open import Cat.Functor.Naturality
open import Cat.Functor.Compose
open import Cat.Functor.Base
open import Cat.Prelude

import Cat.Functor.Reasoning
import Cat.Natural.Reasoning
import Cat.Reasoning

module Cat.Functor.Properties where

private variable
  o h o₁ h₁ : Level
  B C D : Precategory o h
open Precategory
open Functor

Functors🔗

This module defines the most important classes of functors: full, faithful, fully faithful (abbreviated ff), split essentially surjective and (“merely”) essentially surjective.

is-full : Functor C D → Type _
is-full {C = C} {D = D} F = ∀ {x y} → is-surjective (F .F₁ {x = x} {y})

is-faithful : Functor C D → Type _
is-faithful F = ∀ {x y} → injective (F .F₁ {x = x} {y})

module _ {C : Precategory o h} {D : Precategory o₁ h₁} where
  private module _ where
    module C = Cat.Reasoning C
    module D = Cat.Reasoning D
    open Cat.Reasoning using (_≅_ ; Inverses)
    open _≅_ public
    open Inverses public

  faithful→iso-fibre-prop
    : ∀ (F : Functor C D)
    → is-faithful F
    → ∀ {x y} → (f : F · x D.≅ F · y)
    → is-prop (Σ[ g ∈ x C.≅ y ] (F-map-iso F g ≡ f))
  faithful→iso-fibre-prop F faithful f (g , p) (g' , q) =
    Σ-prop-path! $ ext (faithful (ap D.to (p ∙ sym q)))

  is-faithful-∘
    : ∀ {F : Functor C D} {G : Functor B C}
    → is-faithful F → is-faithful G
    → is-faithful (F F∘ G)
  is-faithful-∘ Ff Gf p = Gf (Ff p)

Fully faithful functors🔗

A functor is fully faithful (abbreviated ff) when its action on hom-sets is an equivalence. Since Hom-sets are sets, it suffices for the functor to be full and faithful; Rather than taking this conjunction as a definition, we use the more directly useful data as a definition and prove the conjunction as a theorem.

is-fully-faithful : Functor C D → Type _
is-fully-faithful F = ∀ {x y} → is-equiv (F .F₁ {x = x} {y})

ff→faithful : {F : Functor C D} → is-fully-faithful F → is-faithful F
ff→faithful f = Equiv.injective (_ , f)

ff→full : {F : Functor C D} → is-fully-faithful F → is-full F
ff→full {F = F} ff g = inc (equiv→inverse ff g , equiv→counit ff g)

full+faithful→ff
  : (F : Functor C D) → is-full F → is-faithful F → is-fully-faithful F
full+faithful→ff {C = C} {D = D} F surj inj .is-eqv = p where
  img-is-prop : ∀ {x y} f → is-prop (fibre (F .F₁ {x = x} {y}) f)
  img-is-prop f (g , p) (h , q) = Σ-prop-path (λ _ → D .Hom-set _ _ _ _) (inj (p ∙ sym q))

  p : ∀ {x y} f → is-contr (fibre (F .F₁ {x = x} {y}) f)
  p f .centre = ∥-∥-elim (λ _ → img-is-prop f) (λ x → x) (surj f)
  p f .paths = img-is-prop f _

A very important property of fully faithful functors (like $F\text{)}$ is that they are conservative: If the image of $f:x\to y$ under $F$ is an isomorphism $Fx\cong Fy\text{,}$ then $f$ was really an isomorphism $f:x\cong y\text{.}$

module _ {C : Precategory o h} {D : Precategory o₁ h₁} where
  private
    module C = Precategory C
    module D = Precategory D

  import Cat.Morphism C as Cm
  import Cat.Morphism D as Dm

  is-ff→is-conservative
    : {F : Functor C D} → is-fully-faithful F
    → ∀ {X Y} (f : C.Hom X Y) → Dm.is-invertible (F .F₁ f)
    → Cm.is-invertible f
  is-ff→is-conservative {F = F} ff f isinv = i where
    open Cm.is-invertible
    open Cm.Inverses

Since the functor is ff, we can find a map “$F_1^{-1}(f):y\to x$” in the domain category to serve as an inverse for $f\text{:}$

    g : C.Hom _ _
    g = equiv→inverse ff (isinv .Dm.is-invertible.inv)
    module ff {a} {b} = Equiv (_ , ff {a} {b})

    Ffog =
      F .F₁ (f C.∘ g)     ≡⟨ F .F-∘ _ _ ⟩≡
      F .F₁ f D.∘ F .F₁ g ≡⟨ ap₂ D._∘_ refl (ff.ε _) ∙ isinv .Dm.is-invertible.invl ⟩≡
      D.id                ∎

    Fgof =
      F .F₁ (g C.∘ f)     ≡⟨ F .F-∘ _ _ ⟩≡
      F .F₁ g D.∘ F .F₁ f ≡⟨ ap₂ D._∘_ (ff.ε _) refl ∙ isinv .Dm.is-invertible.invr ⟩≡
      D.id                ∎

    i : Cm.is-invertible _
    i .inv = g
    i .inverses .invl =
      f C.∘ g                     ≡⟨ sym (ff.η _) ⟩≡
      ff.from ⌜ F .F₁ (f C.∘ g) ⌝ ≡⟨ ap! (Ffog ∙ sym (F .F-id)) ⟩≡
      ff.from (F .F₁ C.id)        ≡⟨ ff.η _ ⟩≡
      C.id                        ∎
    i .inverses .invr =
      g C.∘ f                     ≡⟨ sym (ff.η _) ⟩≡
      ff.from ⌜ F .F₁ (g C.∘ f) ⌝ ≡⟨ ap! (Fgof ∙ sym (F .F-id)) ⟩≡
      ff.from (F .F₁ C.id)        ≡⟨ ff.η _ ⟩≡
      C.id                        ∎

  is-ff→essentially-injective
    : {F : Functor C D} → is-fully-faithful F
    → ∀ {X Y} → F .F₀ X Dm.≅ F .F₀ Y
    → X Cm.≅ Y
  is-ff→essentially-injective {F = F} ff im = im' where
    open Dm._≅_ im using (to ; from ; inverses)
    D-inv' : Dm.is-invertible (F .F₁ (equiv→inverse ff to))
    D-inv' .Dm.is-invertible.inv = from
    D-inv' .Dm.is-invertible.inverses =
      subst (λ e → Dm.Inverses e from) (sym (equiv→counit ff _)) inverses

    open Cm.is-invertible (is-ff→is-conservative {F = F} ff (equiv→inverse ff to) D-inv')

    im' : _ Cm.≅ _
    im' .to   = equiv→inverse ff to
    im' .from = inv
    im' .inverses .Cm.Inverses.invl = invl
    im' .inverses .Cm.Inverses.invr = invr

Essential fibres🔗

The essential fibre of a functor $F:C\to D$ over an object $y:D$ is the space of objects of $C$ which $F$ takes, up to isomorphism, to $y\text{.}$

Essential-fibre : Functor C D → D .Ob → Type _
Essential-fibre {C = C} {D = D} F y = Σ[ x ∈ C ] (F · x ≅ y)
  where open import Cat.Morphism D

is-split-eso : Functor C D → Type _
is-split-eso F = ∀ y → Essential-fibre F y

is-eso : Functor C D → Type _
is-eso F = ∀ y → ∥ Essential-fibre F y ∥

module _ {C : Precategory o h} {D : Precategory o₁ h₁} where
  import Cat.Reasoning C as C
  import Cat.Reasoning D as D
  private module _ where
    open import Cat.Reasoning using (_≅_ ; Inverses)
    open _≅_ public
    open Inverses public

  is-ff→F-map-iso-is-equiv
    : {F : Functor C D} → is-fully-faithful F
    → ∀ {X Y} → is-equiv (F-map-iso F {x = X} {Y})
  is-ff→F-map-iso-is-equiv {F = F} ff = is-iso→is-equiv isom where
    isom : is-iso _
    isom .is-iso.inv    = is-ff→essentially-injective {F = F} ff
    isom .is-iso.rinv x = ext (equiv→counit ff _)
    isom .is-iso.linv x = ext (equiv→unit ff _)

If a functor $F:\mathcal{C}\to \mathcal{D}$ is essentially surjective, then the precomposition functor $(-)\circ F:[\mathcal{D},\mathcal{A}]\to [\mathcal{C},\mathcal{A}]$ is faithful for every precategory $\mathcal{A}\text{.}$

  is-eso→precompose-is-faithful
    : ∀ {oa ℓa} {A : Precategory oa ℓa}
    → (F : Functor C D)
    → is-eso F
    → is-faithful (precompose F {D = A})

This is rather abstract, so let’s unfold the statement a bit. If precomposition by $F$ is fully faithful, then we can determine equality of natural transformations $\alpha ,\beta :G\to H$ between functors $G,H:\mathcal{D}\to \mathcal{A}$ by only looking at the components of the form ${\alpha }_{F(c)},{\beta }_{G(c)}:G(F(c))\to H(F(c))$ for every $c:\mathcal{C}\text{.}$

  is-eso→precompose-is-faithful {A = A} F F-eso {G} {H} {α} {β} αL=βL =
    ext λ d → ∥-∥-out! do
      (c , Fc≅d) ← F-eso d
      let module Fc≅d = D._≅_ Fc≅d
      pure $
        α.η d                                             ≡⟨ A.intror (G.annihilate (D.invl Fc≅d)) ⟩≡
        α.η d A.∘ G.₁ Fc≅d.to A.∘ G.₁ Fc≅d.from           ≡⟨ A.extendl (α.is-natural _ _ _) ⟩≡
        H.₁ Fc≅d.to A.∘ ⌜ α.η (F.₀ c) ⌝ A.∘ G.₁ Fc≅d.from ≡⟨ ap! (unext αL=βL c) ⟩≡
        H.₁ Fc≅d.to A.∘ β.η (F.₀ c) A.∘ G.₁ Fc≅d.from     ≡⟨ A.extendl (sym (β.is-natural _ _ _)) ⟩≡
        β.η d A.∘ G.₁ Fc≅d.to A.∘ G.₁ Fc≅d.from           ≡⟨ A.elimr (G.annihilate (D.invl Fc≅d)) ⟩≡
        β.η d ∎
    where
      module A = Cat.Reasoning A
      module F = Cat.Functor.Reasoning F
      module G = Cat.Functor.Reasoning G
      module H = Cat.Functor.Reasoning H
      module α = _=>_ α
      module β = _=>_ β

  _ = C.epic-precomp-embedding

Another way of viewing this result is that it is a higher-dimensional analog of the fact that precomposition with an epi is an embedding.

Additionally, precomposition with an essentially surjective functor is conservative.

  eso→precompose-conservative
    : ∀ {oa ℓa} {A : Precategory oa ℓa}
    → (F : Functor C D)
    → is-eso F
    → {G H : Functor D A} {α : G => H}
    → is-invertibleⁿ (α ◂ F)
    → is-invertibleⁿ α

  eso→precompose-conservative {A = A} F F-eso {G} {H} {α} α⁻¹ =
    invertible→invertibleⁿ α λ d → ∥-∥-out! do
      (c , Fc≅d) ← F-eso d
      let module Fc≅d = D._≅_ Fc≅d
      pure $
        A.make-invertible (G.₁ Fc≅d.to A.∘ α⁻¹.η c A.∘ H.₁ Fc≅d.from)
          (α.pulll (A.cancell (α⁻¹.invl ·ₚ c)) ∙ H.annihilate Fc≅d.invl)
          (A.pullr (α.cancelr (α⁻¹.invr ·ₚ c)) ∙ G.annihilate Fc≅d.invl)
    where
      module A = Cat.Reasoning A
      module F = Cat.Functor.Reasoning F
      module G = Cat.Functor.Reasoning G
      module H = Cat.Functor.Reasoning H

      module α = Cat.Natural.Reasoning α
      module α⁻¹ where
        open is-invertibleⁿ α⁻¹ public
        open Cat.Natural.Reasoning inv hiding (op) public

Pseudomonic functors🔗

A functor is pseudomonic if it is faithful and full on isomorphisms. Pseudomonic functors are arguably the correct notion of subcategory, as they ensure that we are not able to distinguish between isomorphic objects when creating a subcategory.

module _ {C : Precategory o h} {D : Precategory o₁ h₁} where
  import Cat.Reasoning C as C
  import Cat.Reasoning D as D

  is-full-on-isos : Functor C D → Type (o ⊔ h ⊔ h₁)
  is-full-on-isos F =
    ∀ {x y} → (f : F .F₀ x D.≅ F .F₀ y) → ∃[ g ∈ x C.≅ y ] (F-map-iso F g ≡ f)

  record is-pseudomonic (F : Functor C D) : Type (o ⊔ h ⊔ h₁) where
    no-eta-equality
    field
      faithful : is-faithful F
      isos-full : is-full-on-isos F

  open is-pseudomonic

Somewhat surprisingly, pseudomonic functors are conservative. As $F$ is full on isos, there merely exists some iso $g$ in the fibre of $f\text{.}$ However, invertibility is a property of morphisms, so we can untruncate the mere existence. Once we have our hands on the isomorphism, we perform a simple calculation to note that it yields an inverse to $f\text{.}$

  pseudomonic→conservative
    : ∀ {F : Functor C D}
    → is-pseudomonic F
    → ∀ {x y} (f : C.Hom x y) → D.is-invertible (F .F₁ f)
    → C.is-invertible f
  pseudomonic→conservative {F = F} pseudo {x} {y} f inv =
    ∥-∥-rec C.is-invertible-is-prop
      (λ (g , p) →
        C.make-invertible (C.from g)
          (sym (ap (C._∘ _) (pseudo .faithful (ap D.to p))) ∙ C.invl g)
          (sym (ap (_ C.∘_) (pseudo .faithful (ap D.to p))) ∙ C.invr g))
      (pseudo .isos-full (D.invertible→iso _ inv))

In a similar vein, pseudomonic functors are essentially injective. The proof follows a similar path to the prior one, hinging on the fact that faithful functors are an embedding on isos.

  pseudomonic→essentially-injective
    : ∀ {F : Functor C D}
    → is-pseudomonic F
    → ∀ {x y} → F .F₀ x D.≅ F .F₀ y
    → x C.≅ y
  pseudomonic→essentially-injective {F = F} pseudo f =
    ∥-∥-rec (faithful→iso-fibre-prop F (pseudo .faithful) f)
      (λ x → x)
      (pseudo .isos-full f) .fst

Fully faithful functors are pseudomonic, as they are faithful and essentially injective.

  ff→pseudomonic
    : ∀ {F : Functor C D}
    → is-fully-faithful F
    → is-pseudomonic F
  ff→pseudomonic {F} ff .faithful = ff→faithful {F = F} ff
  ff→pseudomonic {F} ff .isos-full f =
    inc (is-ff→essentially-injective {F = F} ff f ,
         ext (equiv→counit ff (D.to f)))

Equivalence on objects functors🔗

A functor $F:\mathcal{C}\to \mathcal{D}$ is an equivalence on objects if its action on objects is an equivalence.

is-equiv-on-objects : (F : Functor C D) → Type _
is-equiv-on-objects F = is-equiv (F .F₀)

If $F$ is an equivalence-on-objects functor, then it is (split) essentially surjective.

equiv-on-objects→split-eso
  : ∀ (F : Functor C D) → is-equiv-on-objects F → is-split-eso F
equiv-on-objects→split-eso {D = D} F eqv y =
  equiv→inverse eqv y , path→iso (equiv→counit eqv y)

equiv-on-objects→eso : ∀ (F : Functor C D) → is-equiv-on-objects F → is-eso F
equiv-on-objects→eso F eqv y = inc (equiv-on-objects→split-eso F eqv y)