morphism of abstract shape

• 23 September 2025
• Mathematics,
• Process book,
• Learning,
• Sketchbook

🌤 I am writing a few notes,
since the semester at the BSc mathematics
started and I am busy with the material.

Mathematics learning/doing
is awesome as ever in my experience!

🌀 I am taking a course called "Function iterations",
and at first I did not know what it was about,
I thought it had to do maybe with recursion and functions,
which I wanted to learn more about.

It turned out to be a really nice theme,
which has N and NNO,
and repeating iterative patterns,
and the textbook starts with exercises
about opennes, closedness,
sort of topological notions.

🍩 I also took a "Visual topology" introductory
topology course and I love the textbooks
and I noticed this lecture was
the first ever which I looked forward to with lots of excitement.
I am trying to preserve, recall this excitement for other lectures as well.

I am learning to write tiny proofs, for example I wrote

a | b iff |a| | |b|

a|b → |a| | |b|

    b=ae  (def_div)

|a|=ra  (r=±1)|b|=sb  (s=±1)
    =s(ae) (1)
    =(sa)e  (mul_assoc)
    =(r|a|)e  (s=r)
    =(|a|r)e  (mul_comm)
    =|a|(re)  (mul_assoc)
|a| | |b| → a|b

    |b|=|a|f  (def_div)

a=ς|a|  (ς=±1)b=υ|b|  (υ=±1)
     =υ(|a|f)  (1)
     =ς(|a||f)  (υ=ς)
     =(ς|a|)f  (mul_assoc)
     =(|a|ς)f  (mul_comm)
     =|a|(ςf)  (mul_assoc)
     =|a|(ςf)  (mul_assoc)
◇🖊

🐾 this morning I stretched my brain wondering about:

f: ⇉ ⟶ 🔀
⇉ and 🔀 are maybe two points, while f is a continuous map between them:
∀ε>0 ∃δ>0, |⇉-🔀|<δ ⇒ |f(⇉)−f(🔀)|<ε

位相 (isō) → topology/phase

圏 (ken) → category

関手 (kanshu) → functor

射 (sha) → morphism

構造 (kōzō) → structure

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