[note dump] Iterations and Actions

[MphLee]

2022, May 07 - bird's-eye view on algebraic iteration (road to reboot of old notes part 2) IMPOVED/COMPLETED

This is the intro to the second part of the notes I was rebooting for this thread but for some reasons I did not include into the 23th April post. The main reason is that I felt it was too abstract and I was going too quickly. That's why I'm going to elaborate a bit on some details of the 28th April's post.
Let's do just a brief recap leading us up to speed. Let \(A\) be a monoid: is is a set of things that you can freely combine \(+_A\) together (closed), where bracketing doesn't matter (associative) and where there is a thing \(0_A\in A\) that when combined produces no change in the result (identity element).

Def. an \(A\)-Action over a set \(Y\) is a map \(F:A\times X\to X\) st for all \(a,b\in A;\, y\in Y\)

1. \(F(0_A,y)=y\)
   
2. \(F(a+_Ab,y)=F(a,F(b,y))\)
   

Def. an \(A\)-iteration over a set \(Y\) is a map \(f:A\to {\rm End}(Y)\) st forall \(a,b\in A\)

1. \(f_{0_A}={\rm id}_Y\)
   
2. \(f_{a+_Ab}=f_a\circ f_b\)
   

Every \(A\)-action defines an unique \(A\)-iteration and every \(A\)-iteration an unique \(A\)-action. So talking about the former is equivalent to talking about the latter.

The moral of the story: is that \(\mathbb N\)-iterations - i.e. the theory of superfunctions - are a special case, a boundary case, of the more general theory of \(A\)-iterations(flows), i.e. dynamics where \(A\) is an arbitrary monoid of time.

^ In the figure, \(A\) is meant to be an arbitrary monoid. So the interior of the blob tries to express the vast territory of all the possible iterations using exotic monoids of time, and that connects integer, real and complex iterations.

The landscape The crucial observation to understand where we are in the landscape of mathematics is: classical dynamics bounds itself to special monoid of time, \(A=\mathbb R,\mathbb C\). There is a good reason for this limitation: that's because those are monoids endowed with a continuous (topological) structure and allows us to describe and study differential and analytical properties (flows<->infinitesimal generators<->vector fields).

We can say that the theory of \(A\)-actions or equivalently, as we have seen, of \(A\)-iterations is the broad land connecting, bridging the gap between, discrete dynamics and continuous dynamics. It is a general theory of dynamics were we allow the monoid of time to be an arbitrary monoid.

^ In the figure, the monoids are depicted as living inside their box, the "world of monoids" and to each of them there is associated an object of a different nature, that lives in the "world of categories" floating above. The monoids in the figure are just examples. \(A\) is just a generic monoid living in this world. This world must not be visualized as a linear progression of successive extensions, from smaller monoids to bigger monoids, but rather as a grandiose starry sky. Each monoid is a star, each of different magnitude and a different hue. Some of those monoids happen, as the stars, to be part of a constellation. The relations between monoids links them toghether as the lines connecting the points of a constallation.
The monoid of natural numbers is like the pole star because allows you to know the position of all the other stars. While the trivial monoid is the sun, because its light it's not useful to learn something about the other stars, it makes them invisible!!!

Terminology: with the term "theory of" I'm rather informal. I don't mean a theory in the model theoretic sense. With a theory of \(A\)-actions I just refer to the complete collection of all the possible \(A\)-actions on every possible set. In addition to that I consider a theory to be the organized collection of all these actions: organized means that I consider all the \(A\)-actions and the network of ways they are in relation with each other. I call this big object
\[A{\rm -Act}\]
and this express all the information of what it means for \(A\) to act on something (a set). The same goes for the theory of \(A\)-iterations: I'll denote that theory with
\[{\rm Set}^{BA}\]
So, when I say that to each monoid \(A\) we are associating the theory of \(A\)-actions and that it is equivalent to the theory of \(A\)-iterations... I precisely mean the procedures that take \(A\) as an input and return these massive objects \(A{\rm -Act}\) and \({\rm Set}^{BA}\) as outputs and that in some sense to be made precise \[A{\rm -Act}\simeq{\rm Set}^{BA}.\]

Key observation: these procedures are contravariant functors, preasheaves of categories to be precise. This means many things. The most immediate thing is that the network of relations between monoids gets reflected, reversing all the directions, by the functors: all the relations connecting monoids become relations connecting their theories of iterations!!!

Great unification: let's consider the big network containing for each possible monoid it's theory of actions. This monstrous "galaxy" of theories includes the theory of \(\mathbb N\)-iterations, of integer and other discrete dynamics but also rational, real and complex iterations. This landscape but also the \(1\)-iterations of the trivial one-element monoid \(1\): this is just the theory of sets \(1{\rm -Act}\simeq {\rm Set}^{B1}\simeq {\rm Set}\), a dynamics where every point is a fixed point, aka the dynamics of the identity!

Example 1 Just to have a glimpse of the richness just imagine a universe where the time resets back to the beginning every 5 units of time. In that universe we would like to study \(\mathbb Z /5\mathbb Z\) iterations. Every iteration of that kind would have inevitably periods of length \(5\) or fixed points.

Example 2 Not only that. That framework is so general that we could study a universe that has as monoid of time the monoid of real numbers under multiplication \(\mathbb R^{\rm mul}:=(\mathbb R,\cdot,1)\). Then you could ask me: "cmon that is just a silly game, give me an example of an \(\mathbb R^{\rm mul}\)-action that is of any interest!!"
I can give an entire class of \(\mathbb R^{\rm mul}\)-actions that is fundamental to classical mathematics: real vector spaces. Sure! to have a vector space you need much more data than an \(\mathbb R^{\rm mul}\)-action structure, but it is true that every \(\mathbb R\)-vector space is an \(\mathbb R^{\rm mul}\)-action. How?
An \(\mathbb R^{\rm mul}\)-action over a set \(V\) produces for each \(\lambda\in \mathbb R^{\rm mul}\) a "scaling function" \(\mu_\lambda:v\mapsto \lambda v\) that we regard as an iteration with "multiplicative time". \[(\lambda\kappa)v=\lambda (\kappa v)\]

In this class of examples, that is called linear algebra, an orbit under the iteration is just a linear one-dimensional subspace and the dynamics can be thought as the dynamic of linear motion of a vector from the origin of the vector space.

A rosetta stone the last example opens a new world of heuristics and analogies that can guide our intuition and that will guide all this thread.
\[\begin{align}
\mathbb N{\bf -iteration} && A{\bf -iteration}&&\mathbb R^{\rm mul}{\bf -action}\\
f^n(y)&&F(a,y) && \lambda v \\
y_0-based\, recursion:\mathbb N\to Y&& y_0-orbit\,map:\, A\to Y && parametrization:\mathbb R\to V\\
n\mapsto f^n(y_0)&& a\mapsto F(a,y_0) && \lambda\mapsto \lambda v\\
orbit\, of \, y:\, \{y,fy,f^2y,...\}\subseteq Y && orbit\, of\, y: \{F(a,y)\}_{a\in A}&&linear\, span\, \langle v\rangle\subseteq V \\
ancestor\,relation && A-reachability&&linear\, dependence \\
y\sqsubseteq_f z\,{\rm iff}\,\exists n.\, f^n(y)=z&&y\sqsubseteq_F z\,{\rm iff}\,\exists a.\, F(a,y)=z && \exists \lambda.\, \lambda v=w \,{\rm or }\,v=\frac{1}{\lambda}w \\
connectedness\,relation && A-connectedness&&linear\, dependence \\
y\frown_f z\,\,{\rm iff}\,\exists n,m.\, f^n(y)=f^m(z)&&y\frown_F z\,{\rm iff}\,\exists a,b.\, F(a,y)=F(b,z) && \exists \lambda,\kappa.\, \lambda v=\kappa w \,{\rm or }\,v=\frac{\kappa}{\lambda}w \\
components\, Y /_{\frown_f}&& components\, Y /_{\frown_f} &&projective\, space \, \mathbb PV \\
??? && ??? && dimension\\
&& && \\
superfunction && A-equivariant \, map && \mathbb R-linear \, map\\
\chi:{(X,g)}\to (Y,f)&& \phi:{(X,G)}\to (Y,F) && M:W\to V\\
\chi(g(x))=f(\chi ( x))&& \phi(G(a,x))=F(a,\phi( x)) && M(\lambda w)=\lambda (M w)\\
\forall n.\,\chi g^n=f^n\chi&& \forall a.\,\phi G_a=F_a\phi && \forall \lambda.\,M \lambda =\lambda M\\
&& && \\
&& {\bf Eigentheory}&&\\
centralizer\, of\, f && centralizer\,of\,\{F_a\}_{a\in A}&& linear\,operators\,over\, V\\
\alpha f=f\alpha&& \forall a.\,\alpha F_a=F_a\alpha && \forall \lambda.\,\Phi(\lambda v) =\lambda \Phi(v)\\
eigenpoint\,y\, of\, \alpha&&  eigenpoint \,y\, of\, \alpha && eigenvector \,v\, of\, \Phi\\
y\,s.t.\, \exists n\in\mathbb N.\alpha(y)=f^n(y)&& y\,s.t.\, \exists a\in A.\alpha(v)=F(a,y) && v\,s.t.\, \exists\lambda\in\mathbb R^{\rm mul}.\Phi(v)=\lambda v\\
eigennumber\,n\,associated\, to\, y&&  eigentime\,a\,associated\, to\, y &&eigenvalue\,\lambda\,associated\, to\, v\\
spectrum\,of\,\alpha &&  spectrum\,of\,\alpha && spectrum\,of\,\Phi \\
\sigma (\alpha)=\{n\,:\,\exists v.\,\alpha(y)=f^n(y)\}&& \sigma(\alpha)=\{a\, eigentime \, of \, \alpha\} && \sigma(\Phi)\\
&& && \\
&& {\bf Geometry} && \\
??\simeq\mathbb N^d&& ??\simeq A^d && affine\, spaces\, X\simeq\mathbb R^d \\
??\subseteq \mathbb N^d&& ??\subseteq A^d && affine\, varieties\, X\subseteq\mathbb R^d \\
??&& ?? && manifolds \\
??&& ?? && homology/cohomology\\
\end{align}\]

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The last lines of this rosetta stone suggests that, in some way, we are extending geometry itself: from real/complex geometry to the geometries of universes where the time is discrete or more exotic monoid \(A\).

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Man, I don't know what kind of weed you're smoking, but that is insane.

i) You think I'm high, I tell you this is just the tip of the iceberg. This is how high we can go:

• ground) Integer iteration: we are limited to \(\mathbb N,\,\mathbb Z\);
  
• 1) Real/ complex iteration: we extend our number system;
  
• 2) We realize that this is not about number systems but about monoids. So we realize that the previous cases are particular instances of a general framework that takes monoids and gives theories of \(A\)-iterations where \(A=\mathbb N, \mathbb R,\mathbb C\);
  
• 3) We realize that all we are doing is in fact considering a monoid \(A\) as a category with a single object. If we relax our understanding of what we accept as the object representing time to other kind of categories, let's say preorders, we obtain a new extension that contains the theories of iterations as special case;
  
• 4) When the object of time is a discrete preorder we obtain a theory of complete recursion and of iterated compositions (omega notation);
  
• 5) If we extend the object of time to continuous preorders, or real intervals, we obtain the algebraic version of compositional integrals.
  
• 6) If we remove all the limitations on what time can be, we just accept arbitrary "categories of time", we obtain the theory of functor categories. That is part of what can be called \(2\)-category theory.
  
• 7) Can we apply this on itself and obtain Goodstein's hyperoperations were ranks are arbitrary? Almost...
  

How high are we atm? We are at point 3). How much THC we are ready to take to get as high as 7)?

One problem I have, and it's dumb, but

ii) About this I added a remark under the figure you mention. I warn you to do not place the object \(A{\rm -Act}\) on the same level of those monoids. So... No, there is not a monoid that comes before, in a strict sense. Remember that every "constellation" in the world of monoids gets projected, reflected like mirror and in a reversed direction, on the world where the theories of \(A\)-iterations live. Just like the following illustration, where I show just one of the infinite constellations and its contravariant reflection:

iii) About the topology on these things. That is crucial... I don't know how to canonically extract topological structure from that. I have notes on how to extract divisibility information, i.e. information on the rational iterates... There are more ways btw. In classical analysis the topological information is extracted from two informations: the complete order relation of the real numbers and from its field structure. So you use the metric and from there the norm and use them to study sequences by defining convergence first and continuity. Here maybe we should maybe extract some partial-order information from the monoids and from the iterations and maybe use it in  some way. But that is the analytic way...

If we instead follow a more Grothendiek inspiration... maybe I should study the mechanism of sheafification and how to extract directly the lattice of open sets from the monoid or from the set equipped with the action and how all of this varies when modulating the monoid of time.

tl,dr: It's too early for me to introduce topology at that general level. I believe I'd better go for Jabotinsky's related business and in how to produce vector fields from \(A\)-actions.