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} \\]