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

The category of elements🔗

The (contravariant¹) category of elements of a presheaf $P : \mathcal{C}^{\mathrm{op}} \to \mathbf{Sets}$ is a means of unpacking the data of the presheaf. Its objects are pairs of an object $x$, and a section $s : P x$.

record Element : Type (o ⊔ s) where
  constructor elem
  field
    ob : Ob
    section : ∣ P.₀ ob ∣

open Element

We can think of this as taking an eraser to the data of $P$. If $P(x) = \{x_0, x_1, x_2\}$, then the category of elements of $P$ will have three objects in place of the one: $(x, x_0)$, $(x, x_1)$, and $(x, x_2)$. We’ve erased all the boundaries of each of the sets associated with $P$, and are left with a big soup of points.

We do something similar for morphisms, and turn functions $P(f) : P(Y) \to P(X)$ into a huge collection of morphisms between points. We do this by defining a morphism $(x, x_0) \to (y, y_0)$ to be a morphism $f : X \to Y$ in $\mathcal{C}$, as well as a proof that $P(f)(y_0) = x_0$ This too can be seen as erasing boundaries, just this time with the data associated with a function. Instead of having a bunch of data bundled together that describes the action of $P(f)$ on each point of $P(Y)$, we have a bunch of tiny bits of data that only describe the action of $P(f)$ on a single point.

record Element-hom (x y : Element) : Type (ℓ ⊔ s) where
  constructor elem-hom
  no-eta-equality
  field
    hom : Hom (x .ob) (y .ob)
    commute : P.₁ hom (y .section) ≡ x .section

open Element-hom

As per usual, we need to prove some helper lemmas that describe the path space of Element-hom

Element-hom-path : {x y : Element} {f g : Element-hom x y} → f .hom ≡ g .hom → f ≡ g
Element-hom-path p i .hom = p i
Element-hom-path {x = x} {y = y} {f = f} {g = g} p i .commute =
  is-prop→pathp (λ j → P.₀ (x .ob) .is-tr (P.₁ (p j) (y .section)) (x .section))
    (f .commute)
    (g .commute) i

One interesting fact is that morphisms $f : X \to Y$ in $C$ induce morphisms in the category of elements for each $py : P(y)$.

induce : ∀ {x y} (f : Hom x y) (py : ∣ P.₀ y ∣)
       → Element-hom (elem x (P.₁ f py)) (elem y py)
induce f _ = elem-hom f refl

Using all this machinery, we can now define the category of elements of $P$!

∫ : Precategory (o ⊔ s) (ℓ ⊔ s)
∫ .Precategory.Ob = Element
∫ .Precategory.Hom = Element-hom
∫ .Precategory.Hom-set = Element-hom-is-set
∫ .Precategory.id {x = x} = elem-hom id  λ i → P.F-id i (x .section)
∫ .Precategory._∘_ {x = x} {y = y} {z = z} f g = elem-hom (f .hom ∘ g .hom) comm
  where
    abstract
      comm : P.₁ (f .hom ∘ g .hom) (z .section) ≡ x .section
      comm =
        P.₁ (f .hom ∘ g .hom) (z .section)       ≡⟨ happly (P.F-∘ (g .hom) (f .hom)) (z .section) ⟩
        P.₁ (g .hom) (P.₁ (f .hom) (z .section)) ≡⟨ ap (P.F₁ (g .hom)) (f .commute)  ⟩
        P.₁ (g .hom) (y .section)                ≡⟨ g .commute ⟩
        x .section ∎
∫ .Precategory.idr f = Element-hom-path (idr (f .hom))
∫ .Precategory.idl f = Element-hom-path (idl (f .hom))
∫ .Precategory.assoc f g h = Element-hom-path (assoc (f .hom) (g .hom) (h .hom))

Projection🔗

The category of elements is equipped with a canonical projection functor $\pi_p : \int P \to C$, which just forgets all of the points and morphism actions.

πₚ : Functor ∫ C
πₚ .F₀ x = x .ob
πₚ .F₁ f = f .hom
πₚ .F-id = refl
πₚ .F-∘ f g = refl

This functor makes it clear that we ought to think of the category of elements as something defined over $\mathcal{C}$. For instance, if we look at the fibre over each $X : \mathcal{C}$, we get back the set $P(X)$!