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".
Version 2.1.8p2
