I read your paper again, and I think I have some more thoughts, but I have more questions. I think I'll formulate a couple of questions and try to explain myself through an air of questioning; and hone the questions better and then ask. First, I thought it warranted to try to talk categorically.
Can we write,
\(
\mathcal{F} = \{f \in \mathcal{C}(\mathbb{R}^+,\mathbb{R}^+),\,f\,\text{is an isomorphism},\,f' \neq 0\}
\)
So that \( f \) is say, a diffeomorphism (I believe that's the word, if not; it's something like that) of \( \mathbb{R}^+ \). Just so my shallow brain can think of a representative of the category; and it's not all up in the air. Let's additionally assume that:
\(
|f(x)| \le Ae^{Bx}
\)
For some constants \( A,B \). Which will make the exponential convergents behave well. And it would imply it's inverse at worse grows like \( \log \) somethin' somethin'. This would be a perfectly good algebraic space where we could derive,
\(
\forall f,g \in \mathcal{F} \exists \phi \in \mathcal{F}
f\phi=\phi g
\)
Now I haven't proven that, not entirely sure how to, but it's manageable--I could probably prove something close enough to continue the discussion.
-------------------------
With that out of the way, I'm going to keep thinking about this as operations on \( \mathcal{F} \) and functors; but to me they make sense as functors on \( \mathcal{F} \); or subgroups, or different versions or whatever. What I mean is, can we think of \( \mathcal{F} \) as an almost IDEAL space. Like the best space possible; where all the algebra is simple. Rather than monsters like \( e^x \) we look at simple amoebas like \( x^2 + x \). And build from the bottom up. Because I agree with a lot of what you are saying. But from a categorical perspective, start simple, no?
Unless I'm missing something drastic. You're paper was the most riveting the 3rd time... Maybe I just got over analytical, lmao
Can we write,
\(
\mathcal{F} = \{f \in \mathcal{C}(\mathbb{R}^+,\mathbb{R}^+),\,f\,\text{is an isomorphism},\,f' \neq 0\}
\)
So that \( f \) is say, a diffeomorphism (I believe that's the word, if not; it's something like that) of \( \mathbb{R}^+ \). Just so my shallow brain can think of a representative of the category; and it's not all up in the air. Let's additionally assume that:
\(
|f(x)| \le Ae^{Bx}
\)
For some constants \( A,B \). Which will make the exponential convergents behave well. And it would imply it's inverse at worse grows like \( \log \) somethin' somethin'. This would be a perfectly good algebraic space where we could derive,
\(
\forall f,g \in \mathcal{F} \exists \phi \in \mathcal{F}
f\phi=\phi g
\)
Now I haven't proven that, not entirely sure how to, but it's manageable--I could probably prove something close enough to continue the discussion.
-------------------------
With that out of the way, I'm going to keep thinking about this as operations on \( \mathcal{F} \) and functors; but to me they make sense as functors on \( \mathcal{F} \); or subgroups, or different versions or whatever. What I mean is, can we think of \( \mathcal{F} \) as an almost IDEAL space. Like the best space possible; where all the algebra is simple. Rather than monsters like \( e^x \) we look at simple amoebas like \( x^2 + x \). And build from the bottom up. Because I agree with a lot of what you are saying. But from a categorical perspective, start simple, no?
Unless I'm missing something drastic. You're paper was the most riveting the 3rd time... Maybe I just got over analytical, lmao