Defining a Matrix Category (with possible errors…)

Let \text{V} be the category for matrices such that \text{Obj}(\text{V})=\mathbb{N}, and \text{Hom}_\text{V}(m,n)=\{m \times n \ \text{matrices} \ : \forall{n},m\in\mathbb{N}\}. We can define a composition function \circ_\text{V} using knowledge of matrix multiplication:

\circ_\text{V}:\text{Hom}_\text{V}(m_a,n_b)\times\text{Hom}_\text{V}(m_c,n_d)\to\text{Hom}_\text{V}(m_a,n_d)

where a, b, c, d \in\mathbb{N}.

Now let f:\text{Hom}_\text{V}(m_1,n_1,), g:\text{Hom}_\text{V}(m_2,n_2), and h:\text{Hom}_\text{V}(m_3,n_3) in that f, g, h \in \text{Obj}(\text{V}). From the composition of n\times m matrices, we can now imply that the category is associative:

(f\circ_\text{V}g)\circ_\text{V}h=

[\text{Hom}_\text{V}(m_1,n_1)\circ_\text{V}\text{Hom}_\text{V}(m_2,n_2)]\circ_\text{V}\text{Hom}_\text{V}(m_3,n_3)=

\text{Hom}_\text{V}(m_1,n_2)\circ_\text{V}\text{Hom}_\text{V}(m_3,n_3)=

\text{Hom}_\text{V}(m_1,n_3)=

\text{Hom}_\text{V}(m_1,n_1)\circ_\text{V}\text{Hom}_\text{V}(m_2,n_3)=

\text{Hom}_\text{V}(m_1,n_1)\circ_\text{V}[\text{Hom}_\text{V}(m_2,n_2)\circ_\text{V}\text{Hom}_\text{V}(m_3,n_3)]=

f\circ_\text{V}(g\circ_\text{V}h).

Here is the communative diagram for the associativity:  

  

Therefore we may show that the identity where \text{I}_m=\text{Hom}_\text{V}(m,m) and \text{I}_n=\text{Hom}_\text{V}(n,n):

\text{I}_mf=f\text{I}_n=f.

\therefore We have defined a category of matrices. (hopefully…)

Note: Please let me know of any errors I have made as I want to be able to correct them as I learn the concepts.

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2 thoughts on “Defining a Matrix Category (with possible errors…)

  1. isospectral

    Just write $\hom(a,b)\times\hom(b,c)\to\hom(a,c)$ to keep things simple – there’s no need to use subscripts with three things. Note that I repeated the letter $b$ twice, instead of writing four letters $a,b,c,d$. In order to compose to morphisms, the codomain of the one must be the domain of the other. A similar thing must be said for your commutative diagram and all the different natural numbers you use. What does $f:\hom$ mean by the way? Why did you replace the letters $f,g,h$ with $\hom$-sets in your symbolic derivation? That makes no sense at all to me. If you had just said there is a category whose objects were natural numbers and whose hom-sets $\hom(m,n)$ were $m\times n$ matrices over a field $F$, because matrix multiplication is associative and the identity matrix acts as the identity, that single sentence would have been enough for me to express everything you intend to express here.

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    • Julian Rachman

      Ah. I see. I believe that I just made it a whole lot harder for myself by writing everything out and assigning separate letters of make everything even more complicated. $f:\Hom$ is just a morphism that I defined to “make things easier for the reader.” But I guess not. The symbolic derivation you mean in the communative diagram? I was just showing associativity. Or am I referring to something different than to what you are thinking of? I understand the rest of what you said. I just need to learn how to simplify my explanations and not make it more complicated than it already is.

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