module Cat.Instances.Slice.Presheaf {o ℓ} {C : Precategory o ℓ} where

Slices of presheaf categories🔗

We prove that slices of a presheaf category are again presheaf categories. Specifically, for PP a presheaf, we have an isomorphism PSh(C)/P≅PSh(∫P)\mathrm{PSh}(\mathcal{C})/P \cong \mathrm{PSh}(\int P), where ∫\int denotes the category of elements of PP.

An object in the slice PSh(C)/P\mathrm{PSh}(\mathcal{C})/P consists of a functor QQ together with a natural transformation η:P→Q\eta : P \to Q. To transform this data into a functor ∫P→Sets\int P \to \mathbf{Sets}, observe that for each element (x,s)(x, s) in ∫P\int P, the fibre ηx∗(s)\eta_x^*(s) is a set. But why this choice in particular? Well, observe that ∫P\int P is essentially the total space of PP — so that what we’re doing here is proving an equivalence between fibrations and dependent functions! This is in line with the existence of object classifiers, and in the 1-categorical level, with slices of Sets.

In fact, since we have Sets=PSh(∗)\mathbf{Sets} = \mathrm{PSh}(*), that latter equivalence is a special case of the one constructed here — where in the calculation below, cc denotes the constant presheaf ∗↩S* \mapsto S. The category of elements of a presheaf ∗↩S* \mapsto S consists of pairs (x,e)(x, e) where x:∗x : *, of which there is only one choice, and e:Se : S.

Sets/S≅PSh(∗)/c(S)≅PSh(∫c(S))≅PSh(Disc(S)) \mathbf{Sets}/S \cong \mathrm{PSh}(*)/c(S) \cong \mathrm{PSh}(\textstyle\int c(S)) \cong \mathrm{PSh}(\mathrm{Disc}(S))

module _ {P : Functor (C ^op) (Sets Îș)} where
  private module P = Functor P

  slice-ob→presheaf
    : Ob (Slice Cat[ C ^op , Sets Îș ] P)
    → Functor (∫ C P ^op) (Sets Îș)
  slice-ob→presheaf sl .F₀ (elem x s) = el! (fibre (sl .map .η x) s)

  slice-ob→presheaf sl .F₁ eh (i , p) =
      sl .domain .F₁ (eh .hom) i
    , happly (sl .map .is-natural _ _ _) _ ·· ap (P.₁ _) p ·· eh .commute

Keeping with the theme, in the other direction, we take a total space rather than a family of fibres, with fibration being the first projection fst:

  presheaf→slice-ob : Functor (∫ C P ^op) (Sets Îș) → Ob (Slice Cat[ C ^op , Sets Îș ] P)
  presheaf→slice-ob y = obj where
    obj : /-Obj {C = Cat[ _ , _ ]} P
    obj .domain .F₀ c = el! (Σ[ sect ∈ ∣ P.₀ c ∣ ] ∣ y .F₀ (elem c sect) ∣)
    obj .domain .F₁ f (x , p) = P.₁ f x , y .F₁ (elem-hom f refl) p
    obj .map .η x = fst

Since the rest of the construction is routine calculation, we present it without comment.

  slice→total : Functor (Slice Cat[ C ^op , Sets Îș ] P) Cat[ (∫ C P) ^op , Sets Îș ]
  slice→total = func where
    func : Functor (Slice Cat[ C ^op , Sets Îș ] P) Cat[ (∫ C P) ^op , Sets Îș ]
    func .F₀ = slice-ob→presheaf
    func .F₁ {x} {y} h .η i arg =
      h .map .η (i .ob) (arg .fst) , h .commutes ηₚ _ $ₚ arg .fst ∙ arg .snd
    func .F₁ {x} {y} h .is-natural _ _ _ = funext λ i →
      ÎŁ-prop-path (λ _ → P.₀ _ .is-tr _ _) (happly (h .map .is-natural _ _ _) _)

    func .F-id    = Nat-path (λ x → funext λ y → ÎŁ-prop-path (λ _ → P.₀ _ .is-tr _ _) refl)
    func .F-∘ f g = Nat-path (λ x → funext λ y → ÎŁ-prop-path (λ _ → P.₀ _ .is-tr _ _) refl)

  slice→total-is-ff : is-fully-faithful slice→total
  slice→total-is-ff {x} {y} = is-iso→is-equiv (iso inv rinv linv) where
    inv : Hom Cat[ ∫ C P ^op , Sets _ ] _ _
        → Slice Cat[ C ^op , Sets _ ] P .Hom _ _
    inv nt .map .η i o = nt .η (elem _ (x .map .η i o)) (o , refl) .fst

    inv nt .map .is-natural _ _ f = funext λ z →
        ap (λ e → nt .η _ e .fst) (ÎŁ-prop-path (λ _ → P.₀ _ .is-tr _ _) refl)
      ∙ ap fst (happly (nt .is-natural _ _
          (elem-hom f (happly (sym (x .map .is-natural _ _ _)) _))) _)

    inv nt .commutes = ext λ z w →
      nt .η (elem _ (x .map .η _ _)) (w , refl) .snd

    rinv : is-right-inverse inv (F₁ slice→total)
    rinv nt = ext λ where
      o (z , p) → ÎŁ-prop-path (λ _ → P.₀ _ .is-tr _ _)
        (λ i → nt .η (elem (o .ob) (p i)) (z , (λ j → p (i ∧ j))) .fst)

    linv : is-left-inverse inv (F₁ slice→total)
    linv sh = trivial!

  open is-precat-iso
  slice→total-is-iso : is-precat-iso slice→total
  slice→total-is-iso .has-is-ff = slice→total-is-ff
  slice→total-is-iso .has-is-iso = is-iso→is-equiv isom where
    open is-iso
    isom : is-iso slice-ob→presheaf
    isom .inv = presheaf→slice-ob

Proving that the constructions presheaf→slice-ob and slice-ob→presheaf are inverses is mosly incredibly fiddly path algebra, so we omit the proof.

    isom .rinv x =
      Functor-path
        (λ i → n-ua (Fibre-equiv (λ a → ∣ x .F₀ (elem (i .ob) a) ∣) (i .section)))
        λ f → ua→ λ { ((a , b) , p) → path→ua-pathp _ (lemma x _ ∙ lemma' _ _ _) }
      where abstract
        lemma'
          : ∀ {o o'} {sect : ∣ P.₀ (o .ob) ∣}
              (f : Hom (∫ C P ^op) o o')
              (b : ∣ x .F₀ (elem (o .ob) sect) ∣)
              (p : sect ≡ o .section)
          → x .F₁ (elem-hom (f .hom) (ap (P.₁ (f .hom)) p ∙ f .commute)) b
          ≡ x .F₁ f (subst (λ e → ∣ x .F₀ (elem (o .ob) e) ∣) p b)
        lemma' {o = o} {o' = o'} f b p =
          J (λ _ p → ∀ f b → x .F₁ (elem-hom (f .hom) (ap (P.₁ (f .hom)) p ∙ f .commute)) b
                           ≡ x .F₁ f (subst (λ e → ∣ x .F₀ (elem (o .ob) e) ∣) p b))
            (λ f b → ap₂ (x .F₁) (Element-hom-path _ _ refl) (sym (transport-refl b)))
            p f b

    isom .linv x =
      /-Obj-path
        (Functor-path (λ x → n-ua (Total-equiv _ e⁻Âč))
          λ f → ua→ λ a → path→ua-pathp _ refl)
        (Nat-pathp _ _ (λ x → ua→ (λ x → sym (x .snd .snd))))

  -- downgrade to an equivalence for continuity/cocontinuity
  slice→total-is-equiv : is-equivalence slice→total
  slice→total-is-equiv = is-precat-iso→is-equivalence slice→total-is-iso

  total→slice : Functor Cat[ (∫ C P) ^op , Sets Îș ] (Slice Cat[ C ^op , Sets Îș ] P)
  total→slice = slice→total-is-equiv .is-equivalence.F⁻Âč