <metaweta@gmail.com>]

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