Hey [Simon](https://forum.azimuthproject.org/profile/1689/Simon%20Willerton),

> Matthew, you say in [#87](https://forum.azimuthproject.org/discussion/comment/18707/#Comment_18707)
>
> I had written in #63 [something involving \$$\le_{\mathcal{V}}\$$]
>
> No you hadn't :-) you wrote something involving \$$\le_{\mathcal{X}}\$$ which I didn't understand and which I questioned.

I admittedly changed my notation because you didn't understand \$$\le_{\mathcal{X}}\$$ and I thought this would be clearer.

I am trying to change my notation in order to clarify myself.

I am sorry for the confusion.

> Anyway, now that you've established that \$$\mathcal{V}\$$ is just a monoid, you can get rid of \$$\le_{\mathcal{V}}\$$ and just write \$$=\$$ instead everywhere.

We still need to keep \$$\otimes_\mathcal{V}\$$ and \$$\otimes_\mathcal{U}\$$ around, since there are different monoids.

Below, I started to use \$$\otimes\$$ and \$$\odot\$$ instead.

Hopefully this clears up some clutter.

> That will simplify your (initial three) axioms somewhat.

Unfortunately those axioms do not work.

I believe we have to use the big axiom in [#87](https://forum.azimuthproject.org/discussion/comment/18707/#Comment_18707).

> Also, you need to be a bit more careful and say precisely what you mean by
> > maps \$$\phi : \mathcal{X} \to \mathcal{Y} \$$

Okay.

I am assuming \$$\mathcal{X}\$$ is enriched in one monoid \$$\mathcal{V}\$$ and \$$\mathcal{Y}\$$ is enriched in \$$\mathcal{U}\$$.

In general, \$$\mathcal{V} \neq \mathcal{U}\$$.

I need some way of doing the book-keeping.

Below is my attempt at a careful formulation.

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

Define a new category \$$\mathbb{M}\mathbb{E}\$$. Here \$$\mathbb{M}\mathbb{E}\$$ stands for *monoid enriched*. Previously I called this **DiscretePosEnrich**, but that name is confusing.

- **Objects**

Objects are \$$\mathcal{V}\$$-enriched categories \$$\mathcal{X}\$$ for various monoids.

For instance there are [\$$\mathbb{Z}/2\mathbb{Z}\$$](https://en.wikipedia.org/wiki/Cyclic_group)-enriched categories in \$$\mathbb{M}\mathbb{E}\$$.

There are also [\$$D_8\$$](https://en.wikipedia.org/wiki/Dihedral_group)-enriched categories in \$$\mathbb{M}\mathbb{E}\$$.

For any [semilattice](https://en.wikipedia.org/wiki/Semilattice) \$$\mathbb{L}\$$ then \$$\mathbb{L}\$$-enriched categories are members of \$$\mathbb{M}\mathbb{E}\$$.

John mentioned how to take any group \$$G\$$ to a \$$G\$$-enriched category \$$\mathcal{G}\$$ in [#81](https://forum.azimuthproject.org/discussion/comment/18687/#Comment_18687). These are members of \$$\mathbb{M}\mathbb{E}\$$.

- **Morphisms**

Let \$$\mathcal{V} = \langle V, I_{\mathcal{V}}, \otimes \rangle\$$ and \$$\mathcal{U} = \langle U, I_{\mathcal{U}}, \odot \rangle\$$ be arbitrary monoids.

Let \$$\mathcal{X}\$$ be \$$\mathcal{V}\$$-enriched.

Let \$$\mathcal{Y}\$$ be \$$\mathcal{U}\$$-enriched.

A morphism in \$$\mathbb{M}\mathbb{E}\$$ is a function \$$\phi_{\mathcal{V},\mathcal{U}} : \mathrm{Obj}(\mathcal{X}) \to \mathrm{Obj}(\mathcal{Y}) \$$ which obeys the following law:

\$\text{if } \$

\$I_{\mathcal{V}} \otimes \mathcal{X}(a,b) \otimes \mathcal{X}(c,d) \otimes \cdots \otimes \mathcal{X}(y,z) = I_{\mathcal{V}} \otimes \mathcal{X}(p,q) \otimes \mathcal{X}(r,s) \otimes \cdots \otimes \mathcal{X}(w,x) \$

\$\text{then} \$

\$I_{\mathcal{U}} \odot \mathcal{Y}(\phi(a),\phi(b)) \odot \mathcal{Y}(\phi(c),\phi(d)) \odot \cdots \odot \mathcal{Y}(\phi(y),\phi(z)) = I_{\mathcal{U}} \odot \mathcal{Y}(\phi(p),\phi(q)) \odot \mathcal{Y}(\phi(r),\phi(s)) \odot \cdots \odot \mathcal{Y}(\phi(w),\phi(x)) \$

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

I know your gut says you want to use just one category for enrichment, Simon. I am sorry I am not doing that.

As I mentioned, I have changed my notation around a bit above in order to clarify what I have in mind.

John already showed how to lift any group \$$G\$$ into \$$\mathbb{M}\mathbb{E}\$$. If you want, I can show how a group homomorphism gives rise to an \$$\mathbb{M E}\$$ morphism.

Thank you for your patience as I attempt to clarify myself and work out the kinks of this idea.