Matthew wrote:

> Define a **DiscretePos**-enriched category to be an enriched category with a discrete poset and an arbitrary symmetric monoid, obeying the rules John wrote in the lecture.

I'm confused by this. So far we've been talking about \\(\mathcal{V}\\)-enriched categories when \\(\mathcal{V}\\) is a monoidal preorder. So, for example, there's a monoidal preorder \\(\mathbf{Bool}\\), and that's what lets us talk about \\(\mathbf{Bool}\\)-enriched categories.

Your phrase "\\(\mathbf{DiscretePos}\\)-enriched category" would thus suggest that you have in mind some specific monoidal preorder \\(\mathbf{DiscretePos}\\). If you did, and I knew what it was, I would instantly know what a \\(\mathbf{DiscretePos}\\)-enriched category was.

But you seem to be using this phrase in a different way.

And when you say "... to be an enriched category", you're not saying what it's enriched _in_, so we don't know enough to do anything. An enriched category that's not enriched _in_ something is not a real thing. It's like a son that's not anyone's son.

> Define a **DiscretePos**-enriched category to be an enriched category with a discrete poset and an arbitrary symmetric monoid, obeying the rules John wrote in the lecture.

I'm confused by this. So far we've been talking about \\(\mathcal{V}\\)-enriched categories when \\(\mathcal{V}\\) is a monoidal preorder. So, for example, there's a monoidal preorder \\(\mathbf{Bool}\\), and that's what lets us talk about \\(\mathbf{Bool}\\)-enriched categories.

Your phrase "\\(\mathbf{DiscretePos}\\)-enriched category" would thus suggest that you have in mind some specific monoidal preorder \\(\mathbf{DiscretePos}\\). If you did, and I knew what it was, I would instantly know what a \\(\mathbf{DiscretePos}\\)-enriched category was.

But you seem to be using this phrase in a different way.

And when you say "... to be an enriched category", you're not saying what it's enriched _in_, so we don't know enough to do anything. An enriched category that's not enriched _in_ something is not a real thing. It's like a son that's not anyone's son.