yoneda-directed

"The dependent Yoneda lemma is a directed analogue of path induction." — Emily Riehl & Michael Shulman

The Key Insight

| Standard HoTT | Directed HoTT | |---------------|---------------| | Path induction | Directed path induction | | Yoneda for ∞-groupoids | Dependent Yoneda for ∞-categories | | Types have identity | Segal types have composition |

Core Definition (Rzk)

#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

Chemputer Semantics

Chemical Interpretation:

• To prove a property of all reaction products from starting material A,
• It suffices to prove it for A itself (the identity "null reaction")
• Directed induction propagates the property along all reaction pathways

GF(3) Triad

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:

• Properties propagate correctly along morphisms
• Base case at identity suffices
• Induction principle is sound

Theorem

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"

References

1. Riehl, E. & Shulman, M. (2017). "A type theory for synthetic ∞-categories." §5.
2. Rzk sHoTT library

CT lattice atlas

Part of: para-mensch-commons (CT lattice family).