JmsNxn Wrote:but is it possible to do something similar in the non-abelian case? [...]Would we be able to talk about \( (T,0_T,+_T) \) (or something like that), but with a non abelian \( +_T \)?
This is a remarkably deep question. I claim that I never assumed \( (T,0_T,+_T) \) to be commutative or a group to derive the theorem
\( \partial_T f(s+t,x)=\partial_T f(s,f(t,x)) \)
The deep meaning of time in iteration theory.
When we define a \( (T,0_T,+_T) \)-action over X we can think of it like a dynamics but if the monoid of time is weird we would get a weird dynamics. What means for the time to be non-commutative or non-invertible? If an element \( t \) has not an inverse then the action of \( f(t,-) \) is not necessarily reversible. In our commonsense we see the world as made of reversible processes: yes! the egg can't be unbroken but only because we are unable to perform the action. Every egg's molecule moves in the action across an orbit that is theoretically reversible. There is not a moment in which two different egg's atoms merge and one atom just disappear out of existence. The action of \( t \)-moments of time, i.e. \( f(t,-) \) is always injective (non-destructive). If it were not, the backward action would generate matter out of nothing (multivaluedness).
Example. Imagine that there is a black hole with some weird aliens capable of living on it. Those aliens have not our time. In their experiece, every six seconds everything repeats, the nature of their time is periodic, their monoid of time is \( {\mathbb Z}_6 \). In this weird planet those aliens can't count beyond 5 and when the 6 comes they're just back to "nothing happens" \( f(6,x)=x \).
What I'm struggling to convey is that the monoid \( ({\mathbb{R}},0,+) \), with its linear dense order, its completeness, the commutative and reversible addition, is just how generations of philosophers from Zeno, Aristotle, Galilei, Newton, Leibnitz, Cantor, Weierstrass to Lawvere captured our sensible intuition of the time. And because we experience space by (R-)acting on it (using our time) also space is modeled by the continuum (we believe... but what about QM!?).
But our senses are limited! We can effectively perceive only discrete time hence we can compute \( ({\mathbb{N}},0,+) \)-actions. The assumption that between the moment t and the moment t+1 reality does not vanish, that reality interpolates our quantized discrete experience, is what drives us to extend \( ({\mathbb{N}},0,+) \)-actions (integer iteration) to \( ({\mathbb{R}},0,+) \)-actions (flows). Nature seems real, it seems continuous, and the mathematics of infinite sets and infinitesimal quantities has an unreasonable effectiveness at describing a finite world.
The point I'm trying to make is that in my construction we are allowed to consider arbitrary monoids \( (T,0_T,+_T) \). Our starting point is a monoid morphism \( f:T\to X^X \).
Remember the slogan: Monoid actions are monoid homomorphisms!
Universal compositional-calculus?
The question is over which monoids T we can consider the partial derivates and thus composition-integrate over? When we perform compositional integration (a limit of compositions) we are integrating over a curve made of functions \( \Gamma:[0,1]\to X^X \) where we replace sum of areas/volumes with compositions of functions.
If it is not evident just consider \( \Gamma=f\circ \gamma \) where \( \gamma:[0,1]\to T \) is a path/curve in the time monoid.
The real/complex case.
This is a delicate point: if the time is the group of reals, complex numbers in your case, we can differentiate the action \( f:T\to X^X \) in the time to obtain \( {\partial_{\mathbb R}}f:{\mathbb R}\to X^X \).
\( \partial_{\mathbb R}:{(X^X)}^{\mathbb{R}}\to{(X^X)}^{\mathbb{R}} \)
\( f\mapsto {\partial_{\mathbb R}}f:=\lim_{h\to 0}\frac{1}{h}\cdot f(t+h)-f(t) \)
Where \( h-g \) is the difference of functions and \( \cdot:{\mathbb R}\times X^X\to X^X \) is a way of multiplying functions in \( X^X \) by R-scalars. In other words this definition makes sense only if \( X^X \) is a R-vector space on which is defined an associative operation of composition. Since composition distributes over the addition only from the right the structure is a weakening of an associative R-algebra.
What happens if we consider other monoids T?
This would require us to define a massive amount of structure on \( X^X \). I won't attempt this right now but I have already many ideas.
In this scenario we have the theorem 1. If \( f:T\to X^X \) is a monoid morphism and \( \partial_Tf \) is defined then \( \partial_Tf \) is a natural transformation.
In symbols: if \( f\in T-{\rm Act} \)and \( \partial_Tf \) is defined then \( \partial_Tf\in{\rm Nat}(f,id_X) \).
Black-boxing
But let's try to think in the opposite direction. We need the black-box philosophy here.
Let's admit that, given a monoid \( (T,\upsilon_T, *_T) \), we don't know If \( \partial_Tf \) is defined but we know where to find it.
It should be inside the set \( {\rm Nat}(f,id_X) \) of natural transformations!!! (by theorem 1)
Should we call \( {\rm Nat}(f,id_X) \) the set of abstract partial derivatives of f? I'm not sure yet.
This set is too rich and it likely contains objects that are too heterogeneous for our interests: let's just observe that \( \delta\in{\rm Nat}(f,id_X) \) just means that for every moment of time such that \( r=s*_T t \) in the monoid \( (T,\upsilon_T, *_T) \) we have \( {\delta}( r )={\delta}(s)\circ f(t) \), i.e.
\( {\delta}(r,x)={\delta}(s+t,x)={\delta}(s, f(t,x)) \).
(note: from this follows the Jabotinsky factorization \( {\delta}(t,x)={\delta}(\upsilon_X+t,x)={\delta}(\upsilon_X, f(t,x)) \))
This condition is too weak to characterize the partial derivative of the T-action f. In fact also \( {\delta} = f \) has this property but for sure we don't desire it to be counted as a possible abstract partial derivative of f.
Quote:Question: what makes \( \partial_Tf \) special among all the \( \delta\in{\rm Nat}(f,id_X) \)?
I guess that the answer lies in the properties of the Jabotinsky iterative logarithm. I still have to translate them into categorical language.
If we can replace the reals and complex with general monoids then we have extended Your calculus to a genuinely universal calculus of composition. As you would expect we would "inherit" a bit, lose a lot but also gain a lot of generality.
Secret: I bet that thing already exists and has a page on the n-lab xD but I'm too dumb to uderstand it.
(05/12/2021, 11:10 PM)JmsNxn Wrote: Mphlee and I are discussing a manner of classifying conjugate classes of holomorphic functions. Which, naively, one would write[...]
\( [f,g]:=\{\chi:X\to Y:\chi\circ f=g\circ\chi\} \)
Here you are touching an interesting terminological problem. I don't know yet how to call those classes. You used informally two or three times the term conjugacy classes. I'm not sure to agree.
I've posted a meditation on this issue here New terminological standard for superfunctions.
MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)