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
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
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
"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.
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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"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
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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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.
participants (6)
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Fred E.J. Linton -
Hugo Macedo -
Mike Stay -
R Brown -
Ross Street -
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