Keith wrote:

> I have a simple question.

Usually when people say that it turns out to be an incredibly vague, subtle question.

> A poset is to a category as a quantale is to what exactly?

Ah, but this time it's actually a simple question.

A quantale is a closed monoidal poset with all joins.

But the analogue for categories of a 'join' is a ['colimit'](https://en.wikipedia.org/wiki/Limit_(category_theory)#Colimits). In particular, every poset is a category, and a colimit in a poset is just a join.

So, the answer to your question is: a closed monoidal category with all colimits.

But a category with all colimits is called [cocomplete](https://en.wikipedia.org/wiki/Complete_category). So, a slicker way to say the same thing is: a cocomplete closed monoidal category.

In short:

_A poset is to a category as a quantale is to a cocomplete closed monoidal category_.

There are lots of cocomplete closed monoidal categories. Every quantale is one. But here's one that's not a quantale: \\( (\mathbf{Set}, \times, 1)\\).

A more representative example is \\( (\mathbf{Vect}_k, \otimes, k) \\) - the category of vector spaces over a field \\(k\\), made monoidal with the tensor product of vector spaces, with \\(k\\) as the unit for the tensor product.

By the way, to be nitpicky, whenever I said "all" above I should have said "all small".

> I have a simple question.

Usually when people say that it turns out to be an incredibly vague, subtle question.

> A poset is to a category as a quantale is to what exactly?

Ah, but this time it's actually a simple question.

A quantale is a closed monoidal poset with all joins.

But the analogue for categories of a 'join' is a ['colimit'](https://en.wikipedia.org/wiki/Limit_(category_theory)#Colimits). In particular, every poset is a category, and a colimit in a poset is just a join.

So, the answer to your question is: a closed monoidal category with all colimits.

But a category with all colimits is called [cocomplete](https://en.wikipedia.org/wiki/Complete_category). So, a slicker way to say the same thing is: a cocomplete closed monoidal category.

In short:

_A poset is to a category as a quantale is to a cocomplete closed monoidal category_.

There are lots of cocomplete closed monoidal categories. Every quantale is one. But here's one that's not a quantale: \\( (\mathbf{Set}, \times, 1)\\).

A more representative example is \\( (\mathbf{Vect}_k, \otimes, k) \\) - the category of vector spaces over a field \\(k\\), made monoidal with the tensor product of vector spaces, with \\(k\\) as the unit for the tensor product.

By the way, to be nitpicky, whenever I said "all" above I should have said "all small".