Nostr notes by rntz

Ah yeah, I just reread my initial post and realized I use ...

2026-06-24T23:35:59Z

In reply to nevent1q…2w6z
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Ah yeah, I just reread my initial post and realized I use "C" without explanation. "C" is meant to stand for a typeclass like Expr, not a functor F, for instance:

Expr ≅ Fa → a for Fa = Int+(a×a)
Stream elem a ≅ a → Fa for Fa = Int × a

so when Ca = Fa → a we get

(1) ∀α. C α → α ≅ ∀α. (Fα → α) → α (Church encoding)

and when Ca = a → Fa we get

(2) ∃α. C α × α ≅ ∃α. (α → Fα) × α (co-Church encoding?)

I don't think (∃x. x -> F x) is actually a F-coalgebra ...

2026-06-24T23:25:47Z

In reply to nevent1q…0d8v
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I don't think (∃x. x -> F x) is actually a F-coalgebra itself! For that I'd need:

(∃x. x -> F x) -> F (∃x. x -> F x)

and I don't see how. The terminal coalgebra is actually (∃x. x * (x -> F x)), I think. So the conatural numbers should be (∃x. x * (x -> Maybe x)), I think.

Ok, I think I worked it out: (1) is an initial algebra when C a ...

2026-06-24T23:09:08Z

In reply to nevent1q…ah7d
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Ok, I think I worked it out:

(1) is an initial algebra when C a ≅ Fa → a for some functor F.
(2) is a terminal coalgebra when C a ≅ a → Fa for some F. Example:

class Stream elem a where
get: a → elem
step: a → a

Stream Int a ≅ a → Int × a. Then (∃a. Stream Int a × a) is a terminal coalgebra for (Int × _).

class Expr a add: a × a → a lit: Int → a The type of ...

2026-06-24T18:35:05Z

class Expr a
add: a × a → a
lit: Int → a

The type of expressions in this language is definable by Church-encoding:

∀α. Expr α → α

Since Expr a ≅ Fa → a for Fa = Int+(a×a), this corresponds to initial F-algebras. But it's definable for *any* class C, not just for F-algebras. And it has a... dual?:

(1) ∀α. C α → α
(2) ∃α. C α × α

I'm used to thinking of (2) as OO "objects". But is this related to terminal coalgebras somehow? If so, how? Or is it a different sort of dual entirely?

It's well known that set intersection is associative. ...

2026-01-18T15:36:47Z

It's well known that set intersection is associative. Unfortunately, it's also false:

import Data.Set
evens = fromList [0, 2 .. 20_000_000]
odds = fromList [1, 3 .. 20_000_000]
ends = fromList [0, 20_000_000]
fast = evens `intersection` (odds `intersection` ends)
slow = (evens `intersection` odds) `intersection` ends

λ> :set +s
λ> size fast
0
(0.00 secs, 70,592 bytes)
λ> size slow
0
(0.68 secs, 1,500,066,912 bytes)

I didn't even realize pointed sets had coproducts until now! ...

2025-02-11T22:03:56Z

In reply to nevent1q…9e2l
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I didn't even realize pointed sets had coproducts until now! But now I see they're "smash" coproducts, ie the coproduct of the underlying sets quotiented by inl ⊥ = inr ⊥ = ⊥. interesting...

sketch of why exponentials don't exist: Find some f,g : Hom(A ...

2025-02-11T21:50:46Z

In reply to nevent1q…3lca
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sketch of why exponentials don't exist:

Find some f,g : Hom(A x B, C) such that f(⊥,x) ≠ g(⊥,x).
Curry f,g to get f*, g* : Hom(A, B ⇒ C).
But then f*(⊥) = g*(⊥) = ⊥ by preserving points.
So it sure seems like we must have f(⊥,x) = g(⊥,x).

I do not see a way to get around this. This is not proper proof, though.

The category of pointed sets (sets w/ a designed "point", ...

2025-02-11T21:43:46Z

The category of pointed sets (sets w/ a designed "point", ⊥) and point-preserving maps (f(⊥) = ⊥) has:

cartesian products A × B: pairs with ⊥ = (⊥,⊥)
tensor/smash products A ⊗ B: quotient A × B by (⊥,y) = (x,⊥) = ⊥
internal homs A ⊸ B: point-preserving maps with ⊥ = λx.⊥, satisfying Hom(A ⊗ B, C) ≅ Hom(A, B ⊸ C)

but AFAIK no exponential A ⇒ B with Hom(A × B, C) ≅ Hom(A, B ⇒ C)

is there an extension of pointed sets that has exponentials w/o losing ⊸? (& thus models bunched implications?)

Nostr event nevent1qqsqvt8uhhsdqpq34rl6zawunstydl55vukfer06c4tnw9aa658vtrgzyr4gzx7ud3u50fw2m57h4h93lw7rylwa97hj0kd7g77v4r3wj6zvydxceml

2024-10-26T16:08:19Z

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