Composition, bullet notation and the general role of categories

[MphLee]

Fucking Beautiful PDF MphLee. Fucking Beautiful. That's just like my notation for compositional integrals. Never even thought for a second it was THAT categorical. Just Fucking, fucking Gordon Ramsey needs to speak to the chef about how FUCKING beautiful it was. I need to time to reallllly think about this. Thank you for the diagrams!

Oh ahahahah xD!
Omg!
I was in the writing mode when you posted this... I missed it... hahahah
ROTFLING

I need to time to reallllly think about this. Thank you for the diagrams!

Take all the time you need. Eventually going down this rabbit hole and while I ascend more to the analytic holy heights we will meet midway! haha!

That's just like my notation for compositional integrals. Never even thought for a second it was THAT categorical.

It is even more than you think. Given a finite sequence of \( n \) consecutive arrows we can think of it like a functor from the ordinal number \( n+1 \) to the context we are working in, e.g. complex numbers



Given an infinite sequence of consecutive arrows we can use your notation to write the n-truncations.



I want to convince you that we have a functor assigning to every pair \( (m,n) \), s.t. \( m< n \), a function \( X_m\to X_{n+1} \)



and this is functorial (I can prove it), i.e. the law that holds for integral has a deep categorical origin.



LET'S GO BACK ON THE POINTS AS MORPHISMS PARADIGM!
If we say that points of a space (elements of a set) are just morphisms from the point to the space we can obtain this





where we get (correct me if I'm wrong)

Now, this can be seen as a sterile change of notation. I believe it isn't, I feel and bet it isn't.
In all of this you see as the index is always a \( j\in{\mathbb N} \)... but what if we replace the naturals with the reals? Or with arbitrary monoids? That's what I want to discover and apply to the superfunction trick.


ps: as a side note that I'll expand soon, from a categorical point of view the outer composition is more natural and the inner composition is contravariant, i.e. it inverts the orientation of arrows.