Hey, Everyone; been a long time...

[MphLee]

Honestly, your discussions of hyper operators were always category theory (iterations within iterations within iterations)

Yep, even without realizing it I was always trying to do category theory. I didn't have, and still I don't have, the right level of education to realize it, and now to use this fact fully. If you vaguely remember me talking about \( \Sigma \) multivalued maps and stuff... I was really obsessed with it back in the days because I was convinced to be on something really big, really monumental that nobody was noticing. I was, but it was not really my discovery, it was just me not knowing enough math: I rediscovered Hom-sets, so basically I was rediscovering category theory from another angle. The funny thing is that what pushed me there were, not counting my ignorance of analytical methods, three posts in this forum: one you made (JmsNxn, dec 2010), the other from (bo198214, aug 2007)  and (Base-Acid Tetration, feb 2009).

Happy to hear you're still working on tetration. Honestly, I'd like to read some of your cool ideas.

Not exactly Tetration but I'm looking for "new" ways to look at the concept of iteration in general, ways that can simplify things in order to work with conditions as weak as possible. All of this may sound weird for people like you, Sheldonison and Tommy. You are monsters in analysis and will see this as "running away from serious stuff and turning instead to trivial matters" and you are right. But for me understanding delicate complex dynamic structures and arguments when to me the basic algebraic/geometric backbone on which the topological/analytic machinery lies on is not clear is like learning the alphabet from the Z.


My primary effort was on rephrasing the language of this forum and the concept of iteration into categorical terms and in this language trying to find formal definitions of gadgets like Hyperoperations and ranks. This effort, as you can imagine, is headed towards the point where my naif first-year undergrad expertise will render me superfluous. But it's kinda exciting when you discover that tons of concepts and definitions where already known for decades under other names and are just easy exercises for competent mathematicians: it means you were not that crazy.

As far as Kan extensions go, I'm clueless. Frankly, all I know from category theory is from group theory; don't know how much help I'd be. But I'd be happy to talk.

Group theory and a bit of linear algebra is enough to give you a taste of it using some slogans. Long story short, the endgame of my crackpotish travels in the lands of math are the following points:

1. Solutions sets of Superfunction/Abel's/Shroeder's/Böttcher's equations, and general recursion, like sums and iterated compositions, are Hom-sets aka Functors (! in the appropriate categories).
   You already know well the hom functors: every cat \( C \) has its own hom functor: \( {\rm Hom}_C:C^{op}\times C\to {\bf Set} \). Take two groups \( {\rm Hom}_{\rm Grp}(G,H) \) is the set of homomorphisms, take two vec. \( k \)-spaces \( {\rm Hom}__{{\rm Vec}_k}(V,W) \) are all the linear applications, take two sets and you get the set functions \( Y^X \), take two top. spaces and you get the continuous maps . The key point is that that a morphism between two structures is a function that solves  some conditions, e.g. respecting the group operations. Now, an "iteration" \( (X,s)\to (Y,f) \) is a function that respects the successor and the dynamics of an endomap (using Kouznetsov terms: it respects the transfer function) \( \phi(s(x))=f(\phi(x)) \). So, in short, the Superfunctions/Abel's/Shroeder's/Böttcher's equation solutions sets are spaces of morphisms in the right category, e.g. the category of endomaps. To have a glimpse of how this is interesting look at the particular case of the Homsets \( C^0(X,{\mathbb R}) \) and how thet're stratified into the smoothness classes, well in some sense there is a context where this phenomenon happens but where we get the rank-classes.
   
2. Don't focus on the "concrete" functions, but abstract away the algebraic structure, i. e. study monoids and categories.
   Tl;dr: don't study directly iterations and continuous iterations but rather groups and monoid actions and their categories.
   
3. Extension of the iteration "is like" extension of scalars for vector spaces.
   
4. Goodstein Hyperoperations are star-shaped diagrams in categories of monoid actions.
   

If needed I can add some context for my slogans. I had prepared few paragraphs of elementary explainations for some of them but I cutted them because the post was getting lengthy.
I wont say too much in this post because I'm preparing trivial question on the the limiting trick you use.