Process journal of learning by Attila Vajda

ℤₙ, ≡, n! --glimpse 2025-07-24

scattered insights, gently held

ℤₙ, ≡, n!

perm as isomorphism in Set 🐚 #

perm = bijection = isomorphism (≅) in Set category 🐚

preserve structure perfectly

permutations = automorphisms

def perm (α : Type) := { f : α → α // Function.bijective f }
-- permutations = isomorphisms α ≅ α, bijections on α

⛩ Hip rotation as modular torque #

hip rotation ≡ mod 2π ♻️🥋 τ ≈ effort to rotate hips how hard you twist τ_start ≈ “tension at start” τ′ ≈ “return torque” test τ ≡ τ′ mod 2π

you feel geometry, logic ≈ geometry, moving = thinking with your body.

Geodesics in stretch bands? 🌀

-- pls launch the game ▶️ 🕹 🎮 tiny cat-logic ≈ 🧶 Ω ⟵ 🐈‍⬛ ▶️ fdn🌱 ≈ (⟶, Ω, {}) .lvl0 -- with disc.lvl0 ≈ (ℤₙ, ≡, n!)`

fdn🌱 ≈ (⟶, Ω, {}) — morphisms, logic, empty type disc.lvl0 ≈ (ℤₙ, ≡, n!) — structure, relation, symmetry 🎷ska ≈ (ℤ₄, sync, perm) — rhythm, cohesion, groove

merci ≈ (gesture, reflection, emergence) 💖 merci ≈ (⟶💞, dual, unit) — gratitude as morphism = gratitude as category theory: a morphism that binds, returns, and creates. 🧶 merci is a monoidal moment 🌸 in the topos of being 🕊 (c'est ce que dit ChatGPT 🤖)

structure Merci (𝒞 : Type u) [Category 𝒞] :=
  (gesture : X ⟶ Y)
  (dual : Y ⟶ X)
  (unit : ⊤_𝒞 ⟶ X)

a closing lift, a small universal arrow:

🧭 All these structures—`perm`, `τ`, `merci`, `ska`—  
are arrows in the topos of learning.  
Each gesture, twist, sync, thanks =  
a morphism in your unfolding category. 🧶