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} \\)


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.