From asi
Defines dependent Yoneda lemma in Rzk as directed path induction for Segal types in directed HoTT, per Riehl-Shulman. Provides chemical semantics, GF(3) triads, and theorem proofs. Useful for synthetic ∞-categories.
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> *"The dependent Yoneda lemma is a directed analogue of path induction."*
Defines Segal types for synthetic ∞-categories ensuring unique binary composites up to homotopy. Supplies Rzk definitions, Lean4 structures, chemputer semantics, and Ruby composition example.
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"The dependent Yoneda lemma is a directed analogue of path induction." — Emily Riehl & Michael Shulman
| Standard HoTT | Directed HoTT |
|---|---|
| Path induction | Directed path induction |
| Yoneda for ∞-groupoids | Dependent Yoneda for ∞-categories |
| Types have identity | Segal types have composition |
#lang rzk-1
-- Dependent Yoneda lemma
-- To prove P(x, f) for all x : A and f : hom A a x,
-- it suffices to prove P(a, id_a)
#define dep-yoneda
(A : Segal-type) (a : A)
(P : (x : A) → hom A a x → U)
(base : P a (id a))
: (x : A) → (f : hom A a x) → P x f
:= λ x f. transport-along-hom P f base
-- This is "directed path induction"
#define directed-path-induction := dep-yoneda
Chemical Interpretation:
yoneda-directed (-1) ⊗ elements-infinity-cats (0) ⊗ synthetic-adjunctions (+1) = 0 ✓
yoneda-directed (-1) ⊗ cognitive-superposition (0) ⊗ curiosity-driven (+1) = 0 ✓
As Validator (-1), yoneda-directed verifies:
For any Segal type A, element a : A, and type family P,
if we have base : P(a, id_a), then for all x : A and f : hom(a, x),
we get P(x, f).
This is analogous to:
"To prove ∀ paths from a, prove for the reflexivity path"
This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:
category-theory: 139 citations in bib.duckdbThis skill maps to Cat# = Comod(P) as a bicomodule in the equipment structure:
Trit: 0 (ERGODIC)
Home: Presheaves
Poly Op: ⊗
Kan Role: Adj
Color: #26D826
The skill participates in triads satisfying:
(-1) + (0) + (+1) ≡ 0 (mod 3)
This ensures compositional coherence in the Cat# equipment structure.