Process journal of learning by Attila Vajda

keep noticing

growth mindset praise of the day by computer agent: #

Keep noticing stuff like this β€” it’s category-theoretic catnip. 🐾🌟
You're doing beautifully.

blackboard installed #

we installed a blackboard in my studio/atelier a01.png

β˜‘οΈ Black Hole: Made #

I made a meta-functorial black hole, a recursive c… Oh I forgot it's name. Wait a second.

this looks cool, but I don't yet understand it #

I don't yet understand this:
-- ℒ𝒾𝓂 ← π’œα΅’  β‰œ π’œ  ⇔ sheaf β„± : 𝒰ᡒ ↦ π’œα΅’
-- βˆƒ (π’œ : Type) , π’œ β‰… lim ← (π’œα΅’)
-- ✧ gluing ⇔ colim ⟢ consis. obs.
-- π’œα΅’ : ℰₒ𝒷ₛ(Uα΅’), ⊒ π’œα΅’ βŠ‚ π’œ βˆ€α΅’
-- QFT: presheaf β†’ sheafify β†’ glue

linear algebra connection? #

i i % 2 i % 3 Beat?
0 0 0 βœ… βœ…
1 1 1 βœ–οΈ
2 0 2 βœ…
3 1 0 βœ…
4 0 1 βœ…
5 1 2 βœ–οΈ

Is there a linear algebra connection here? This is a 3:2 polyrhythm. Computer agent asked me if I can define 3:2 polyrhythm,.

πŸŽ‰ 3:2 as pushout #

c/a asked if I could identify pushout in 3:2 polyrythm, so I wrote:

the pushout in 3:2, I don't yet know

o-ooo-
([1]) 2 [3] (4) [5] 6  -- [] for 3, and () for two,
β„€_6, mod 2 and mod 3!
βˆͺ Z6 % 2, Z6 % 3
βˆ‘ Z6 % 2 Z6 % 3 pushout 3:2,  maybe

#publish #interesting

gluing together ℀₃ and β„€β‚‚ #

to ℀₆, where both patterns coexist! #insight #publish #interesting #curious

def Z6 := Fin 6
def mod2 := Ξ» i : Z6 => i.val % 2
def mod3 := Ξ» i : Z6 => i.val % 3
def pushout32 := [(i, mod2 i, mod3 i) | i in List.range 6]

def pushout32 := List.range 6 |>.map (λ i => (⟨i, by decide⟩, i % 2, i % 3))

def combo := λ i => (i % 2 == 0) ∨ (i % 3 == 0) #

π₁ : ℀₆ Γ— {2,3} β†’ ℀₆ ℀₆ Γ— {mod 2, mod 3}

collapsing local layers (%2, %3) ⟢ global trace

this sort of stuff seems very energising

projection ⟡ product ⟡ coproduct what is projection, product and coproduct in 3:2 polyrhtyhm ?

mod 2:   ●   ●    ●  
mod 3:   ●     ●   

combo:   ●   ● ●  ●   (superposition)

#interesting #publish

the upper diagram seems to be like #

points written in a 2d cartesian coordinate system,

mod 2:   ●   ●    ●  
mod 3:   ●     ●

while combo is like projection to the x axis maybe

combo:   ●   ● ●  ●   (superposition)

projection to time? is vertical~local, horisontal~global? the sound of local global? #interesting #publish

is combo the pullback over ℀₆? #

#interesting #publish so there is subobject classifier and pullback square somewhere hidden in this pattern?

[ \documentclass{standalone} \usepackage{tikz-cd} \begin{document} \begin{tikzcd} A \arrow[r, "\phi"] \arrow[d, red] & B \arrow[d, "\psi" red] \ C \arrow[r, red, "\eta" blue] & |[blue, rotate=-15]| D \end{tikzcd} \end{document} ]

here is a nice espanso snippet: #

for the upper diagram, the diagram overhead, of previous diagram really:

  - trigger: :\sq
    replace: |
      ```tikz
      \usepackage{tikz-cd}
      \begin{document}
      \begin{tikzcd}
        A \arrow[r, "\phi"] \arrow[d, red]
          & B \arrow[d, "\psi" red] \\
        C \arrow[r, red, "\eta" blue] 
          & |[blue, rotate=-15]| D 
      \end{tikzcd}
      \end{document}
      ```

"I wonder..." #

def iWonder (P : Prop) := ¬proved P ∧ curious P

journal page #

j01