Fully faithful functors between precategories

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2023-10-20.
Last modified on 2024-01-27.

module category-theory.fully-faithful-functors-precategories where

open import category-theory.conservative-functors-precategories
open import category-theory.essentially-injective-functors-precategories
open import category-theory.faithful-functors-precategories
open import category-theory.full-functors-precategories
open import category-theory.fully-faithful-maps-precategories
open import category-theory.functors-precategories
open import category-theory.isomorphisms-in-precategories
open import category-theory.precategories
open import category-theory.pseudomonic-functors-precategories

open import foundation.action-on-identifications-binary-functions
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.function-types
open import foundation.identity-types
open import foundation.propositional-truncations
open import foundation.propositions
open import foundation.subtypes
open import foundation.surjective-maps
open import foundation.universe-levels

Idea

A functor between precategories C and D is fully faithful if it’s an equivalence on hom-sets, or equivalently, a full and faithful functor on precategories.

Definitions

The predicate on functors between precategories of being fully faithful

module _
  {l1 l2 l3 l4 : Level}
  (C : Precategory l1 l2)
  (D : Precategory l3 l4)
  (F : functor-Precategory C D)
  where

  is-fully-faithful-functor-Precategory : UU (l1 ⊔ l2 ⊔ l4)
  is-fully-faithful-functor-Precategory =
    is-fully-faithful-map-Precategory C D (map-functor-Precategory C D F)

  is-prop-is-fully-faithful-functor-Precategory :
    is-prop is-fully-faithful-functor-Precategory
  is-prop-is-fully-faithful-functor-Precategory =
    is-prop-is-fully-faithful-map-Precategory C D
      ( map-functor-Precategory C D F)

  is-fully-faithful-prop-functor-Precategory : Prop (l1 ⊔ l2 ⊔ l4)
  is-fully-faithful-prop-functor-Precategory =
    is-fully-faithful-prop-map-Precategory C D
      ( map-functor-Precategory C D F)

  equiv-hom-is-fully-faithful-functor-Precategory :
    is-fully-faithful-functor-Precategory → {x y : obj-Precategory C} →
    hom-Precategory C x y ≃
    hom-Precategory D
      ( obj-functor-Precategory C D F x)
      ( obj-functor-Precategory C D F y)
  equiv-hom-is-fully-faithful-functor-Precategory =
    equiv-hom-is-fully-faithful-map-Precategory C D
      ( map-functor-Precategory C D F)

  inv-equiv-hom-is-fully-faithful-functor-Precategory :
    is-fully-faithful-functor-Precategory →
    {x y : obj-Precategory C} →
    hom-Precategory D
      ( obj-functor-Precategory C D F x)
      ( obj-functor-Precategory C D F y) ≃
    hom-Precategory C x y
  inv-equiv-hom-is-fully-faithful-functor-Precategory is-ff-F =
    inv-equiv (equiv-hom-is-fully-faithful-functor-Precategory is-ff-F)

  map-inv-hom-is-fully-faithful-functor-Precategory :
    is-fully-faithful-functor-Precategory →
    {x y : obj-Precategory C} →
    hom-Precategory D
      ( obj-functor-Precategory C D F x)
      ( obj-functor-Precategory C D F y) →
    hom-Precategory C x y
  map-inv-hom-is-fully-faithful-functor-Precategory is-ff-F =
    map-equiv (inv-equiv-hom-is-fully-faithful-functor-Precategory is-ff-F)

The type of fully faithful functors between two precategories

module _
  {l1 l2 l3 l4 : Level}
  (C : Precategory l1 l2)
  (D : Precategory l3 l4)
  where

  fully-faithful-functor-Precategory : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
  fully-faithful-functor-Precategory =
    Σ (functor-Precategory C D) (is-fully-faithful-functor-Precategory C D)

  functor-fully-faithful-functor-Precategory :
    fully-faithful-functor-Precategory → functor-Precategory C D
  functor-fully-faithful-functor-Precategory = pr1

  is-fully-faithful-fully-faithful-functor-Precategory :
    (F : fully-faithful-functor-Precategory) →
    is-fully-faithful-functor-Precategory C D
      ( functor-fully-faithful-functor-Precategory F)
  is-fully-faithful-fully-faithful-functor-Precategory = pr2

  obj-fully-faithful-functor-Precategory :
    fully-faithful-functor-Precategory → obj-Precategory C → obj-Precategory D
  obj-fully-faithful-functor-Precategory =
    obj-functor-Precategory C D ∘ functor-fully-faithful-functor-Precategory

  hom-fully-faithful-functor-Precategory :
    (F : fully-faithful-functor-Precategory)
    {x y : obj-Precategory C} →
    hom-Precategory C x y →
    hom-Precategory D
      ( obj-fully-faithful-functor-Precategory F x)
      ( obj-fully-faithful-functor-Precategory F y)
  hom-fully-faithful-functor-Precategory =
    hom-functor-Precategory C D ∘ functor-fully-faithful-functor-Precategory

  equiv-hom-fully-faithful-functor-Precategory :
    (F : fully-faithful-functor-Precategory)
    {x y : obj-Precategory C} →
    hom-Precategory C x y ≃
    hom-Precategory D
      ( obj-fully-faithful-functor-Precategory F x)
      ( obj-fully-faithful-functor-Precategory F y)
  equiv-hom-fully-faithful-functor-Precategory F =
    equiv-hom-is-fully-faithful-functor-Precategory C D
      ( functor-fully-faithful-functor-Precategory F)
      ( is-fully-faithful-fully-faithful-functor-Precategory F)

  inv-equiv-hom-fully-faithful-functor-Precategory :
    (F : fully-faithful-functor-Precategory)
    {x y : obj-Precategory C} →
    hom-Precategory D
      ( obj-fully-faithful-functor-Precategory F x)
      ( obj-fully-faithful-functor-Precategory F y) ≃
    hom-Precategory C x y
  inv-equiv-hom-fully-faithful-functor-Precategory F =
    inv-equiv (equiv-hom-fully-faithful-functor-Precategory F)

  map-inv-hom-fully-faithful-functor-Precategory :
    (F : fully-faithful-functor-Precategory)
    {x y : obj-Precategory C} →
    hom-Precategory D
      ( obj-fully-faithful-functor-Precategory F x)
      ( obj-fully-faithful-functor-Precategory F y) →
    hom-Precategory C x y
  map-inv-hom-fully-faithful-functor-Precategory F =
    map-equiv (inv-equiv-hom-fully-faithful-functor-Precategory F)

Properties

Fully faithful functors are the same as full and faithful functors

module _
  {l1 l2 l3 l4 : Level}
  (C : Precategory l1 l2)
  (D : Precategory l3 l4)
  (F : functor-Precategory C D)
  where

  is-full-is-fully-faithful-functor-Precategory :
    is-fully-faithful-functor-Precategory C D F →
    is-full-functor-Precategory C D F
  is-full-is-fully-faithful-functor-Precategory is-ff-F x y =
    is-surjective-is-equiv (is-ff-F x y)

  full-functor-is-fully-faithful-functor-Precategory :
    is-fully-faithful-functor-Precategory C D F → full-functor-Precategory C D
  pr1 (full-functor-is-fully-faithful-functor-Precategory is-ff-F) = F
  pr2 (full-functor-is-fully-faithful-functor-Precategory is-ff-F) =
    is-full-is-fully-faithful-functor-Precategory is-ff-F

  is-faithful-is-fully-faithful-functor-Precategory :
    is-fully-faithful-functor-Precategory C D F →
    is-faithful-functor-Precategory C D F
  is-faithful-is-fully-faithful-functor-Precategory is-ff-F x y =
    is-emb-is-equiv (is-ff-F x y)

  faithful-functor-is-fully-faithful-functor-Precategory :
    is-fully-faithful-functor-Precategory C D F →
    faithful-functor-Precategory C D
  pr1 (faithful-functor-is-fully-faithful-functor-Precategory is-ff-F) = F
  pr2 (faithful-functor-is-fully-faithful-functor-Precategory is-ff-F) =
    is-faithful-is-fully-faithful-functor-Precategory is-ff-F

  is-fully-faithful-is-full-is-faithful-functor-Precategory :
    is-full-functor-Precategory C D F →
    is-faithful-functor-Precategory C D F →
    is-fully-faithful-functor-Precategory C D F
  is-fully-faithful-is-full-is-faithful-functor-Precategory
    is-full-F is-faithful-F x y =
    is-equiv-is-emb-is-surjective (is-full-F x y) (is-faithful-F x y)

  fully-faithful-functor-is-full-is-faithful-functor-Precategory :
    is-full-functor-Precategory C D F →
    is-faithful-functor-Precategory C D F →
    fully-faithful-functor-Precategory C D
  pr1
    ( fully-faithful-functor-is-full-is-faithful-functor-Precategory
      is-full-F is-faithful-F) =
    F
  pr2
    ( fully-faithful-functor-is-full-is-faithful-functor-Precategory
      is-full-F is-faithful-F) =
    is-fully-faithful-is-full-is-faithful-functor-Precategory
      ( is-full-F) (is-faithful-F)

module _
  {l1 l2 l3 l4 : Level}
  (C : Precategory l1 l2)
  (D : Precategory l3 l4)
  (F : fully-faithful-functor-Precategory C D)
  where

  full-functor-fully-faithful-functor-Precategory : full-functor-Precategory C D
  full-functor-fully-faithful-functor-Precategory =
    full-functor-is-fully-faithful-functor-Precategory C D
      ( functor-fully-faithful-functor-Precategory C D F)
      ( is-fully-faithful-fully-faithful-functor-Precategory C D F)

  faithful-functor-fully-faithful-functor-Precategory :
    faithful-functor-Precategory C D
  faithful-functor-fully-faithful-functor-Precategory =
    faithful-functor-is-fully-faithful-functor-Precategory C D
      ( functor-fully-faithful-functor-Precategory C D F)
      ( is-fully-faithful-fully-faithful-functor-Precategory C D F)

Fully faithful functors are essentially injective

module _
  {l1 l2 l3 l4 : Level}
  (C : Precategory l1 l2)
  (D : Precategory l3 l4)
  (F : functor-Precategory C D)
  (is-ff-F : is-fully-faithful-functor-Precategory C D F)
  {x y : obj-Precategory C}
  ((e , e' , l , r) :
      iso-Precategory D
        ( obj-functor-Precategory C D F x)
        ( obj-functor-Precategory C D F y))
  where

  hom-iso-is-essentially-injective-is-fully-faithful-functor-Precategory :
    hom-Precategory C x y
  hom-iso-is-essentially-injective-is-fully-faithful-functor-Precategory =
    map-inv-hom-is-fully-faithful-functor-Precategory C D F is-ff-F e

  hom-inv-iso-is-essentially-injective-is-fully-faithful-functor-Precategory :
    hom-Precategory C y x
  hom-inv-iso-is-essentially-injective-is-fully-faithful-functor-Precategory =
    map-inv-hom-is-fully-faithful-functor-Precategory C D F is-ff-F e'

  is-right-inverse-hom-inv-iso-is-essentially-injective-is-fully-faithful-functor-Precategory :
    ( comp-hom-Precategory C
      ( hom-iso-is-essentially-injective-is-fully-faithful-functor-Precategory)
      ( hom-inv-iso-is-essentially-injective-is-fully-faithful-functor-Precategory)) ＝
    ( id-hom-Precategory C)
  is-right-inverse-hom-inv-iso-is-essentially-injective-is-fully-faithful-functor-Precategory =
    ( inv
      ( is-retraction-map-inv-is-equiv
        ( is-ff-F y y)
        ( comp-hom-Precategory C
          ( map-inv-is-equiv (is-ff-F x y) e)
          ( map-inv-is-equiv (is-ff-F y x) e')))) ∙
    ( ap
      ( map-inv-hom-is-fully-faithful-functor-Precategory C D F is-ff-F)
      ( ( preserves-comp-functor-Precategory C D F
          ( map-inv-is-equiv (is-ff-F x y) e)
          ( map-inv-is-equiv (is-ff-F y x) e')) ∙
        ( ap-binary
          ( comp-hom-Precategory D)
          ( is-section-map-inv-is-equiv (is-ff-F x y) e)
          ( is-section-map-inv-is-equiv (is-ff-F y x) e')) ∙
        ( l) ∙
        ( inv (preserves-id-functor-Precategory C D F y)))) ∙
    ( is-retraction-map-inv-is-equiv (is-ff-F y y) (id-hom-Precategory C))

  is-left-inverse-hom-inv-iso-is-essentially-injective-is-fully-faithful-functor-Precategory :
    ( comp-hom-Precategory C
      ( hom-inv-iso-is-essentially-injective-is-fully-faithful-functor-Precategory)
      ( hom-iso-is-essentially-injective-is-fully-faithful-functor-Precategory)) ＝
    ( id-hom-Precategory C)
  is-left-inverse-hom-inv-iso-is-essentially-injective-is-fully-faithful-functor-Precategory =
    ( inv
      ( is-retraction-map-inv-is-equiv
        ( is-ff-F x x)
        ( comp-hom-Precategory C
          ( map-inv-is-equiv (is-ff-F y x) e')
          ( map-inv-is-equiv (is-ff-F x y) e)))) ∙
    ( ap
      ( map-inv-hom-is-fully-faithful-functor-Precategory C D F is-ff-F)
      ( ( preserves-comp-functor-Precategory C D F
          ( map-inv-is-equiv (is-ff-F y x) e')
          ( map-inv-is-equiv (is-ff-F x y) e)) ∙
        ( ap-binary
          (comp-hom-Precategory D)
          ( is-section-map-inv-is-equiv (is-ff-F y x) e')
          ( is-section-map-inv-is-equiv (is-ff-F x y) e)) ∙
        ( r) ∙
        ( inv (preserves-id-functor-Precategory C D F x)))) ∙
    ( is-retraction-map-inv-is-equiv (is-ff-F x x) (id-hom-Precategory C))

  is-iso-iso-is-essentially-injective-is-fully-faithful-functor-Precategory :
    is-iso-Precategory C
      ( hom-iso-is-essentially-injective-is-fully-faithful-functor-Precategory)
  pr1
    is-iso-iso-is-essentially-injective-is-fully-faithful-functor-Precategory =
    hom-inv-iso-is-essentially-injective-is-fully-faithful-functor-Precategory
  pr1
    ( pr2
        is-iso-iso-is-essentially-injective-is-fully-faithful-functor-Precategory) =
    is-right-inverse-hom-inv-iso-is-essentially-injective-is-fully-faithful-functor-Precategory
  pr2
    ( pr2
        is-iso-iso-is-essentially-injective-is-fully-faithful-functor-Precategory) =
    is-left-inverse-hom-inv-iso-is-essentially-injective-is-fully-faithful-functor-Precategory

module _
  {l1 l2 l3 l4 : Level}
  (C : Precategory l1 l2)
  (D : Precategory l3 l4)
  (F : functor-Precategory C D)
  (is-ff-F : is-fully-faithful-functor-Precategory C D F)
  where

  is-essentially-injective-is-fully-faithful-functor-Precategory :
    is-essentially-injective-functor-Precategory C D F
  pr1 (is-essentially-injective-is-fully-faithful-functor-Precategory x y e) =
    hom-iso-is-essentially-injective-is-fully-faithful-functor-Precategory
      C D F is-ff-F e
  pr2 (is-essentially-injective-is-fully-faithful-functor-Precategory x y e) =
    is-iso-iso-is-essentially-injective-is-fully-faithful-functor-Precategory
      C D F is-ff-F e

Fully faithful functors are pseudomonic

module _
  {l1 l2 l3 l4 : Level}
  (C : Precategory l1 l2)
  (D : Precategory l3 l4)
  (F : functor-Precategory C D)
  (is-ff-F : is-fully-faithful-functor-Precategory C D F)
  where

  is-full-on-isos-is-fully-faithful-functor-Precategory :
    (x y : obj-Precategory C) →
    is-surjective (preserves-iso-functor-Precategory C D F {x} {y})
  is-full-on-isos-is-fully-faithful-functor-Precategory x y e =
    unit-trunc-Prop
      ( ( is-essentially-injective-is-fully-faithful-functor-Precategory
            C D F is-ff-F x y e) ,
        eq-type-subtype
          ( is-iso-prop-Precategory D)
          ( is-section-map-inv-is-equiv (is-ff-F x y) (pr1 e)))

  is-pseudomonic-is-fully-faithful-functor-Precategory :
    is-pseudomonic-functor-Precategory C D F
  pr1 is-pseudomonic-is-fully-faithful-functor-Precategory =
    is-faithful-is-fully-faithful-functor-Precategory C D F is-ff-F
  pr2 is-pseudomonic-is-fully-faithful-functor-Precategory =
    is-full-on-isos-is-fully-faithful-functor-Precategory

Fully faithful functors are conservative

module _
  {l1 l2 l3 l4 : Level}
  (C : Precategory l1 l2)
  (D : Precategory l3 l4)
  (F : functor-Precategory C D)
  (is-ff-F : is-fully-faithful-functor-Precategory C D F)
  where

  is-conservative-is-fully-faithful-functor-Precategory :
    is-conservative-functor-Precategory C D F
  is-conservative-is-fully-faithful-functor-Precategory {x} {y} =
    is-conservative-is-pseudomonic-functor-Precategory C D F
      ( is-pseudomonic-is-fully-faithful-functor-Precategory C D F is-ff-F)
      { x} {y}

External links

• Fully Faithful Functors at 1lab
• full and faithful functor at $n$Lab
• Full and faithful functors at Wikipedia
• fully faithful functor at Wikidata

Recent changes

• 2024-01-27. Egbert Rijke. Fix “The predicate of” section headers (#1010).
• 2023-11-09. Fredrik Bakke. Typeset nlab as $n$Lab (#911).
• 2023-11-01. Fredrik Bakke. Fun with functors (#886).
• 2023-10-20. Fredrik Bakke and Egbert Rijke. Small subcategories (#861).