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?