Michael said

> Can the base of enrichment be an enriched category?

That's a very good question, one that causes much confusion and one that I'm not sure that John has answered, yet. I'm sure John will have plenty to say about this, but, at the risk of adding to the confusion, let me say some things here.

You've identified the following problem.

- The base of enrichment, \\(\mathcal{V}\\), needs to be a monoidal preorder (at least in the generality we're in here).

- There is no canonical way to make a preorder into a \\(\mathcal{V}\\)-category.

- A \\(\mathcal{V}\\)-functor goes between \\(\mathcal{V}\\)-categories.

So asking for a \\(\mathcal{V}\\)-functor \\(\Phi : C^{\text{op}} \times D \to \mathcal{V}\\) gives a type error. The codomain is a monoidal preorder not a \\(\mathcal{V}\\)-category.

However, in many cases of interest one can 'turn \\(\mathcal{V}\\) into a \\(\mathcal{V}\\)-category'. Many authors will use the same symbol for both the monoidal preorder and the enriched category, which as I said, can lead to confusion. People talk of \\(\mathcal{V}\\) being 'enriched over itself', which has a fundamental type error built into it! This is okay for experts who understand the subtleties here, but I've seen experienced mathematicians from other areas getting tangled up over these things. (I could add that a lot of the terminology around enriched categories goes back to the days of 'concrete enrichment', but I shouldn't get sidetracked.)

[In case you come across these terms, other ways of describing this process which are perhaps less confusing, include making \\(\mathcal{V}\\) into a 'closed monoidal category', or definining 'internal homs'.]

In the case of a commutative monoidal *poset* there is only one sensible way to do this. [I'm going to use \\(\rtimes\\) for the order on \\(\mathcal{V}\\) to avoid the confusion of switching between \\(\ge\\) and \\(\le\\) when going to \\(\textbf{Cost}\\).] I'm going to write \\(\overline{\mathcal{V}}\\) for the \\(\mathcal{V}\\)-enriched version of \\(\mathcal{V}\\). This will be a \\(\mathcal{V}\\)-category with the same objects as \\(\mathcal{V}\\). So I want a function

\[\\text{ob}\mathcal{V}\times\text{ob}\mathcal{V}\to \text{ob}\mathcal{V} ; \qquad x,y\mapsto \overline{\mathcal{V}}(x,y).\]

You take

\[

\overline{\mathcal{V}}(x,y)=\bigvee\\{z\mid z\otimes x \rtimes y\\}.

\]

This will work provided that the poset has the requisite joins, and that \\(\otimes\\) distributes over joins. And because it is a *poset* rather than a preorder, the join will be unique.

In the case of \\(\textbf{Cost}\\) you get the following.

\[

\overline{\textbf{Cost}}(x,y)

=\inf\\{z\mid z + x \ge y\\}

=\max(y-x, 0).

\]

So this is a very asymmetric version of the usual metric on the half interval \\([0,\infty]\\). But you can check that this is a \\(\textbf{Cost}\\)-category, \\(\overline{\textbf{Cost}}\\).

> Can the base of enrichment be an enriched category?

That's a very good question, one that causes much confusion and one that I'm not sure that John has answered, yet. I'm sure John will have plenty to say about this, but, at the risk of adding to the confusion, let me say some things here.

You've identified the following problem.

- The base of enrichment, \\(\mathcal{V}\\), needs to be a monoidal preorder (at least in the generality we're in here).

- There is no canonical way to make a preorder into a \\(\mathcal{V}\\)-category.

- A \\(\mathcal{V}\\)-functor goes between \\(\mathcal{V}\\)-categories.

So asking for a \\(\mathcal{V}\\)-functor \\(\Phi : C^{\text{op}} \times D \to \mathcal{V}\\) gives a type error. The codomain is a monoidal preorder not a \\(\mathcal{V}\\)-category.

However, in many cases of interest one can 'turn \\(\mathcal{V}\\) into a \\(\mathcal{V}\\)-category'. Many authors will use the same symbol for both the monoidal preorder and the enriched category, which as I said, can lead to confusion. People talk of \\(\mathcal{V}\\) being 'enriched over itself', which has a fundamental type error built into it! This is okay for experts who understand the subtleties here, but I've seen experienced mathematicians from other areas getting tangled up over these things. (I could add that a lot of the terminology around enriched categories goes back to the days of 'concrete enrichment', but I shouldn't get sidetracked.)

[In case you come across these terms, other ways of describing this process which are perhaps less confusing, include making \\(\mathcal{V}\\) into a 'closed monoidal category', or definining 'internal homs'.]

In the case of a commutative monoidal *poset* there is only one sensible way to do this. [I'm going to use \\(\rtimes\\) for the order on \\(\mathcal{V}\\) to avoid the confusion of switching between \\(\ge\\) and \\(\le\\) when going to \\(\textbf{Cost}\\).] I'm going to write \\(\overline{\mathcal{V}}\\) for the \\(\mathcal{V}\\)-enriched version of \\(\mathcal{V}\\). This will be a \\(\mathcal{V}\\)-category with the same objects as \\(\mathcal{V}\\). So I want a function

\[\\text{ob}\mathcal{V}\times\text{ob}\mathcal{V}\to \text{ob}\mathcal{V} ; \qquad x,y\mapsto \overline{\mathcal{V}}(x,y).\]

You take

\[

\overline{\mathcal{V}}(x,y)=\bigvee\\{z\mid z\otimes x \rtimes y\\}.

\]

This will work provided that the poset has the requisite joins, and that \\(\otimes\\) distributes over joins. And because it is a *poset* rather than a preorder, the join will be unique.

In the case of \\(\textbf{Cost}\\) you get the following.

\[

\overline{\textbf{Cost}}(x,y)

=\inf\\{z\mid z + x \ge y\\}

=\max(y-x, 0).

\]

So this is a very asymmetric version of the usual metric on the half interval \\([0,\infty]\\). But you can check that this is a \\(\textbf{Cost}\\)-category, \\(\overline{\textbf{Cost}}\\).