open import Cat.Displayed.Base
open import Cat.Prelude

module Cat.Displayed.Instances.Elements {o ℓ s} (B : Precategory o ℓ)
  (P : Functor (B ^op) (Sets s)) where

open import Cat.Reasoning B
open Functor

private
  module P = Functor P

The Displayed Category of Elements🔗

It is useful to view the category of elements of a presheaf P as a displayed category. Instead of considering pairs of objects $X$ and sections $s$, we instead think of the set of sections as displayed over $X$. The story is similar for morphisms; instead of taking pairs of morphisms $f$ and fragments of data that $P(f)(x) = y$, we place those fragments over the morphism $f$.

In a sense, this is the more natural presentation of the category of elements, as we obtain the more traditional definition by taking the total category of ∫.

∫ : Displayed B s s
Displayed.Ob[ ∫ ] X = ∣ P.₀ X ∣
Displayed.Hom[ ∫ ] f P[X] P[Y] = P.₁ f P[Y] ≡ P[X]
Displayed.Hom[ ∫ ]-set _ _ _ = hlevel!
∫ .Displayed.id′ = happly P.F-id _
∫ .Displayed._∘′_ {x = x} {y = y} {z = z} {f = f} {g = g} p q = pf where abstract
  pf : P.₁ (f ∘ g) z ≡ x
  pf =
    P.₁ (f ∘ g) z   ≡⟨ happly (P.F-∘ g f) z ⟩≡
    P.₁ g (P.₁ f z) ≡⟨ ap (P.₁ g) p ⟩≡
    P.₁ g y         ≡⟨ q ⟩≡
    x               ∎
∫ .Displayed.idr′ _ = to-pathp (P.₀ _ .is-tr _ _ _ _)
∫ .Displayed.idl′ _ = to-pathp (P.₀ _ .is-tr _ _ _ _)
∫ .Displayed.assoc′ _ _ _ = to-pathp (P.₀ _ .is-tr _ _ _ _)