Discrete Universe Evolution : what is the structure preserving map?

Question

Hello,

This is going to be a question on defining how a discrete universe should evolve.

Panangaden and Martin showed that spacetimes are equivalent to interval domains.

I have recently shown that we can see these domain maps as functors and all of the following.  The map on your local laboratory is an endomap, and thus an endofunctor.  I have also postulated that the map from an early universe, into your local lab, is a functor with an adjoint.  I further state that the experiment you are doing in the local lab is a monad and a comonad and satisfies the frobenius law.  We see in Heunen's work that the Frobenius Monads on Hilb are precisely the quantum measurements, which bolsters my theory.

Panangaden et alias had an interesting take on a quantum universe on a discrete causal background.  I would say that a picture of a universe that you get from this paper is one of quantum systems connected by a causal structure.

In my research, I take the more general view that you can have any system type you want at the nodes of the causal structure, with some restrictions.

I am taking a universe to be systems of types connected by a causal structure.  Thus, a universe is an object in the category of interval domains, but it is also a diagram in an a category.  The most general evolution of a universe, then, is just a domain map (a morphism in the category of domains).  What Panangaden and Martin did not write out, explicitly, is what special constraints the map should have.  This is important.  Defining the extra structure on the maps actually goes to defining what a universe is.  So, I am proposing in this question a way to give more details on what defines a "Universe" via understanding the maps a universe would obey.

Given that, when seen as a functor, the structure preserving map of a universe is an ambidextrous adjunction, what, if any, extra constraints does this put on the domain map?

Comments

Interesting, maybe @UrsSchreiber can say something about this as he likes to work in the categorical framework too?

---

How do you define an universe in this theory? What is exactly mapped? How is the indistinguishability handled? Is there an intresic range of clearness like in QM and SR ? It is very interesting

---

@igael  I will attempt to answer your questions.  I think I need to add another reference, the main one I worked on in grad school! (lol).  What is "range of clearness"?  By indistinguishability, do you mean that of systems of the same type?  Like, Hilb supports a symmetric product so swapping systems of the same type looks like $A \otimes A \rightarrow A \otimes A$, so you can't tell apart systems of the same time?

Answer 1

I am not sure what the substance of the question is. The article by Panangaden and Martin is along the lines of well known old theorems, which show that much of the structure of Lorentzian manifolds is governed just by their combinatorial causal structure.

It seems to me somewhere along the lines of the question the word "spacetime" gets replaced by "universe" which is then taken to suggest deep implications. But it seems to be just words.

Also, it is clear that a poset map is a functor if we regard the poset as a category.

Generally, category theory underlies all of mathematics, much like complex numbers underly much of mathematics. Not every appearance of a complex number is a theory of everything, and the same is true for the appearance of categories.

Comments

One of the interesting things about the work I am proposing is that the type system you are observing can be different from the type system of your apparatus.  We map causal structures in the system under study (SUS) into causal diagrams in one's apparatus via a functor.  This is made deeply meaningful by the idea that this is exactly an evolutionary step in your universe (a causal structure mapping to a causal structure).

As to my question, if we take the type system to be Hilbert spaces at both ends, we know that the endofunctor at the apparatus is a Frobenius monad, meaning that the evolutionary map of the universe had to be an ambidextrous adjunction.  We can push this down on the the discrete causal structure as data about the domain map.

---

Sorry @BenSprott, I can''t see how this is not just rambling.

You look at an embedding of a map of one causal set into another, which is something as mundane as the order-preserving injection of the naturals into the reals, and the words you choose for this are: "[...] deeply meaningful [...] this is exactly an evolutionary step in your universe".