yoneda-directed

Directed Yoneda induction proves properties P(x, f) for all x and morphisms f in a Segal type by establishing a base case P(a, id_a), then transporting along directed paths. Provide a Segal type A, element a, predicate P over pairs (x, morphism a→x), and the base proof; the induction derives P for every x and f: hom A a x.

A Segal type is a type-theoretic model of directed ∞-categories satisfying composition coherence. Directed Yoneda requires Segal types because they formalize the homomorphism structure and path composition needed to transport proofs along directed morphisms from a base element.

Yes, Rzk provides executable notation for directed Yoneda. Write the dependent type (x:A) → hom A a x → U for your property P, supply base: P a (id a), and Rzk's directed-path-induction constructs a proof function returning P x f for all x and f in your Segal type.

You need a Segal type A (formalized in synthetic ∞-category notation), a distinguished element a: A, a dependent predicate P parameterized by target x and morphism hom A a x, and a base proof P(a, id_a). Familiarity with type theory and directed path induction strengthens application.

Directed Yoneda extends classical Yoneda to directed ∞-categories by replacing undirected natural isomorphisms with directed path induction along morphisms from a fixed source. It operates on Segal types with explicit hom-types and composition, enabling proofs about directed homotopy structure rather than set-level functors.

Directed Yoneda requires explicit construction of the predicate P and base case P(a, id_a) for each proof goal; it does not automate large-scale category reasoning. Scalability depends on your proof assistant's performance with dependent types and your ability to formalize coherence conditions for complex Segal types.