open import Cat.Diagram.Product.Solver
open import Cat.Internal.Opposite
open import Cat.Diagram.Pullback
open import Cat.Diagram.Terminal
open import Cat.Diagram.Product
open import Cat.Prelude

import Cat.Internal.Reasoning
import Cat.Internal.Base
import Cat.Reasoning

module Cat.Internal.Functor.Outer
  {o ℓ} {C : Precategory o ℓ}
  where

open import Cat.Reasoning C
open import Cat.Internal.Base C
open Internal-hom

Outer functors🔗

The category $\mathbf{Sets}$ is not strict, so it is not internal to any category of sets, even setting aside size issues. However, $\mathbf{Sets}$ still plays an important role in (strict) category theory, by passing to the copresheaf categories $\mathcal{C} \to \mathbf{Sets}$.

We wish to relativize this to an arbitrary base category $\mathcal{C}$, not just $\mathbf{Sets}$. Specifically, if $\mathbb{C}$ is a category internal to $\mathcal{C}$, we want to define “functor from $\mathbb{C} \to \mathcal{C}$” — which we call an outer functor¹. We will employ the fibration/family correspondence: a family of sets $B \to \mathbf{Sets}$ is the same as a function $E \to B$ — and we know how to relativise functions!

module _ (ℂ : Internal-cat) where
  open Cat.Internal.Reasoning ℂ

  record Outer-functor : Type (o ⊔ ℓ) where
    no-eta-equality

The first piece of data we require is some object $P : \mathcal{C}$ that will act like the disjoint union of the image of our functor. If the base category were $\mathbf{Sets}$, this would be the type $\Sigma_(x : \mathbb{C}) P(x)$. The next piece of data, $P_0$, corresponds to the first projection morphism: it assigns (generalised) objects of $P$ to objects of $\mathbb{C}$.

    field
      ∫P : Ob
      P₀ : ∀ {Γ} → Hom Γ ∫P → Hom Γ C₀

The next data captures how $\mathbb{C}$’s morphisms act on the fibres. Since the family-fibration correspondence takes dependency to sectioning, we require an assigment sending maps $f : P_0(x) \to y$ to maps $P_1(f) : \Gamma \to P$, which satisfy $y = P_0(P_1(f))$.

      P₁ : ∀ {Γ} (px : Hom Γ ∫P) {y : Hom Γ C₀} → Homi (P₀ px) y → Hom Γ ∫P
      P₁-tgt : ∀ {Γ} (px : Hom Γ ∫P) {y : Hom Γ C₀}
             → (f : Homi (P₀ px) y) → y ≡ P₀ (P₁ px f)

Next, we have the functoriality conditions, which are straightforward (modulo indexing).

      P-id : ∀ {Γ} (px : Hom Γ ∫P) → P₁ px (idi (P₀ px)) ≡ px
      P-∘ : ∀ {Γ} (px : Hom Γ ∫P) {y z : Hom Γ C₀} (f : Homi y z) (g : Homi (P₀ px) y)
          → P₁ px (f ∘i g) ≡ P₁ (P₁ px g) (adjusti (P₁-tgt px g) refl f)

Finally, the naturality conditions that allow us to work using generalized objects.

      P₀-nat : ∀ {Γ Δ} → (px : Hom Δ ∫P) → (σ : Hom Γ Δ) → P₀ px ∘ σ ≡ P₀ (px ∘ σ)
      P₁-nat : ∀ {Γ Δ} (px : Hom Δ ∫P) {y : Hom Δ C₀} (f : Homi (P₀ px) y) (σ : Hom Γ Δ)
        → P₁ px f ∘ σ ≡ P₁ (px ∘ σ) (adjusti (P₀-nat px σ) refl (f [ σ ]))

open Outer-functor

Another perspective on outer functors is that they are the internal discrete opfibrations over $\mathbb{C}$. The object $P$ is the total category, the map $P_0$ is the fibration itself, and $P_1$ captures the lifting into opcartesian maps.

We can obtain internal discrete fibrations by looking at outer functors from the internal opposite category of $\mathbb{C}$.

Internal Hom functors🔗

The canonical example of an outer functor is the internal analog of the hom functor $\mathcal{C}(X,-)$. We require the following data: $\mathcal{C}$ is finitely complete, $\mathbb{C}$ is a category internal to $\mathcal{C}$, and $x : \mathcal{C}(\top, \mathbb{C}_0)$ is a global object of $\mathbb{C}$ — an object of $\mathbb{C}$ in the empty context.

module _ (pb : has-pullbacks C) (term : Terminal C) (ℂ : Internal-cat) where
  open Cat.Internal.Reasoning ℂ
  open Pullbacks C pb
  open Terminal term

  Internal-hom-from : (x : Hom top C₀) → Outer-functor ℂ
  Internal-hom-from x = outf where

Recall that defining an outer functor on $\mathbb{C}$ requires choosing an object $P : \mathcal{C}$ to play the role of total space. For the outer functor corresponding to a Hom-functor, this ought to be the object of morphisms with domain $x$. We can encode this internally with the following pullback:

The projection from the total space to $C_0$ takes a generalized element of $C_x$ to its codomain. The lifting morphism $P_1$ follows from the universal property of the pullback.

    open Pullback (pb src x) renaming (apex to Cₓ)

    outf : Outer-functor ℂ
    outf .∫P = Cₓ
    outf .P₀ fₓ = tgt ∘ p₁ ∘ fₓ
    outf .P₁ fₓ {y = y} g = universal coh
      module hom-from-action where abstract
        coh : src ∘ (g ∘i homi (p₁ ∘ fₓ)) .ihom ≡ x ∘ !
        coh =
          src ∘ (g ∘i homi (p₁ ∘ fₓ)) .ihom ≡⟨ (g ∘i homi (p₁ ∘ fₓ)) .has-src ⟩≡
          src ∘ p₁ ∘ fₓ ≡⟨ pulll square ⟩≡
          (x ∘ p₂) ∘ fₓ ≡⟨ pullr (sym (!-unique _)) ⟩≡
          x ∘ ! ∎

    outf .P₁-tgt fₓ {y = y} g = tgt-coh where abstract
      tgt-coh : y ≡ tgt ∘ p₁ ∘ universal (hom-from-action.coh fₓ g)
      tgt-coh =
        y                                 ≡˘⟨ (g ∘i homi (p₁ ∘ fₓ)) .has-tgt ⟩≡˘
        tgt ∘ (g ∘i homi (p₁ ∘ fₓ)) .ihom ≡˘⟨ ap (tgt ∘_) p₁∘universal ⟩≡˘
        tgt ∘ p₁ ∘ universal _            ∎
    outf .P-id fₓ =
      sym $ unique (sym (ap ihom (idli _))) (sym (!-unique _))
    outf .P-∘ fₓ g h = unique
      (p₁∘universal
      ∙ ap ihom (sym $ associ _ _ _)
      ∙ ∘i-ihom
          (sym (ap (src ∘_) p₁∘universal ∙ (h ∘i homi (p₁ ∘ fₓ)) .has-src))
          (sym (ap (tgt ∘_) p₁∘universal ∙ (h ∘i homi (p₁ ∘ fₓ)) .has-tgt))
          refl refl (sym p₁∘universal))
      p₂∘universal
    outf .P₀-nat fₓ σ = sym (assoc _ _ _) ∙ ap (tgt ∘_) (sym (assoc _ _ _))
    outf .P₁-nat fₓ g σ = unique
      (pulll p₁∘universal
        ∙ ap ihom (∘i-nat g (homi (p₁ ∘ fₓ)) σ)
        ∙ ∘i-ihom
            (sym (assoc _ _ _) ∙ ap (src ∘_) (sym (assoc _ _ _)))
            (sym (assoc _ _ _) ∙ ap (tgt ∘_) (sym (assoc _ _ _)))
            refl refl (sym (assoc _ _ _)))
      (sym (!-unique _))

The contravariant internal $\mathbf{Hom}$ functor is defined by duality, which carries “pullback along $\operatorname{src}$” to “pullback along $\operatorname{tgt}$.”. This outer functor plays the role of the Yoneda embedding.

  Internal-hom-into : (x : Hom top C₀) → Outer-functor (ℂ ^opi)
  Internal-hom-into x = outf where
    open Pullback (pb tgt x) renaming (apex to Cₓ)

    outf : Outer-functor (ℂ ^opi)
    outf .∫P = Cₓ
    outf .P₀ fₓ = src ∘ p₁ ∘ fₓ
    outf .P₁ fₓ g = universal coh
      module hom-into-action where abstract
        coh : tgt ∘ (homi (p₁ ∘ fₓ) ∘i op-ihom g) .ihom ≡ x ∘ !
        coh =
          tgt ∘ (homi (p₁ ∘ fₓ) ∘i op-ihom g) .ihom ≡⟨ (homi (p₁ ∘ fₓ) ∘i op-ihom g) .has-tgt ⟩≡
          tgt ∘ p₁ ∘ fₓ ≡⟨ pulll square ⟩≡
          (x ∘ p₂) ∘ fₓ ≡⟨ pullr (sym (!-unique _)) ⟩≡
          x ∘ ! ∎
    outf .P₁-tgt fₓ {y} g = src-coh where abstract
      src-coh : y ≡ src ∘ p₁ ∘ universal (hom-into-action.coh fₓ g)
      src-coh =
        sym (ap (src ∘_) p₁∘universal
        ∙ (homi (p₁ ∘ fₓ) ∘i op-ihom g) .has-src)
    outf .P-id fₓ =
      sym $ unique (sym (ap ihom (idri _))) (sym (!-unique _))
    outf .P-∘ fₓ g h =
      unique
        (p₁∘universal
         ∙ ap ihom (associ _ _ _)
         ∙ ∘i-ihom refl
             (sym (ap (src ∘_) p₁∘universal ∙ (homi (p₁ ∘ fₓ) ∘i op-ihom h) .has-src))
             (sym (ap (tgt ∘_) p₁∘universal ∙ (homi (p₁ ∘ fₓ) ∘i op-ihom h) .has-tgt))
             (sym p₁∘universal) refl)
        p₂∘universal
    outf .P₀-nat fₓ σ =
      sym (assoc _ _ _)
      ∙ ap (src ∘_) (sym (assoc _ _ _))
    outf .P₁-nat fₓ g σ =
      unique
        (pulll p₁∘universal
        ∙ ap ihom (∘i-nat _ _ _)
        ∙ ∘i-ihom refl
             (sym (assoc _ _ _) ∙ ap (src ∘_) (sym (assoc _ _ _)))
             (sym (assoc _ _ _) ∙ ap (tgt ∘_) (sym (assoc _ _ _)))
             (sym (assoc _ _ _)) refl)
        (sym (!-unique _))

Outer natural transformations🔗

If $\mathcal{C}$ is a category, $\mathbb{C}$ is internal to $\mathcal{C}$, and $P, Q$ are two outer functors on $\mathbb{C}$, we can define the outer natural transformations $P \to Q$: They are (mostly) given by transformations $\mathcal{C}(-, \int P)$ to $\mathcal{C}(-, \int Q)$, together with a bit of naturality data.

module _ {ℂ : Internal-cat} where
  open Internal-cat ℂ
  record _=>o_ (P Q : Outer-functor ℂ) : Type (o ⊔ ℓ) where
    no-eta-equality
    field
      ηo : ∀ {Γ} → Hom Γ (P .∫P) → Hom Γ (Q .∫P)

The first condition corresponds to indexing: Outer natural transformations map elements of $\int P$ over $x$ to elements of $\int Q$ over $x$.

      ηo-fib : ∀ {Γ} (px : Hom Γ (P .∫P)) → Q .P₀ (ηo px) ≡ P .P₀ px

Over this coherence, we can state the naturality condition. It should be familiar, since, modulo the aforementioned coherence, it says $\eta P_1(f) = Q_1(f\eta)$.

      is-naturalo : ∀ {Γ : Ob} (px : Hom Γ (P .∫P)) (y : Hom Γ C₀)
        → (f : Homi (P .P₀ px) y)
        → ηo (P .P₁ px f) ≡ Q .P₁ (ηo px) (adjusti (sym (ηo-fib px)) refl f)

The final naturality condition is stability under substitution, allowing us to work in the internal language of $\mathcal{C}$.

      ηo-nat : ∀ {Γ Δ} (px : Hom Δ (P .∫P)) (σ : Hom Γ Δ) → ηo px ∘ σ ≡ ηo (px ∘ σ)

  open _=>o_

  Outer-nat-path
    : ∀ {F G : Outer-functor ℂ} {α β : F =>o G}
    → (∀ {Γ} (px : Hom Γ (F .∫P)) → α .ηo px ≡ β .ηo px)
    → α ≡ β
  Outer-nat-path p i .ηo px = p px i
  Outer-nat-path {G = G} {α = α} {β = β} p i .ηo-fib px =
    is-prop→pathp (λ i → Hom-set _ _ (G .P₀ (p px i)) _)
      (α .ηo-fib px)
      (β .ηo-fib px) i
  Outer-nat-path {F = F} {G = G} {α = α} {β = β} p i .is-naturalo px y f j =
    is-set→squarep (λ i j → Hom-set _ _)
      (p (F .P₁ px f))
      (α .is-naturalo px y f)
      (β .is-naturalo px y f)
      (λ i → G .P₁ (p px i)
        (adjusti (sym (Outer-nat-path {α = α} {β = β} p i .ηo-fib px)) refl f))
      i j
  Outer-nat-path {α = α} {β = β} p i .ηo-nat px σ =
    is-prop→pathp (λ i → Hom-set _ _ (p px i ∘ σ) (p (px ∘ σ) i))
      (α .ηo-nat px σ)
      (β .ηo-nat px σ) i

  private unquoteDecl nat-eqv = declare-record-iso nat-eqv (quote _=>o_)

  Outer-nat-is-set
    : ∀ {F G : Outer-functor ℂ} → is-set (F =>o G)
  Outer-nat-is-set = Iso→is-hlevel  nat-eqv hlevel!

instance
  H-Level-Outer-nat
    : ∀ {ℂ : Internal-cat} {F G : Outer-functor ℂ} {n}
    → H-Level (F =>o G) ( + n)
  H-Level-Outer-nat = basic-instance  Outer-nat-is-set

Some outer functors🔗

One important outer functor is the constant outer functor on an object $X : \mathcal{C}$, which can be constructed if $\mathcal{C}$ has products. This is the internalized version of the chaotic bifibration.

module _ (prods : has-products C) {ℂ : Internal-cat} where
  open Binary-products C prods
  open Internal-cat ℂ
  open _=>o_

The total space of this functor is the product $X \times \mathbb{C}_0$, and the projection map is simply the second projection.

  ConstO : (X : Ob) → Outer-functor ℂ
  ConstO X .∫P = X ⊗₀ C₀
  ConstO X .P₀ f = π₂ ∘ f

Lifting is given by the universal map into the product.

  ConstO X .P₁ px {y} f = ⟨ π₁ ∘ px , y ⟩
  ConstO X .P₁-tgt px {y = y} f = sym π₂∘⟨⟩

For once, the naturality constraints are not egregious: In fact, since they are all facts about products, they can all be solved automatically.

  ConstO X .P-id px = products! C prods
  ConstO X .P-∘ px f g = products! C prods
  ConstO X .P₀-nat px σ = sym (assoc _ _ _)
  ConstO X .P₁-nat px f σ = products! C prods

If $x \to y$ is a map in $\mathcal{C}$, then it defines an outer natural transformation between the associated constant outer functors. Here too we can apply automation to satisfy the coherence constraints.

  const-nato : ∀ {X Y : Ob} → Hom X Y → ConstO X =>o ConstO Y
  const-nato f .ηo g = ⟨ f ∘ π₁ ∘ g , π₂ ∘ g ⟩
  const-nato f .ηo-fib px          = products! C prods
  const-nato f .is-naturalo px y g = products! C prods
  const-nato f .ηo-nat px σ        = products! C prods