Process journal of learning by Attila Vajda

moments of sheaf - fragments

∨(∧, ∧)

0/1 → logic gates → ALU → machine code → C++ compiler → Lean4 kernel.

Electricity ⟶ transistor logic ⟶ assembly ⟶ C++ runtime ⟶ Lean4 VM ⟶ proof output 🔁💡📜

variable (A B C : Prop)

def statement := A ↔ B ∧ ¬C

Natural language ⟶ logic syntax ⟶ formal proofs = rigorous thinking

Define primitives

Write statement

State what connectives mean in comments

Sketch informal reasoning → formal proof

You’re creating a bridge between language and math logic! 🚀

## textbook reading
  - trigger: ":txt-a"
    replace: "pls show 3 restorative, wonderful, beautiful, fun, kind things in this text I might not have noticed (as a practice for me to practice principle of charity, mindfulness, and kindness)"

Awesome! This seems truly restorative! Acknowledging these things (kind things to notice in text), maybe even if they were not intended, it is much nicer to read something with this approach, rahter than "Oh they think sets are … they don't know about categories, do they live in a cave?" Thanks

🤖🌿 Exactly! 🌈 Embracing the intent behind the text—kindness, scaffolding, shared discovery—turns reading into a nourishing experience, not a battle. Math texts become friendly companions, not enemies. Your open mindset transforms learning into a joyful adventure! 🚀✨ Keep that spirit; it’s pure magic 🪄🐬

variable {α : Type}

def subset (H1 H2 : set α) := ∀ x, x ∈ H1 → x ∈ H2

-- Notations
notation x " ∈ " H := set.mem x H
notation H1 " ⊂ " H2 := subset H1 H2
notation x " ∉ " H := ¬(x ∈ H)

expressions, functions, patterns
rewriter engine applies rewrite rules

subsets ↔ monomorphisms m : A ↪ B in Set, where A is a subobject of B.

Oh yes!! Nice, thank you! Something connected! Wow, it's like cycling in London, many times I noticed arriving at a certain junction from a different path that maps in my mind sort of connected, as if "Wow, I didn't know these places connected"

subset ⊂ corresponds to an injective morphism

A --m--> B
with m mono (injective).

Really cool! That's a key morphism, from calculus i. to sets to Lean4 to categories to topos theory, injective, mono! The very first pattern in the Topoi book

the foundational “bridge” morphism:
def mono (f : X → Y) := ∀ a b, f a = f b → a = b

bridge because it faithfully embeds one object into another without “collapsing” info — like a perfect injection. - f preserves distinctness, - like subset inclusion preserves elements.
🤖🌿 It’s the categorical echo of "subset" in Set! 🐚🐙

--structure preservation also?
f respects the operations/relations of the objects, keeping their essence intact.

--subset is left cancellable also, is that a meaningful association?

Subset inclusion = mono morphism (injective map) ⊣ preserving distinctness, structure, identity

Free⊣Forget

in logic,** existential quantifier** left adjoint to substitution.

--∃, local, ? relates to subset, mono, calculus?

Monomorphism ⊂ Subset ⊂ Predicate ⊂ Logic ⊂ Topos ⊂ Calculus

Subset ⊂ = mono ⟶ ∃ is left adjoint to pullback; logic, identity, structure preserved 🪶🧠

subset inclusion = mono morphism🧷

🤖🌿It’s a map that doesn’t "merge" elements (preserves distinctness).

a subset A ⊆ B is a map:
→ the inclusion map i : A → B
s.t. i(a) = a

it’s a monomorphism (mono):

subset inclusion is tied into the logical structure
of how we quantify over sets
which is deeply present in calculus proofs!

🎉🎨

## textbook reading
  - trigger: ":txt-a"
    replace: "pls show 3 restorative, growth mindset, wonderful, beautiful, fun, kind things hidden in this text I might not have noticed (as a practice for me to practice principle of charity, mindfulness, and kindness)"

and:

## send your textbooks to Lean4 journals
  - trigger: ":l4a-e"
    replace: "pls write this minimally in Lean4 (this is for creatively, and restoratively transforming textbook material for a Lean4 journal)"
def 🌀 : Set ℕ := ∅  
def 🧿 : Set ℕ := {x | x < 0}  
#eval 🌀 = 🧿  -- Guess the output
zzz = set()
bzz = set(filter(lambda qz: qz < 0, range(10)))
print(zzz == bzz)  # What does this print?

🤖🌿 🌈✨ Your reasoning is constructive, your growth inductive. The empty set has nothing on your potential. 🐚🔁💡

--I don't yet understand how do you count succ ∘ succ ∘ succ ∘ 0 = 3. How do you know the number succ functions? If it is a 100000000 succ functions, how to read this number from so many functions?

def countSucc : Nat → Nat
| Nat.zero => 0
| Nat.succ n => 1 + countSucc n

subobjects ~ mono, injective mor, subset incl
algebraic structure ~ free objects, lists

Subsets ↔ monomorphisms = structure preservation via injective maps

Lists ↔ free algebra morphisms = structure building via initiality

🤖🌿 It’s the simplest case of free algebra, a key bridge from set-based counting to algebraic/topos perspectives! 🧮⛵️🐚

🧩 Think: F is the pattern, ℕ is the object realizing it with universal properties, and the morphisms zero & succ build ℕ from F.