covariant-fibrations

npx skills add majiayu000/claude-skill-registry --skill covariant-fibrations --agent claude-code

Same command for any agent — swap --agent for codex, cursor, copilot.

# Covariant Fibrations Skill: Directed Transport **Status**: ✅ Production Ready **Trit**: -1 (MINUS - validator/constraint) **Color**: #2626D8 (Blue) **Principle**: Type families respect directed morphisms **Frame**: Covariant transport along 2-arrows --- ## Overview **Covariant Fibrations** are type families B : A → U where transport goes *with* the direction of morphisms. In directed type theory, this ensures type families correctly propagate along the directed interval 𝟚. 1. **Directed interval 𝟚**: Type with 0 → 1 (not invertible) 2. **Covariant transport**: f : a → a' induces B(a) → B(a') 3. **Segal condition**: Composition witness for ∞-categories 4. **Fibration condition**: Lift existence (not uniqueness) ## Core Formula ``` For P : A → U covariant fibration: transport_P : (f : Hom_A(a, a')) → P(a) → P(a') Covariance: transport respects composition transport_{g∘f} = transport_g ∘ transport_f ``` ```haskell -- Directed type theory (Narya-style) covariant_fibration : (A : Type) → (P : A → Type) → Type covariant_fibration A P = (a a' : A) → (f : Hom A a a') → P a → P a' ``` ## Key Concepts ### 1. Covariant Transport ```agda -- Transport along directed morphisms cov-transport

Validate covariance conditions
just covariant-check fibration.rzk
Compute cocartesian lifts
just cocartesian-lift base-morphism.rzk
Generate transport terms
just cov-transport source target