Jonathan wrote:

> A monotone map is a functor between particularly special kinds of categories, right?

Right! A preorder is precisely a category in which every diagram commutes, and a monotone function is precisely a functor between such categories. A monoidal preorder is a monoidal category in which every diagram commutes, and a monoidal monotone is precisely a monoidal functor between such categories.

So, in this course so far, we are doing category theory in the degenerate case where we never need to check that any diagram commutes!

> Is it correct that the fact that a monoidal monotone also preserves the monoidal structure doesn't actually gain us anything, as far as the factorization goes that Tobias mentioned?

I'm not 100% sure what you mean, but I can guess. All this stuff works even in the non-monoidal situation.

Suppose \\(f : A \to D\\) is a monotone function between preorders. Borrowing words from Tobias, then \\(f\\) can be factored as:

* A monotone \\(f_1: A \to B\\) that only adds capabilities: that is, one that acts the identity map \\(x\mapsto x\\) between preorders with the same underlying set. In other words, we only add further inequalities while not doing anything else. For example, the discovery of a new technology is often like this: the set of resources and how they combine does not change, but suddenly you can do stuff that you could not do before!

* A monotone \\(f_2 : B \to C\\) that only changes the underlying set: that is, one where \\(b\leq b'\\) holds if and only if \\(f(b)\leq f(b')\\), and for every \\(c\in C\\) there is \\(b\in B\\) with \\(c\leq f(b)\leq c\\). For example, introducing a new currency may be an example of this: when you include a new currency in your resource theory, then you've expanded the theory, so that the original one is included in the new one. But it's still pretty much the same theory, since the new currency is perfectly interconvertible with your original one.

* A monotone \\(f_3: C \to D\\) that only discovers new kinds of resources: this is inclusion map of some \\(C\\) as a subset of preorder \\(D\\), where the preorder on \\(C\\) is simply given by taking the given preorder on \\(D\\) and restricting it. For example, when your company hires a new employee, then the time of that employee becomes a new type of resource. But the things that you can do without using the time of that employee are still exactly the same.

This is a special case of a fact about categories. Suppose \\(f : A \to D\\) is a functor between categories. Then \\(f\\) can be factored as:

* a functor \\(f_1 : A \to B\\) that is essentially surjective and full (but not necessarily faithful).

* a functor \\(f_2 : B \to C\\) that is essentially surjective and faithful (but not necessarily full).

* a functor \\(f_3: C \to D\\) that is faithful and full (but not necessarily essentially surjective).

This in turn is a generalization of the usual fact that any function can be factored as an onto function followed by a 1-1 function. All this stuff is studied in vastly more detail in my [Lectures on \\(n\\)-categories and cohomology](http://math.ucr.edu/home/baez/cohomology.pdf), in the part about "Postnikov towers".

> A monotone map is a functor between particularly special kinds of categories, right?

Right! A preorder is precisely a category in which every diagram commutes, and a monotone function is precisely a functor between such categories. A monoidal preorder is a monoidal category in which every diagram commutes, and a monoidal monotone is precisely a monoidal functor between such categories.

So, in this course so far, we are doing category theory in the degenerate case where we never need to check that any diagram commutes!

> Is it correct that the fact that a monoidal monotone also preserves the monoidal structure doesn't actually gain us anything, as far as the factorization goes that Tobias mentioned?

I'm not 100% sure what you mean, but I can guess. All this stuff works even in the non-monoidal situation.

Suppose \\(f : A \to D\\) is a monotone function between preorders. Borrowing words from Tobias, then \\(f\\) can be factored as:

* A monotone \\(f_1: A \to B\\) that only adds capabilities: that is, one that acts the identity map \\(x\mapsto x\\) between preorders with the same underlying set. In other words, we only add further inequalities while not doing anything else. For example, the discovery of a new technology is often like this: the set of resources and how they combine does not change, but suddenly you can do stuff that you could not do before!

* A monotone \\(f_2 : B \to C\\) that only changes the underlying set: that is, one where \\(b\leq b'\\) holds if and only if \\(f(b)\leq f(b')\\), and for every \\(c\in C\\) there is \\(b\in B\\) with \\(c\leq f(b)\leq c\\). For example, introducing a new currency may be an example of this: when you include a new currency in your resource theory, then you've expanded the theory, so that the original one is included in the new one. But it's still pretty much the same theory, since the new currency is perfectly interconvertible with your original one.

* A monotone \\(f_3: C \to D\\) that only discovers new kinds of resources: this is inclusion map of some \\(C\\) as a subset of preorder \\(D\\), where the preorder on \\(C\\) is simply given by taking the given preorder on \\(D\\) and restricting it. For example, when your company hires a new employee, then the time of that employee becomes a new type of resource. But the things that you can do without using the time of that employee are still exactly the same.

This is a special case of a fact about categories. Suppose \\(f : A \to D\\) is a functor between categories. Then \\(f\\) can be factored as:

* a functor \\(f_1 : A \to B\\) that is essentially surjective and full (but not necessarily faithful).

* a functor \\(f_2 : B \to C\\) that is essentially surjective and faithful (but not necessarily full).

* a functor \\(f_3: C \to D\\) that is faithful and full (but not necessarily essentially surjective).

This in turn is a generalization of the usual fact that any function can be factored as an onto function followed by a 1-1 function. All this stuff is studied in vastly more detail in my [Lectures on \\(n\\)-categories and cohomology](http://math.ucr.edu/home/baez/cohomology.pdf), in the part about "Postnikov towers".