module Cat.Displayed.Cartesian.Right {o ℓ o′ ℓ′} {ℬ : Precategory o ℓ} (ℰ : Displayed ℬ o′ ℓ′) where open Cat.Reasoning ℬ open Displayed ℰ open Cat.Displayed.Cartesian ℰ open Cat.Displayed.Morphism ℰ open Cat.Displayed.Reasoning ℰ

# Right Fibrations🔗

A cartesian
fibration
$\mathcal{E}$
is said to be a **right fibration** if every morphism in
$\mathcal{E}$
is cartesian.

record Right-fibration : Type (o ⊔ ℓ ⊔ o′ ⊔ ℓ′) where no-eta-equality field is-fibration : Cartesian-fibration cartesian : ∀ {x y} {f : Hom x y} → ∀ {x′ y′} (f′ : Hom[ f ] x′ y′) → is-cartesian f f′ open Cartesian-fibration is-fibration public

One notable fact is every vertical morphism of a right fibrations is invertible. In other words, if $\mathcal{E}$ is a right fibration, then all of its fibre categories are groupoids. This follows directly from the fact that vertical cartesian maps are invertible.

right-fibration→vertical-invertible : Right-fibration → ∀ {x} {x′ x″ : Ob[ x ]} → (f′ : Hom[ id ] x′ x″) → is-invertible↓ f′ right-fibration→vertical-invertible rfib f′ = vertical+cartesian→invertible (Right-fibration.cartesian rfib f′)

More notably, this is an exact characterization of categories fibred in groupoids! If $\mathcal{E}$ is a cartesian fibration where all vertical morphisms are invertible, then it must be a right fibration.

vertical-invertible+fibration→right-fibration : Cartesian-fibration → (∀ {x} {x′ x″ : Ob[ x ]} → (f′ : Hom[ id ] x′ x″) → is-invertible↓ f′) → Right-fibration vertical-invertible+fibration→right-fibration fib vert-inv .Right-fibration.is-fibration = fib vertical-invertible+fibration→right-fibration fib vert-inv .Right-fibration.cartesian {x = x} {f = f} {x′ = x′} {y′ = y′} f′ = f-cart where open Cartesian-fibration fib open Cartesian-lift renaming (x′ to x-lift)

To see this, recall that cartesian morphisms are stable under vertical retractions. The cartesian lift $f^{*}$ of $f$ is obviously cartesian, so it suffices to show that there is a vertical retraction $x^{*} \to x'$. To construct this retraction, we shall factorize $f'$ over $f \cdot id$; which yields a vertical morphism $i^{*} : x' \to x^{*}$. By our hypotheses, $i^{*}$ is invertible, and thus is a retraction. What remains to be shown is that the inverse to $i^{*}$ factors $f'$ and $f^{*}$; this follows from the factorisation of $f'$ and the fact that $i^{*}$ is invertible.

x* : Ob[ x ] x* = has-lift f y′ .x-lift f* : Hom[ f ] x* y′ f* = has-lift f y′ .lifting module f* = is-cartesian (has-lift f y′ .cartesian) i* : Hom[ id ] x′ x* i* = f*.universal′ (idr f) f′ module i*-inv = is-invertible[_] (vert-inv i*) i*⁻¹ : Hom[ id ] x* x′ i*⁻¹ = i*-inv.inv′ factors : f′ ∘′ i*⁻¹ ≡[ idr f ] f* factors = to-pathp⁻ $ f′ ∘′ i*⁻¹ ≡⟨ shiftr _ (pushl′ _ (symP $ f*.commutesp (idr f) f′) {q = ap (f ∘_) (sym (idl _))}) ⟩≡ hom[] (f* ∘′ i* ∘′ i*⁻¹) ≡⟨ weave _ (elimr (idl id)) _ (elimr′ _ i*-inv.invl′) ⟩≡ hom[] f* ∎ f-cart : is-cartesian f f′ f-cart = cartesian-vertical-retraction-stable (has-lift f y′ .cartesian) (inverses[]→from-has-section[] i*-inv.inverses′) factors

As a corollary, we get that all discrete fibrations are right fibrations. Intuitively, this is true, as sets are 0-groupoids.

discrete→right-fibration : Discrete-fibration ℰ → Right-fibration discrete→right-fibration dfib = vertical-invertible+fibration→right-fibration (discrete→cartesian ℰ dfib) (discrete→vertical-invertible ℰ dfib)

## Fibred Functors and Right Fibrations🔗

As every map in a right fibration is cartesian, every displayed functor into a right fibration is automatically fibred.

functor+right-fibration→fibred : ∀ {o₂ ℓ₂ o₂′ ℓ₂′} → {𝒟 : Precategory o₂ ℓ₂} → {ℱ : Displayed 𝒟 o₂′ ℓ₂′} → {F : Functor 𝒟 ℬ} → Right-fibration → (F′ : Displayed-functor ℱ ℰ F) → Fibred-functor ℱ ℰ F functor+right-fibration→fibred rfib F′ .Fibred-functor.disp = F′ functor+right-fibration→fibred rfib F′ .Fibred-functor.F-cartesian f′ _ = Right-fibration.cartesian rfib (F₁′ f′) where open Displayed-functor F′

Specifically, this implies that all displayed functors into a discrete fibrations are fibred. This means that we can drop the fibred condition when working with functors between discrete fibrations.

functor+discrete→fibred : ∀ {o₂ ℓ₂ o₂′ ℓ₂′} → {𝒟 : Precategory o₂ ℓ₂} → {ℱ : Displayed 𝒟 o₂′ ℓ₂′} → {F : Functor 𝒟 ℬ} → Discrete-fibration ℰ → (F′ : Displayed-functor ℱ ℰ F) → Fibred-functor ℱ ℰ F functor+discrete→fibred disc F′ = functor+right-fibration→fibred (discrete→right-fibration disc) F′