covariant-fibrations

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

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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

-- 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

-- Transport along directed morphisms
cov-transport : {A : Type} {P : A → Type} 
              → is-covariant P
              → {a a' : A} → Hom A a a' → P a → P a'
cov-transport cov f pa = cov.transport f pa

-- Functoriality
cov-comp : cov-transport (g ∘ f) ≡ cov-transport g ∘ cov-transport f

2. Cocartesian Lifts

-- Cocartesian lift characterizes covariant fibrations
is-cocartesian : {E B : Type} (p : E → B) 
               → {e : E} {b' : B} → Hom B (p e) b' → Type
is-cocartesian p {e} {b'} f = 
  Σ (e' : E), Σ (f̃ : Hom E e e'), (p f̃ ≡ f) × is-initial(f̃)

3. Segal Types with Covariance

-- Covariant families over Segal types
covariant-segal : (A : Segal) → (P : A → Type) → Type
covariant-segal A P = 
  (x y z : A) → (f : Hom x y) → (g : Hom y z) →
  cov-transport (g ∘ f) ≡ cov-transport g ∘ cov-transport f