open import Cat.Displayed.Cartesian.Indexing
open import Cat.Displayed.Instances.Pullback
open import Cat.Displayed.Instances.Lifting
open import Cat.Displayed.Cartesian
open import Cat.Functor.Equivalence
open import Cat.Displayed.Functor
open import Cat.Instances.Functor
open import Cat.Displayed.Fibre
open import Cat.Displayed.Base
open import Cat.Prelude

import Cat.Displayed.Reasoning
import Cat.Reasoning

module Cat.Displayed.Instances.Diagrams
  {o ℓ o' ℓ'}
  {B : Precategory o ℓ}
  (E : Displayed B o' ℓ')
  where

open Cat.Reasoning B
open Displayed E
open Cat.Displayed.Reasoning E
open Functor
open _=>_

The Diagram Fibration🔗

The appropriate notion of structure for a displayed category $\mathcal{E} \mathrel{\htmlClass{liesover}{\hspace{1.366em}}} \mathcal{B}$ is fibrewise structure: structure found in each fibre category, preserved by the reindexing functors when $\mathcal{E}$ is an (op)fibration.

For instance, the correct notion of $\mathcal{J}$-shaped limit in $\mathcal{E}$ are the fibred limits: where every fibre category has limits of shape $\mathcal{J}$, and these are preserved by reindexing. Unfortunately, proof assistants: since none of the commutativity conditions for limits are definitional, this definition condemns the formaliser to transport hell.

Instead, we opt for a more abstract approach, which starts with a reorganization of what a fibrewise diagram in $\mathcal{E}$ is. Recall that the fibration of liftings describes liftings of functors $\mathcal{J} \to \mathcal{B}$ along the projection functor $\pi : \int \mathcal{E} \to \mathcal{B}$. If we focus on liftings along a constant functor $\Delta_{x} : \mathcal{J} \to \mathcal{B}$, we get a diagram in $\mathcal{E}$ that lies entirely in the fibre $\mathcal{E}_{x}$: a fibrewise diagram!

This allows us to concisely define the fibration of fibrewise diagrams as the base change of $\mathcal{E} \to \mathcal{B}$ along the functor $\mathcal{B} \to [\mathcal{J}, \mathcal{B}]$ that takes an object to the constant diagram on that object.

Diagrams
 : ∀ {oj ℓj} (J : Precategory oj ℓj)
 → Displayed B _ _
Diagrams J = Change-of-base ConstD (Liftings E J)

When $\mathcal{E}$ is a fibration, then so is the fibration of diagrams.

module _ {oj ℓj} (J : Precategory oj ℓj) where
  private module J = Precategory J
  open Lifting
  open _=[_]=>l_

  Diagram-fibration : Cartesian-fibration E → Cartesian-fibration (Diagrams J)
  Diagram-fibration fib =
    Change-of-base-fibration ConstD (Liftings E _)
      (Liftings-fibration E _ fib)

The constant fibrewise diagram functor🔗

Crucially, we have a “constant fibrewise diagram functor” that takes an object $x' : E_{x}$ to the constant diagram. However, defining this functor will require a small bit of machinery.

To begin, we characterize liftings of the constant functor, and natural transformations between them.

  ConstL : ∀ {x} → Ob[ x ] → Lifting {J = J} E (Const x)
  ConstL x' .F₀′ _ = x'
  ConstL x' .F₁′ _ = id′
  ConstL x' .F-id′ = refl
  ConstL x' .F-∘′ _ _ = symP (idr′ _)

  const-ntl
    : ∀ {x y x' y'} {f : Hom x y} → Hom[ f ] x' y'
    → (ConstL x') =[ const-nt f ]=>l (ConstL y')
  const-ntl f' .η′ _ = f'
  const-ntl f' .is-natural′ _ _ _ =
    idr′ _ ∙[] symP (idl′ _)

We also have a vertical functor from $\mathcal{E}$ to the fibration of diagrams of shape $\mathcal{J}$, which takes an $x'$ to the constant diagram.

  ConstFibD : Vertical-functor E (Diagrams J)
  ConstFibD .Vertical-functor.F₀′ = ConstL
  ConstFibD .Vertical-functor.F₁′ = const-ntl
  ConstFibD .Vertical-functor.F-id′ =
    Nat-lift-pathp (λ x → sym (transport-refl _))
  ConstFibD .Vertical-functor.F-∘′ =
    Nat-lift-pathp (λ x → sym (transport-refl _))

Next, we note that liftings of the constant functor correspond with diagrams in fibre categories.

  ConstL→Diagram
    : ∀ {x} → Lifting {J = J} E (Const x) → Functor J (Fibre E x)
  Diagram→ConstL
    : ∀ {x} → Functor J (Fibre E x) → Lifting {J = J} E (Const x)

  ConstL→Diagram F' .F₀ = F' .F₀′
  ConstL→Diagram F' .F₁ = F' .F₁′
  ConstL→Diagram F' .F-id = F' .F-id′
  ConstL→Diagram F' .F-∘ f g =
    from-pathp⁻ $ cast[] {q = sym (idl _)} (F' .F-∘′ f g)

  Diagram→ConstL F .F₀′ = F .F₀
  Diagram→ConstL F .F₁′ = F .F₁
  Diagram→ConstL F .F-id′ = F .F-id
  Diagram→ConstL F .F-∘′ f g =
    cast[] {p = sym (idl _)} $ to-pathp⁻ (F .F-∘ f g)

Furthermore, natural transformations between diagrams in a fibre of $E$ correspond to natural transformations between liftings of a constant functor.

  ConstL-natl→Diagram-nat
    : ∀ {x} {F G : Functor J (Fibre E x)}
    → Diagram→ConstL F =[ const-nt id ]=>l Diagram→ConstL G
    → F => G

  Diagram-nat→ConstL-natl
    : ∀ {x} {F G : Functor J (Fibre E x)}
    → F => G
    → Diagram→ConstL F =[ const-nt id ]=>l Diagram→ConstL G

  ConstL-natl→Diagram-nat α' .η = α' .η′
  ConstL-natl→Diagram-nat α' .is-natural x y f =
    ap hom[] (cast[] $ α' .is-natural′ x y f)

  Diagram-nat→ConstL-natl α .η′ = α .η
  Diagram-nat→ConstL-natl {F = F} {G = G} α .is-natural′ x y f =
    cast[] $
      to-pathp (α .is-natural x y f)
      ∙[] symP (transport-filler (λ i → Hom[ idl id i ] _ _) (F₁ G f ∘′ α .η x))

Fibre Categories🔗

The fibre category of the fibration of diagrams are equivalent to functor categories $[\mathcal{J}, \mathcal{E}_x]$.

  Fibrewise-diagram : ∀ {x} → Functor Cat[ J , Fibre E x ] (Fibre (Diagrams J) x)
  Fibrewise-diagram .F₀ = Diagram→ConstL
  Fibrewise-diagram .F₁ = Diagram-nat→ConstL-natl
  Fibrewise-diagram .F-id = Nat-lift-pathp λ _ → sym Regularity.reduce!
  Fibrewise-diagram .F-∘ _ _ = Nat-lift-pathp λ _ → sym Regularity.reduce!

Again, this isomorphism is almost definitional.

  Fibrewise-diagram-is-iso : ∀ {x} → is-precat-iso (Fibrewise-diagram {x})
  Fibrewise-diagram-is-iso .is-precat-iso.has-is-ff =
    is-iso→is-equiv $ iso
      (ConstL-natl→Diagram-nat)
      (λ α' → Nat-lift-pathp (λ _ → refl))
      (λ α → Nat-path (λ _ → refl))
  Fibrewise-diagram-is-iso .is-precat-iso.has-is-iso =
    is-iso→is-equiv $ iso
      ConstL→Diagram
      (λ F' → Lifting-pathp E refl (λ _ → refl) (λ _ → refl))
      (λ F → Functor-path (λ _ → refl) (λ _ → refl))