Frederick Eisele wrote in [#79](https://forum.azimuthproject.org/discussion/comment/19011/#Comment_19011)

> We have a monoidal skeleton-category
labeled \$$\mathbf{Matrix}_{skel} \$$
and two categories
\$$\mathbf{Matrix}\_{\mathbb{N}} \$$
and
\$$\mathbf{Matrix}\_{multiset}\$$ .
> Here are example objects of those categories.
> $\left( \begin{array}{cc} 1 & 1 \\\\ 1 & 0 \end{array} \right)$
>
> and
>
> $\left( \begin{array}{cc} \lbrace f \rbrace & \lbrace g \rbrace \\\\ \lbrace h \rbrace & \emptyset \end{array} \right)$
>
> The two categories differ in the type of their cells and the definitions of their monoidal
operators but they are both similar to their skeleton in the same way.
>
> What do we call these similarities?

**TL;DR**: I would say you are looking at a *\$$\mathbf{Rig}\$$-homomorphism* \$$\lVert \cdot \rVert : \mathbf{Multiset}_{\mathfrak{M}} \to \mathbb{N}\$$. You also are looking at \$$\mathbf{Rig}\$$-endofunctor called \$$\mathbf{Matrix}^{N \times N}\$$. The map \$$U\$$ Keith shows us in [#61](https://forum.azimuthproject.org/discussion/comment/18948/#Comment_18948) is \$$\mathbf{Matrix}^{N \times N}\_{ \lVert \cdot \rVert }\$$.

(I fixed up your LaTeX and noted that one of sets of matrices was just over \$$\mathbb{N}\$$, rather than \$$\mathbb{R}\$$).

-------------------------------------------------

\$$\mathbf{Rig}\$$ is what John Baez calls the category of [*semirings*](https://en.wikipedia.org/wiki/Semiring). He has mentioned them elsewhere.

One familiar rig is \$$\langle \mathbb{N}, 0, +, 1, \cdot \rangle\$$

If you have a monoid \$$\mathfrak{M} = \langle S, I, ; \rangle\$$, we can construct a rig \$$\mathbf{Multiset}\_{\mathfrak{M}}\$$
of *finite multisets* of \$$\mathfrak{M}\$$. I am following the ideas you gave, Frederick, in
[#55](https://forum.azimuthproject.org/discussion/comment/18938/#Comment_18938), [#56](https://forum.azimuthproject.org/discussion/comment/18942/#Comment_18942) and [#59](https://forum.azimuthproject.org/discussion/comment/18945/#Comment_18945).

The rig \$$\mathbf{Multiset}_{\mathfrak{M}}\$$ is conceptually like \$$\langle \mathbb{N}, 0, +, 1, \cdot \rangle \$$. It has the following differences:

- \$$0\$$ is replaced with \$$\emptyset\$$
- \$$+\$$ is replaced with \$$\cup\$$
- \$$1\$$ is replaced with \$$\\{I\\}\$$
- \$$\cdot\$$ is replaced with \$$X \otimes Y := \lbrace x ; y \; : \; x \in X \text{ and } y \in Y \rbrace\$$

Let \$$\lVert \cdot \rVert : \mathbf{Multiset}\_{\mathfrak{M}} \to \mathbb{N}\$$ measure the cardinality of a multiset. This is a rig-homomorphism between \$$\mathbf{Multiset}\_{\mathfrak{M}}\$$ and \$$\mathbb{N}\$$.

If \$$\mathfrak{R}\$$ is a rig, we can make a new rig \$$\mathbf{Matrix}^{N \times N}_{\mathfrak{R}}\$$ of finite \$$N \times N\$$ square matrices. Addition is defined to be element-wise addition, just like in linear algebra. Matrix multiplication is defined to be sums of products.

\$$\mathbf{Matrix}^{N \times N} \$$ is also a functor. If there a rig-homomorphism between \$$\phi : \mathfrak{R} \to \mathfrak{S}\$$, then \$$\mathbf{Matrix}^{N \times N}\_{\phi} : \mathbf{Matrix}^{N \times N}\_{\mathfrak{R}} \to \mathbf{Matrix}^{N \times N}_{\mathfrak{S}} \$$ just acts element-wise, mapping \$$a\_{ij} \mapsto \phi(a)\_{ij}\$$.

**Example**: In the special case of

\$A^3 = \left( \begin{array}{cc} \lbrace f;f;f, \; g;h;f, \; f;g;h \rbrace & \lbrace f;f;g, \; g;h;g \rbrace \\\\ \lbrace h;f;f, \; h;g;h \rbrace & \lbrace h;f;g \rbrace \end{array} \right) \$

We can see what \$$\mathbf{Matrix}^{2 \times 2}\_{ \lVert \cdot \rVert }\$$ does, and how it is a rig-homomorphism:

\\begin{align} \mathbf{Matrix}^{2 \times 2}\_{ \lVert \cdot \rVert }(A^3) & = \left( \begin{array}{cc} 3 & 2 \\\\ 2 & 1 \end{array} \right) \\\\ & = \left( \begin{array}{cc} 1 & 1 \\\\ 1 & 0 \end{array} \right)^3 \\\\ & = \left( \begin{array}{cc} \lVert\lbrace f \rbrace\rVert & \lVert\lbrace g \rbrace\rVert \\\\ \lVert\lbrace h \rbrace\rVert & \lVert \emptyset \rVert \end{array} \right)^3 \\\\ & = \left( \mathbf{Matrix}^{2 \times 2}\_{ \lVert \cdot \rVert }(A) \right)^3 \\\\ \end{align} \