[note dump] Iterations and Actions

[MphLee]

Comment on the quaternionic extension example. Just thinking about it... I realize all that is missing from my treatment, atm, is the sheaf business: I work globally/algebraically but I don't have a clue yet on how to organize local solutions. But in many cases this means I'm excluding non-trivial solutions.
Using sheafs means to consider both local and global solutions and how to patch them. This is what subconsciously one does in analysis and what algebraic geometers and algebraic topologists do with schemes/sheaves and manifolds/sheaves of analytic/holomorphic functions.
This is the reason, every time I read old posts in the forum declaring to compute fractional iterates using solutions to abel and superfunction equations or Schroeder and inverse Schroeder functions, I'm puzzled. If you really have a solution of that kind \(\chi(x+1)=f(\chi(x))\) and you can invert \(\chi\) globally, this means that the dynamic of the successor is isomorphic to the dynamics of the function \(f\) you are iterating, e.g. of \(f=\exp\).
Clearly this is not always the case. In fact, you all then start talking about neighborhood of points and equations holding locally on some domain.

---

In the last part I tried to build a narrative filling my notes with more explanations and comments: this time I'll comment less. The following notes continue the meditation on how to travel from functions, \(\mathbb N\)-iterations and \(A\)-iterations and backwards.

2022, april 23 - actions, iterations and generalized elements PART 3 (reboot of old notes)

In the previous discussion, given a monoid \(A\), we were discussing the collection \(A{\rm -Act}\simeq {\rm Set}^A\) of all the possible pairs \((Y,\varphi)\) where \(Y\) is a set and \(\varphi_a(y)\) is an \(A\)-action/\(A\)-iteration over the set \(Y\), for all the possible sets.

Now we fix a set \(Y\), and let the monoid vary. We want to focus only on how to iterate the functions \(f:Y\to Y\) over \(Y\). As a convention \({\rm Hom}(A,B)\) is the set of monoid morphisms \(\varphi:A\to B\).

Definition 1 denote with \(A{\rm -act}_Y\) the set of \(A\)-actions over \(Y\). It is the set of object of the fiber of the forgetful functor at \(Y\) so it results by pulling back the forgetful functor, sending an action to it's underlying set, by the inclusion of \(Y\) into sets.

^ Pullback

Question is \(A{\rm -Act}\) related to the family of \(A{\rm -act}_Y\) via a something related to the Grothendieck construction? Note that, abusing the notation and only in what regards objects
\[A{\rm -Act}=\bigcup_{Y\in{\rm Set}}A{\rm -act}_Y\]

Here we define what we mean with the term \(A\)-iteration over a given set.

Definition 2 denote with \({\rm ite}(A;Y)\) the set of what we want to call \(A\)-iterations over \(Y\), i.e. it is defined as \({\rm Hom}(A,{\rm End}(Y))\).
\[{\rm ite}(A;Y):={\rm Hom}(A,{\rm End}(Y))\]
It is the set of object of the fiber of the forgetful functor at \(Y\):

^ Pullback 2

Remark \({\rm Hom}(A,{\rm End}(A;Y))\) is also the set of objects of the functor category \({B{\rm End}(Y)}^{BA}\). Here \(BA\) is the category naturally associated to the monoid \(A\) (see delooping). Note that, abusing the notation and only in what regards objects
\[{B{\rm Set}}^{BA}=\bigcup_{Y\in{\rm Set}}{B{\rm End}(Y)}^{BA}\]

Lemma As expected \(A\)-actions over \(Y\) are in bijection with \(A\)-iterations over \(Y\): for every monoid \(A\)
\[ A{\rm -act}_Y\simeq {\rm ite}(A;Y)\]
this bijection is natural in a technical sense.

We are going to use this fact as a key element of the proof that there is a natural isomorphism of contravariant functors:

^Globe diagram

Proposition The presheaf assigning to each monoid the \(A\)-actions over \(Y\) is a representable presheaf, represented by \({\rm End}(Y)\): for every monoid morphism \(j:B\to A\) the following diagram commute

Proof: We need to show that given an arbitrary \(j:B\to A\) and for every \(A\)-action \(\varphi(a,y)\), for every \(b\in B,\,y\in Y\)
\[{\rm curry}_Y(\bar{j}(\varphi))_b(y)=j^*({\rm curry}_Y(\varphi))_b(y)\]
Just unwrap the definitions
\[\begin{align}
{\rm curry}_Y(\bar{j}(\varphi))_b(y)&=j^*({\rm curry}_Y(\varphi))_b(y)\\
\bar{j}(\varphi)(b,y)&={\rm curry}_Y(\varphi)_{j(b)}(y) &&&\\
\varphi(j(b),y)&=\varphi(j(b),y)&&&
\end{align}\]

Being representable means that iterations and actions are really the same and in a natural way: independently of monoid where the time takes values in.

Conjecture (when time is commutative) since \({\rm Act}_Y:{\rm Mon}^{op}\to {\rm Set}\), \(A\mapsto A{\rm Act}_Y\) ,is a representable functor
\[{\rm act}_Y\simeq {\rm ite}(-;Y)\]
I expect it to send direct products to cartesian products, or better to send colimit into limits, when the monoids we are working with are commutative. In other words, let \(L,M\) be commutative monoids
\[(L\times M){\rm -act}_Y={\rm Hom}(L\times M,{\rm End}(Y))\simeq {\rm Hom}(L,{\rm End}(Y))\times {\rm Hom}(M,{\rm End}(Y))=L{\rm -act}_Y\times M{\rm -act}_Y\]
I guess the way to prove amounts to proving that every morphism \(f:L\times M\to A\) determines, and is determined, uniquely a pair of morphisms \(f_0:L\to A\) and \(f_1:M\to A\).

Reality check set \(A=\mathbb C\) the additive abelian group of complex numbers. Since there is an isomorphism of abelian groups \(\mathbb C\simeq \mathbb R\oplus \mathbb R\), i.e. \(z\mapsto ( \Re(z), \Im(z))\) and \((a,b)\mapsto a+ib\), this means that
\[{\rm Ite}(\mathbb C;Y)\simeq {\rm Ite}(\mathbb R;Y)\times {\rm Ite}(\mathbb R;Y)\]

This says that every complex-iteration \(\varphi_z\) is completely determined by \(1\) and \(i\), or by any pair \(w_0,w_1\) of complex numbers that generates -is a base for- all the complex plane. It seems trivial but I'd like to check the details.
All the stuff above also provide us the power to:

• transform every morphism \(A\to\mathbb C\) into a way to convert \(\mathbb C\)-iterations over \(Y\) into \(A\)-iterations over \(Y\) and
  
• transform every morphism \(\mathbb C\to A\) into a way to convert \(A\)-iterations over \(Y\) into \(\mathbb C\)-iterations over \(Y\).
  

To be concrete here two trivial examples:

• The canonical real part surjective morphism \(\Re:\mathbb C\to \mathbb R\) can convert real iterations over \(Y\) to complex iterations over \(Y\) by computing \( (\Re^* \varphi)_z(y):=\varphi_{\Re z}(y)\);
  
• Fix a complex number \(z\in\mathbb C\), consider the map \(m_z:\mathbb Q\to\mathbb Z\) that sends the rational \(q\in\mathbb Q\) to \(zq\in\mathbb C\). It is a morphism \(m_z:\mathbb Q\to \mathbb C\) and we can convert complex iterations over \(Y\) to rational iterations over \(Y\) by computing \( ((m_z)^*\varphi)^{q}(y):=\varphi^{zq}(y)\);
  

Extracting and repackaging the information of the iterates

Philosophy Fix a set \(Y\), the functor \({\rm ite}(-;Y)\) assigns to every monoid all the possible iterations with time in that monoid. We can say that \({\rm ite}(-;Y)\) is a gadget that organizes and hold in a single place all the information of what it means to iterate functions \(f:Y\to Y\), and with that all the possible way to extend the various iterations and the obstruction to the existence of extensions. It's a very rich gadget. I expect the study of this functor is crucial.

we can also visualize this as follows:

Crucial observation This makes evident that all the information in this functor is "in some sense contained" by the category of bundles over the monoid \({\rm End}(Y)\), i.e. the slice category \({\rm Mon}/{\rm End}(Y)\). Maybe it is possible to prove the isomorphism of categories of the slice to some Grothendiek-related construction. But the really mysterious thing is that the slice construction records a different kind of information: that of divisibility; while the normal category of iterations we record the information about equivariance and preservation of dynamical properties. So in the slice category the relations are relations of divisibility while in the category of actions the relations are superfunctions/equivariances.

I'm tempted to describe this as saying that the original view records the dynamical information, regarding addition of time, composition of processes; while the slice construction makes the mutiplicative/fractional iterative information visible.

Intrinsic iterates This means that every monoid \(A\) has not a total freedom on the way it can iterate function over \(Y\): to be precise it has a limited set of ways. In fact every \(A\)-iteration squishes \(A\) into \({\rm End}(Y)\) unless \(a\mapsto \varphi^a\) is injective as exponentiation, i.e. \(A\) is a submonoid. So the information on the obstructions on how can we iterated is written in the lattice of submonoids of \({\rm End}(Y)\) and that is completely determined by the cardinality of \(Y\) by Cantorian arguments.

Observation it is really important to underline the fact that \({\rm ite}(-;Y)\) is functorial as the monoid time varies but it IS NOT FUNCTORIAL as the set \(Y\) varies. The meaning of this?

• it is easy to transform \(A\)-iterations over \(Y\) to \(B\)-iterations over \(Y\) in a coherent way;
  
• it is not possible to transform and relate \(A\)-iterations over \(Y\) to \(A\)-iterations over \(X\) in a coherent way;
  

The slogan can be: we can relate coherently iterations with different times, but not over different sets - change of base is hard.

---

The next part is the 4th and last and is the missing link between the bird's-eye view and the attempt to recover fractional iterates. After that I only have notes about divisibility of iterations, some notes about my first approach to cohomology and the ones about how to generalize Bennet hyperoperations accordingly.