open import Cat.Instances.Functor
open import Cat.Functor.Adjoint
open import Cat.Functor.Base
open import Cat.Univalent
open import Cat.Prelude

import Cat.Functor.Reasoning as Fr
import Cat.Reasoning

module Cat.Functor.Equivalence where

Equivalences🔗

A functor $F : \mathcal{C} \to \mathcal{D}$ is an equivalence of categories when it has a right adjoint $G : \mathcal{D} \to \mathcal{D}$, with the unit and counit natural transformations being natural isomorphisms. This immediately implies that our adjoint pair $F \dashv G$ extends to an adjoint triple $F \dashv G \dashv F$.

record is-equivalence (F : Functor C D) : Type (adj-level C D) where
  private
    module C = Cat.Reasoning C
    module D = Cat.Reasoning D
    module [C,C] = Cat.Reasoning Cat[ C , C ]
    module [D,D] = Cat.Reasoning Cat[ D , D ]

  field
    F⁻¹      : Functor D C
    F⊣F⁻¹    : F ⊣ F⁻¹

  open _⊣_ F⊣F⁻¹ public

  field
    unit-iso   : ∀ x → C.is-invertible (unit.η x)
    counit-iso : ∀ x → D.is-invertible (counit.ε x)

The first thing we note is that having a natural family of invertible morphisms gives isomorphisms in the respective functor categories:

  F∘F⁻¹≅Id : (F F∘ F⁻¹) [D,D].≅ Id
  F∘F⁻¹≅Id =
    [D,D].invertible→iso counit
      (componentwise-invertible→invertible _ counit-iso)

  Id≅F⁻¹∘F : Id [C,C].≅ (F⁻¹ F∘ F)
  Id≅F⁻¹∘F =
    [C,C].invertible→iso unit
      (componentwise-invertible→invertible _ unit-iso)

  unit⁻¹ = [C,C]._≅_.from Id≅F⁻¹∘F
  counit⁻¹ = [D,D]._≅_.from F∘F⁻¹≅Id

We chose, for definiteness, the above definition of equivalence of categories, since it provides convenient access to the most useful data: The induced natural isomorphisms, the adjunction unit/counit, and the triangle identities. It is a lot of data to come up with by hand, though, so we provide some alternatives:

Fully faithful, essentially surjective🔗

Any fully faithful and (split!) essentially surjective functor determines an equivalence of precategories. Recall that “split essentially surjective” means we have some determined procedure for picking out an essential fibre over any object $d : \mathcal{D}$: an object $F^*(d) : \mathcal{C}$ together with a specified isomorphism $F^*(d) \cong d$.

module _ {F : Functor C D} (ff : is-fully-faithful F) (eso : is-split-eso F) where
  import Cat.Reasoning C as C
  import Cat.Reasoning D as D
  private module di = D._≅_

  private
    ff⁻¹ : ∀ {x y} → D.Hom (F .F₀ x) (F .F₀ y) → C.Hom _ _
    ff⁻¹ = equiv→inverse ff
    module ff {x} {y} = Equiv (_ , ff {x} {y})

It remains to show that, when $F$ is fully faithful, this assignment of essential fibres extends to a functor $\mathcal{D} \to \mathcal{C}$. For the object part, we send $x$ to the specified preimage. For the morphism part, the splitting gives us isomorphisms $F^*(x) \cong x$ and $F^*(y) \cong y$, so that we may form the composite $F^*(x) \to x \to y \to F^*(y)$; Fullness then completes the construction.

  ff+split-eso→inverse : Functor D C
  ff+split-eso→inverse .F₀ x         = eso x .fst
  ff+split-eso→inverse .F₁ {x} {y} f =
    ff⁻¹ (f*y-iso .D._≅_.from D.∘ f D.∘ f*x-iso .D._≅_.to)
    where
      open Σ (eso x) renaming (fst to f*x ; snd to f*x-iso)
      open Σ (eso y) renaming (fst to f*y ; snd to f*y-iso)

  ff+split-eso→inverse .F-id {x} =
    ff⁻¹ (f*x-iso .di.from D.∘ ⌜ D.id D.∘ f*x-iso .di.to ⌝) ≡⟨ ap! (D.idl _) ⟩≡
    ff⁻¹ (f*x-iso .di.from D.∘ f*x-iso .di.to)              ≡⟨ ap ff⁻¹ (f*x-iso .di.invr) ⟩≡
    ff⁻¹ D.id                                               ≡˘⟨ ap ff⁻¹ (F-id F) ⟩≡˘
    ff⁻¹ (F₁ F C.id)                                        ≡⟨ ff.η _ ⟩≡
    C.id                                                    ∎
    where open Σ (eso x) renaming (fst to f*x ; snd to f*x-iso)

  ff+split-eso→inverse .F-∘ {x} {y} {z} f g =
    fully-faithful→faithful {F = F} ff (
      F₁ F (ff⁻¹ (ffz D.∘ (f D.∘ g) D.∘ ftx))      ≡⟨ ff.ε _ ⟩≡
      ffz D.∘ (f D.∘ g) D.∘ ftx                    ≡⟨ cat! D ⟩≡
      ffz D.∘ f D.∘ ⌜ D.id ⌝ D.∘ g D.∘ ftx         ≡˘⟨ ap¡ (f*y-iso .di.invl) ⟩≡˘
      ffz D.∘ f D.∘ (fty D.∘ ffy) D.∘ g D.∘ ftx    ≡⟨ cat! D ⟩≡
      (ffz D.∘ f D.∘ fty) D.∘ (ffy D.∘ g D.∘ ftx)  ≡˘⟨ ap₂ D._∘_ (ff.ε _) (ff.ε _) ⟩≡˘
      F₁ F (ff⁻¹ _) D.∘ F₁ F (ff⁻¹ _)              ≡˘⟨ F-∘ F _ _ ⟩≡˘
      F₁ F (ff⁻¹ _ C.∘ ff⁻¹ _)                     ∎
    )
    where
      open Σ (eso x) renaming (fst to f*x ; snd to f*x-iso)
      open Σ (eso y) renaming (fst to f*y ; snd to f*y-iso)
      open Σ (eso z) renaming (fst to f*z ; snd to f*z-iso)

      ffz = f*z-iso .di.from
      ftz = f*z-iso .di.to
      ffy = f*y-iso .di.from
      fty = f*y-iso .di.to
      ffx = f*x-iso .di.from
      ftx = f*x-iso .di.to

We will, for brevity, refer to the functor we’ve just built as $G$, rather than its “proper name” ff+split-eso→inverse. Hercules now only has 11 labours to go: We must construct unit and counit natural transformations, prove that they satisfy the triangle identities, and prove that the unit/counit we define are componentwise invertible. I’ll keep the proofs of naturality in <details> tags since.. they’re rough.

  private
    G = ff+split-eso→inverse

For the unit, we have an object $x : \mathcal{C}$ and we’re asked to provide a morphism $x \to F^*F(x)$ — where, recall, the notation $F^*(x)$ represents the chosen essential fibre of $F$ over $x$. By fullness, it suffices to provide a morphism $F(x) \to FF^*F(x)$; But recall that the essential fibre $F^*F(x)$ comes equipped with an isomorphism $FF^*F(x) \cong F(x)$.

  ff+split-eso→unit : Id => (G F∘ F)
  ff+split-eso→unit .η x = ff⁻¹ (f*x-iso .di.from)
    where open Σ (eso (F₀ F x)) renaming (fst to f*x ; snd to f*x-iso)

  ff+split-eso→unit .is-natural x y f =
    fully-faithful→faithful {F = F} ff (
      F₁ F (ff⁻¹ ffy C.∘ f)                                    ≡⟨ F-∘ F _ _ ⟩≡
      ⌜ F₁ F (ff⁻¹ ffy) ⌝ D.∘ F₁ F f                           ≡⟨ ap! (ff.ε _) ⟩≡
      ffy D.∘ ⌜ F₁ F f ⌝                                       ≡⟨ ap! (sym (D.idr _) ∙ ap (F₁ F f D.∘_) (sym (f*x-iso .di.invl))) ⟩≡
      ffy D.∘ F₁ F f D.∘ ftx D.∘ ffx                           ≡⟨ cat! D ⟩≡
      (ffy D.∘ F₁ F f D.∘ ftx) D.∘ ffx                         ≡˘⟨ ap₂ D._∘_ (ff.ε _) (ff.ε _) ⟩≡˘
      F₁ F (ff⁻¹ (ffy D.∘ F₁ F f D.∘ ftx)) D.∘ F₁ F (ff⁻¹ ffx) ≡˘⟨ F-∘ F _ _ ⟩≡˘
      F₁ F (ff⁻¹ (ffy D.∘ F₁ F f D.∘ ftx) C.∘ ff⁻¹ ffx)        ≡⟨⟩
      F₁ F (F₁ (G F∘ F) f C.∘ x→f*x)                           ∎
    )
    where
      open Σ (eso (F₀ F x)) renaming (fst to f*x ; snd to f*x-iso)
      open Σ (eso (F₀ F y)) renaming (fst to f*y ; snd to f*y-iso)

      ffy = f*y-iso .di.from
      fty = f*y-iso .di.to
      ffx = f*x-iso .di.from
      ftx = f*x-iso .di.to

      x→f*x : C.Hom x f*x
      x→f*x = ff⁻¹ (f*x-iso .di.from)

      y→f*y : C.Hom y f*y
      y→f*y = ff⁻¹ (f*y-iso .di.from)

For the counit, we have to provide a morphism $FF^*(x) \to x$; We can again pick the given isomorphism.

  ff+split-eso→counit : (F F∘ G) => Id
  ff+split-eso→counit .η x = f*x-iso .di.to
    where open Σ (eso x) renaming (fst to f*x ; snd to f*x-iso)

  ff+split-eso→counit .is-natural x y f =
    fty D.∘ ⌜ F₁ F (ff⁻¹ (ffy D.∘ f D.∘ ftx)) ⌝ ≡⟨ ap! (ff.ε _) ⟩≡
    fty D.∘ ffy D.∘ f D.∘ ftx                   ≡⟨ D.cancell (f*y-iso .di.invl) ⟩≡
    f D.∘ ftx                                   ∎
    where
      open Σ (eso x) renaming (fst to f*x ; snd to f*x-iso)
      open Σ (eso y) renaming (fst to f*y ; snd to f*y-iso)

      ffy = f*y-iso .di.from
      fty = f*y-iso .di.to
      ftx = f*x-iso .di.to

Checking the triangle identities, and that the adjunction unit/counit defined above are natural isomorphisms, is routine. We present the calculations without commentary:

  open _⊣_

  ff+split-eso→F⊣inverse : F ⊣ G
  ff+split-eso→F⊣inverse .unit    = ff+split-eso→unit
  ff+split-eso→F⊣inverse .counit  = ff+split-eso→counit
  ff+split-eso→F⊣inverse .zig {x} =
    ftx D.∘ F₁ F (ff⁻¹ ffx) ≡⟨ ap (ftx D.∘_) (ff.ε _) ⟩≡
    ftx D.∘ ffx             ≡⟨ f*x-iso .di.invl ⟩≡
    D.id                    ∎

The zag identity needs an appeal to faithfulness:

  ff+split-eso→F⊣inverse .zag {x} =
    fully-faithful→faithful {F = F} ff (
      F₁ F (ff⁻¹ (ffx D.∘ ftx D.∘ fftx) C.∘ ff⁻¹ fffx)        ≡⟨ F-∘ F _ _ ⟩≡
      F₁ F (ff⁻¹ (ffx D.∘ ftx D.∘ fftx)) D.∘ F₁ F (ff⁻¹ fffx) ≡⟨ ap₂ D._∘_ (ff.ε _) (ff.ε _) ⟩≡
      (ffx D.∘ ftx D.∘ fftx) D.∘ fffx                         ≡⟨ cat! D ⟩≡
      (ffx D.∘ ftx) D.∘ (fftx D.∘ fffx)                       ≡⟨ ap₂ D._∘_ (f*x-iso .di.invr) (f*f*x-iso .di.invl) ⟩≡
      D.id D.∘ D.id                                           ≡⟨ D.idl _ ∙ sym (F-id F) ⟩≡
      F₁ F C.id                                               ∎
    )

Now to show they are componentwise invertible:

  open is-equivalence

  ff+split-eso→is-equivalence : is-equivalence F
  ff+split-eso→is-equivalence .F⁻¹ = G
  ff+split-eso→is-equivalence .F⊣F⁻¹ = ff+split-eso→F⊣inverse
  ff+split-eso→is-equivalence .counit-iso x = record
    { inv      = f*x-iso .di.from
    ; inverses = record
      { invl = f*x-iso .di.invl
      ; invr = f*x-iso .di.invr }
    }
    where open Σ (eso x) renaming (fst to f*x ; snd to f*x-iso)

Since the unit is defined in terms of fullness, showing it is invertible needs an appeal to faithfulness (two, actually):

  ff+split-eso→is-equivalence .unit-iso x = record
    { inv      = ff⁻¹ (f*x-iso .di.to)
    ; inverses = record
      { invl = fully-faithful→faithful {F = F} ff (
          F₁ F (ff⁻¹ ffx C.∘ ff⁻¹ ftx)        ≡⟨ F-∘ F _ _ ⟩≡
          F₁ F (ff⁻¹ ffx) D.∘ F₁ F (ff⁻¹ ftx) ≡⟨ ap₂ D._∘_ (ff.ε _) (ff.ε _) ⟩≡
          ffx D.∘ ftx                         ≡⟨ f*x-iso .di.invr ⟩≡
          D.id                                ≡˘⟨ F-id F ⟩≡˘
          F₁ F C.id                           ∎)
      ; invr = fully-faithful→faithful {F = F} ff (
          F₁ F (ff⁻¹ ftx C.∘ ff⁻¹ ffx)        ≡⟨ F-∘ F _ _ ⟩≡
          F₁ F (ff⁻¹ ftx) D.∘ F₁ F (ff⁻¹ ffx) ≡⟨ ap₂ D._∘_ (ff.ε _) (ff.ε _) ⟩≡
          ftx D.∘ ffx                         ≡⟨ f*x-iso .di.invl ⟩≡
          D.id                                ≡˘⟨ F-id F ⟩≡˘
          F₁ F C.id                           ∎)
      }
    }
    where
      open Σ (eso (F₀ F x)) renaming (fst to f*x ; snd to f*x-iso)
      ffx = f*x-iso .di.from
      ftx = f*x-iso .di.to

Between categories🔗

Above, we made an equivalence out of any fully faithful and split essentially surjective functor. In set-theoretic mathematics (and indeed between strict categories), the splitting condition can not be lifted constructively: the statement “every (ff, eso) functor between strict categories is an equivalence” is equivalent to the axiom of choice.

However, between univalent categories, the situation is different: Any essentially surjective functor splits. In particular, any functor between univalent categories has propositional essential fibres, so a “mere” essential surjection is automatically split. However, note that both the domain and codomain have to be categories for the argument to go through.

module
  _ (F : Functor C D) (ccat : is-category C) (dcat : is-category D)
    (ff : is-fully-faithful F)
  where
  private
    module C = Cat.Reasoning C
    module D = Cat.Reasoning D

So, suppose we have categories $\mathcal{C}$ and $\mathcal{D}$, together with a fully faithful functor $F : \mathcal{C} \to \mathcal{D}$. For any $y : \mathcal{D}$, we’re given an inhabitant of $\| \sum_{x : \mathcal{C}} F(x) \cong y \|$, which we want to “get out” from under the truncation. For this, we’ll show that the type being truncated is a proposition, so that we may “untruncate” it.

  Essential-fibre-between-cats-is-prop : ∀ y → is-prop (Essential-fibre F y)
  Essential-fibre-between-cats-is-prop z (x , i) (y , j) = they're-equal where

For this magic trick, assume we’re given a $z : \mathcal{D}$, together with objects $x, y : \mathcal{C}$ and isomorphisms $i : F(x) \cong z$ and $j : F(y) \cong z$. We must show that $x \equiv y$, and that over this path, $i = j$. Since $F$ is fully faithful, we can find an isomorphism $x \cong y$ in $\mathcal{C}$, which $F$ sends back to $i \circ j^{-1}$.

    Fx≅Fy : F₀ F x D.≅ F₀ F y
    Fx≅Fy = i D.∘Iso (j D.Iso⁻¹)

    x≅y : x C.≅ y
    x≅y = is-ff→essentially-injective {F = F} ff Fx≅Fy

Furthermore, since we’re working with categories, these isomorphisms restrict to paths $x \equiv y$ and $F(x) \equiv F(y)$. We’re half-done: we’ve shown that some $p : x \equiv y$ exists, and it remains to show that over this path we have $i \equiv j$. More specifically, we must give a path $i \equiv j$ laying over ${\mathrm{ap}}(F)(p)$.

    x≡y : x ≡ y
    x≡y = ccat .to-path x≅y

    Fx≡Fy : F₀ F x ≡ F₀ F y
    Fx≡Fy = dcat .to-path Fx≅Fy

Rather than showing it over $p : x\equiv y$ directly, we’ll show it over the path $F(x) \equiv F(y)$ we constructed independently. This is because we can use the helper Hom-pathp-reflr-iso to establish the result with far less computation:

    over′ : PathP (λ i → Fx≡Fy i D.≅ z) i j
    over′ = D.≅-pathp Fx≡Fy refl
      (Univalent.Hom-pathp-refll-iso dcat (D.cancell (i .D._≅_.invl)))

We must then connect ${\mathrm{ap}}(F)(p)$ with this path $F(x) \cong F(y)$. But since we originally got $p$ by full faithfulness of $F$, they are indeed the same path:

    abstract
      square : ap (F₀ F) x≡y ≡ Fx≡Fy
      square =
        ap (F₀ F) x≡y                     ≡⟨ F-map-path F x≅y ccat dcat ⟩≡
        dcat .to-path ⌜ F-map-iso F x≅y ⌝ ≡⟨ ap! (equiv→counit (is-ff→F-map-iso-is-equiv {F = F} ff) _)  ⟩≡
        dcat .to-path Fx≅Fy               ∎

    over : PathP (λ i → F₀ F (x≡y i) D.≅ z) i j
    over = transport (λ l → PathP (λ m → square (~ l) m D.≅ z) i j) over′

Hence — blink and you’ll miss it — the essential fibres of $F$ over any $z : \mathcal{D}$ are propositions, so it suffices for them to be merely inhabited for the functor to be split eso. With tongue firmly in cheek we call this result the theorem of choice.

    they're-equal = Σ-pathp x≡y over

  Theorem-of-choice : is-eso F → is-split-eso F
  Theorem-of-choice eso y =
    ∥-∥-elim (λ _ → Essential-fibre-between-cats-is-prop y)
      (λ x → x) (eso y)

This theorem implies that any fully faithful, “merely” essentially surjective functor between categories is an equivalence:

  ff+eso→is-equivalence : is-eso F → is-equivalence F
  ff+eso→is-equivalence eso = ff+split-eso→is-equivalence ff (Theorem-of-choice eso)

Isomorphisms🔗

Another, more direct way of proving that a functor is an equivalence of precategories is proving that it is an isomorphism of precategories: It’s fully faithful, thus a typal equivalence of morphisms, and in addition its action on objects is an equivalence of types.

record is-precat-iso (F : Functor C D) : Type (adj-level C D) where
  no-eta-equality
  constructor iso
  field
    has-is-ff  : is-fully-faithful F
    has-is-iso : is-equiv (F₀ F)

Such a functor is (immediately) fully faithful, and the inverse has-is-iso means that it is split essentially surjective; For given $y : D$, the inverse of $F_0$ gives us an object $F^{-1}(y)$; We must then provide an isomorphism $F(F^{-1}(y)) \cong y$, but those are identical, hence isomorphic.

module _ {F : Functor C D} (p : is-precat-iso F) where
  open is-precat-iso p

  is-precat-iso→is-split-eso : is-split-eso F
  is-precat-iso→is-split-eso ob = equiv→inverse has-is-iso ob , isom
    where isom = path→iso {C = D} (equiv→counit has-is-iso _)

Thus, by the theorem above, $F$ is an adjoint equivalence of precategories.

  is-precat-iso→is-equivalence : is-equivalence F
  is-precat-iso→is-equivalence =
    ff+split-eso→is-equivalence has-is-ff is-precat-iso→is-split-eso