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 Precategory 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 $s\text{,}$ we instead think of the set of sections as displayed over $X\text{.}$ The story is similar for morphisms; instead of taking pairs of morphisms $f$ and fragments of data that $P(f)(x)=y\text{,}$ we place those fragments over the morphism $f\text{.}$

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 ʻ 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} {z} {f} {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'   _     = prop!
∫ .Displayed.idl'   _     = prop!
∫ .Displayed.assoc' _ _ _ = prop!