(04/28/2022, 07:10 AM)JmsNxn Wrote: God you have horrible handwriting, I thought I was bad. I showed your handwriting to my friend, and it's worse than his, and all my friends make fun of his chicken scratch. Took me longer but I got through it.
Hahaha, I'd like to be offended by this but, tbh, I often have hard times reading my own handwriting... it's not writing... is more like some performance art where I use my hand to write ideas directly into my brain and then forget about the paper. xD
Anyways I apologize, I'll try to laTex it, for respect: these are notes written hastily using a wacom tablet for pc after an eureka moment using a language that is not my native one. I apologize also for the quality (grammar/typos) because under normal circumstances it takes me 4-5 check to make something I write in Italian acceptable... and it takes me 6-7 checks with grammar tools to make my English text just meh-level... so I'm aware that what I'm doing may be a little too much for the poor reader. But I do mostly just to leave something recorded.
For this reason I appreciate even more you kind words James: thanks! You just have to skim through... if only one single lemma or one philosophical observation here will be able to inspire someone I'll know that all of this was meaningful.
PS: during 2021 I extended partially all of this to cover, and give algebraic foundation to, your theory of iterated composition (the omega notation)! So all of this can be coherently extended to the case we are composing a sequence of functions.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)