For a while I've imagined that higher orders of logic "quantify over sentences" of lower orders of logic.
In Dr. Eugenia Cheng's lecture in Sydney today, she showed a slide saying that category theory is the mathematics of mathematics:
Category theory is the logical study of
the logical study of
- how logical things work.
This reminded me of that notion. Is there a connection here? Is this what Category theory is doing? Taking maths to a level of abstraction where the operators are quantifying over, and acting on, themselves, in such a way that no higher level of abstraction is possible? Does category theory effectively quantify over its own sentences?
I believe you are being mislead by Dr. Cheng's motto.
As everyother motto it should not be taken to formally.
What Dr. Cheng is trying to say is that mathematics amounts to the study of logical structres and transformations and relations between them, these data (these structures and relations) are of course logical structures in their own way and have their own transformations and relations, hence they are object of study of mathematics itself. Category theory is the mathematical study if this structures of structures.
Of course here the word logical is not to be interpreted in the formal way (it does not have to do with propositions, connectives and quantification, although category theory can be applied to the study of these very specific structures). Here the word logical is used in an informal way to specify that the objects considered are abstract entities that live in the human mind.
So in short, no category theory does not quantify over propositions (well actually it does when you are working in a category whose objects or morphisms are logical formulas but that is another story for another day).