(06/03/2021, 03:10 AM)JmsNxn Wrote: Does this imply the "rank up" operation is a functorial by inversion?
That's a great question! The answer is: no, but maybe yes. The point is subtle but is relevant for the problem of chosing canonically a superfunction among all possible solutions.
It is a long but interesting story, I hope.
The fact that subfunction is functorial is amazing: it just means that it has some kind of "naturality" because it respect structure and is well behaved (eg. continuous functions respect the given topology). Not only that! It is functorial for every group and for every base function g (not only for g=successor) is beautiful imho. Why?
If we restrict ourselves to general groups then subfunction is never surjective (the identiy can not have a superfunction) and the best case we can get is when our group is superfunction-complete. In that case the only function that cannot have a superfunction is the identity.
The other obstacle for invertibility is that subfunction is never injective because subfunction of the identity is a the fixed point of subfunction

unless the group has only two element. So subfunction is never invertible in a strict sense. But it has many injective inverses!
Now we are forced to make a choice (

). Every possible inversion is a possible choice of superfunction. We can define concretely a choice as any function

such that performing the choice and then applying the subfunction brings us back where we started (mathematicians call it a section).

thus
Now as long as we are dealing with mere sets, and subfunction is just a set theoretic function we are in deep shit! Because in set theory every element of a set is the same and we have no preferences over elements, no systematic way of performing a choice. We needed more structure on the preimages and on the operator itself to hope in a universal way to invert it! So this is just the beginning of the research.
Brief note on why this is important.
Sadly Subfunction does not respect group structure and it is not clear if it is an important operation at all.
This is in fact a hint that maybe subfunction is not relevant to real mathematics...this is exaclty the feeling we get by reading Qiaochu Yuan answer here https://math.stackexchange.com/questions...c-f-circ-1 . It seems that subfunction is the wrong way to look at conjugation. And conjugation is the real important concept here.
This would imply that maybe iterating it (hyperoperations) is just an artifact, there is nothing deep about it.
This proof of functoriality changes this narrative! It says that subfunction respect the structure of two categories. At this point one could make an objection... well, giving to a set the hand-made ad-hoc topology you can turn every set theoretic function into a continuous one relatively to that made up topology, but we, serious people, only consider the good old classic topologies, the only topologies that are meaningful.
It turn out that the category structure I had to define to make subfunction functorial is pretty natural and it comes straight from the dynamical information contained (trapped?) in the group of functions considered! So extracting that dynamical information and organizing it under the shape of a category automatically makes the subfunction operator "natural" and well behaved!
This is so exciting!
Back to why now we have new hopes. The functor itself is not invertible in a straight way, for the same reason set-theoretic subfunction is not. The functor maps many different objects (dynamical systems) to the same one (not injective on objects). For example take the two S-dynamical systems
)
and
)
. Here I mean that we have
={\rm Add}_b(\phi(z)))
and
={\rm Add}_b(\psi(z)))
. Those two objects are elements of the category that makes the subfunction operator a functor. We haven't injectivity for the subfunction functor (I should name it.. maybe the subdynamics functor?)
To check this consider that
But now we have a functor! And now the preimages have a richer structure.
An elementary example.
To make clear why this could be the case consider basic undergraduate point-set topology.
A topological space is a set equipped with a system of open subsets
)
. So to every top, space we can assign a set, the set of its points.
That is a functor that forgets information, it forgets the topology, the "
shape of the space"

.
Now, can we invert it? Can we restore the lost information? Can we assign a topology to a set of points? Of course we can in an infinite number of ways (how to make a choice?).
All the topologies on the same set can be ordered by the the
coarser/finer relation. The preimage of a set under the point functor is the collection of all the possible topologies on a given set. Just like the preimage of a function under the subfunction operator is the set of all its superfunctions.
With a small but crucial difference. The collection of topologies on a given set is a lattice ordered by coarser/finer relation and has a top and a bottom, two special elements: the discrete and the trivial topology.
So in the case of the functor of points we have two canonical ways to invert it: two canonical way to assign a topology to given set, i.e. its trivial (co-discrete) and it's discrete topology.
Those two "sections" are not exactly inverse functors, but a weaker concept that is possible only when we replace functions with functors. They are called adjoint functors of the Pts functor.
So the next question we should ask is: does the subfunction functor have adjoints??
looks very promising... but that is just the beginning. Now we can finally unleash the full power of category theory on the theory of hyperoperations!