Hey, Mphlee!
I don't want to comment too much as of yet; I haven't absorbed everything here; but I think this is unbelievably fascinating. Honestly:
\[
\text{A-Act} = \text{Set}^{\text{BA}}\\
\]
Then you say:
\[
\text{1-Act} = \text{Set}\\
\]
Man, I don't know what kind of weed you're smoking, but that is insane. That's Godelian thinking at its finest. The trivial monoid produces the standard theory of sets. If I ever make it to Italy you got to give me your plug.
One problem I have, and it's dumb, but:
In one of your diagrams \(\mathbb{N} \to \text{A-Act} \to \mathbb{Q} \to \mathbb{R}\to\mathbb{C}\). Shouldn't \(\text{A-act}\) come even before \(\mathbb{N}\).
Also, if you have any notes on the topology of these things--the idea of creating a continuous topology to \(A-Act\), please share them. Because that is insane. I understand now, when you say this relates deeply to tetration. Because we have an explicit example of \(A-Act\) in a non trivial environment (\(\exp\)); which deriving a continuous extension is almost unforeseeable with the way you've described it in the general case.
I'd like to really say, what hits it home for me, is the concept of \(\text{tet}(q)\) for \(q\) a quaternion. Despite how absurd this sounds, and when you first wrote it, I thought it was a little whacky. But this expansion perfectly exists. Quaternions work similarly through Taylor expansions (you have to finesse it a bit), but I wouldn't be surprised if there's a Kneser Tetration that works for Quaternions. This certainly works for non-singular square matrices, so I wouldn't be surprised. Locality would be whacked out a bit, and you'd get quaternion whacky branch cuts, but it's probably doable.
This is absolutely fascinating.
I think the continuous topology problem will probably be the hardest part. Have you read much on Algebraic Topology, and the topology of Algebraic Extensions//Grothendiek shit?
This reminds me a lot of the "it's so abstract it's nonsense", "but all the numbers work".
Regards, James
I don't want to comment too much as of yet; I haven't absorbed everything here; but I think this is unbelievably fascinating. Honestly:
\[
\text{A-Act} = \text{Set}^{\text{BA}}\\
\]
Then you say:
\[
\text{1-Act} = \text{Set}\\
\]
Man, I don't know what kind of weed you're smoking, but that is insane. That's Godelian thinking at its finest. The trivial monoid produces the standard theory of sets. If I ever make it to Italy you got to give me your plug.
One problem I have, and it's dumb, but:
In one of your diagrams \(\mathbb{N} \to \text{A-Act} \to \mathbb{Q} \to \mathbb{R}\to\mathbb{C}\). Shouldn't \(\text{A-act}\) come even before \(\mathbb{N}\).
Also, if you have any notes on the topology of these things--the idea of creating a continuous topology to \(A-Act\), please share them. Because that is insane. I understand now, when you say this relates deeply to tetration. Because we have an explicit example of \(A-Act\) in a non trivial environment (\(\exp\)); which deriving a continuous extension is almost unforeseeable with the way you've described it in the general case.
I'd like to really say, what hits it home for me, is the concept of \(\text{tet}(q)\) for \(q\) a quaternion. Despite how absurd this sounds, and when you first wrote it, I thought it was a little whacky. But this expansion perfectly exists. Quaternions work similarly through Taylor expansions (you have to finesse it a bit), but I wouldn't be surprised if there's a Kneser Tetration that works for Quaternions. This certainly works for non-singular square matrices, so I wouldn't be surprised. Locality would be whacked out a bit, and you'd get quaternion whacky branch cuts, but it's probably doable.
This is absolutely fascinating.
I think the continuous topology problem will probably be the hardest part. Have you read much on Algebraic Topology, and the topology of Algebraic Extensions//Grothendiek shit?
This reminds me a lot of the "it's so abstract it's nonsense", "but all the numbers work".
Regards, James