covariant-fibrations

Covariant transport takes a morphism f: a → a' and lifts it to a function P(a) → P(a') for a dependent type P. This preserves the direction of the morphism, ensuring consistent propagation through type families in directed type theory.

Cocartesian lifts are universal properties that guarantee every morphism in the base can be lifted to a special morphism in the total space of a fibration. They enable Segal-type compositions and are essential for directed transport to satisfy functorial composition requirements.

Covariant transport respects functorial composition: transport_{g∘f} equals transport_g ∘ transport_f. This ensures that composing two transports along chained morphisms equals transporting along their composition, maintaining type-theoretic coherence.

Yes. Use the covariant-check and cov-transport commands to validate that a dependent type family satisfies covariance with a simple transport example, confirming correct morphism propagation in Agda-like frameworks.

Apply directed transport over the directed interval 𝟚 when modeling path-dependent types in synthetic ∞-categories, ensuring values transform correctly along directed paths rather than undirected equivalences.

Covariant transport reverses the morphism direction: f: a → a' lifts to P(a) → P(a'). Contravariant transport inverts it: f: a → a' lifts to P(a') → P(a). Covariant is used when the family respects morphism direction.