Dear Chris
Thanks for your comment. I´ve read your essay and, as I said, I deeply impressed!
''So observation is itself a functor, from a category in which the objects are quantum states and the morphisms are unitary transformations to a category in which the objects are descriptions encoded in classical information and the morphisms are formal operations defined on those descriptions.''
I see we have slightly different views. To describe a ''quantum system'' or a ''classical system'' we need to use structures like space and time. However, these structures may come with a large degree of redundancy, depending on how we conceive motion in the first place, as I have explained in my essay. So there should be a functor connecting all those semantically ambiguous states, and the outcome of physical process should not depend on how we describe it. This is where the diagram commutes in my view. And the functor that connects all the semantically ambiguous states should be built by using a criterion of ''meaning upon observation'' to fulfill the principle that ''empirical indiscernibles are physical indiscernibles'' (as Robert Spekkens put). My hope is that this would have relational physics and GR as a sub-product. But I see the process I have in mind could be greatly enhanced by first characterizing observation as a functor in the first place, as you said. I will have to think more about that.
Best regards,
Daniel