Pseudomonic functors between precategories

Content created by Fredrik Bakke.

Created on 2023-11-01.
Last modified on 2024-03-11.

module category-theory.pseudomonic-functors-precategories where
Imports 
open import category-theory.conservative-functors-precategories
open import category-theory.faithful-functors-precategories
open import category-theory.functors-precategories
open import category-theory.isomorphisms-in-precategories
open import category-theory.precategories

open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.identity-types
open import foundation.injective-maps
open import foundation.iterated-dependent-product-types
open import foundation.propositional-truncations
open import foundation.propositions
open import foundation.surjective-maps
open import foundation.universe-levels

Idea

A functor between precategories C and D is pseudomonic if it is faithful on all morphism-sets and full on isomorphisms. In particular, this means it induces an equivalence on isomorphism-sets.

Pseudomonic functors present replete subprecategories, which is the “right notion” of subcategory with respect to the principle of invariance under equivalences.

Definition

The predicate on isomorphisms of being full

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

  is-full-on-isos-functor-Precategory : UU (l1 ⊔ l2 ⊔ l4)
  is-full-on-isos-functor-Precategory =
    (x y : obj-Precategory C) →
    is-surjective (preserves-iso-functor-Precategory C D F {x} {y})

  is-prop-is-full-on-isos-functor-Precategory :
    is-prop is-full-on-isos-functor-Precategory
  is-prop-is-full-on-isos-functor-Precategory =
    is-prop-iterated-Π 2
      ( λ x y → is-prop-is-surjective (preserves-iso-functor-Precategory C D F))

  is-full-on-isos-prop-functor-Precategory : Prop (l1 ⊔ l2 ⊔ l4)
  pr1 is-full-on-isos-prop-functor-Precategory =
    is-full-on-isos-functor-Precategory
  pr2 is-full-on-isos-prop-functor-Precategory =
    is-prop-is-full-on-isos-functor-Precategory

The predicate of being pseudomonic

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

  is-pseudomonic-prop-functor-Precategory : Prop (l1 ⊔ l2 ⊔ l4)
  is-pseudomonic-prop-functor-Precategory =
    product-Prop
      ( is-faithful-prop-functor-Precategory C D F)
      ( is-full-on-isos-prop-functor-Precategory C D F)

  is-pseudomonic-functor-Precategory : UU (l1 ⊔ l2 ⊔ l4)
  is-pseudomonic-functor-Precategory =
    type-Prop is-pseudomonic-prop-functor-Precategory

  is-prop-is-pseudomonic-functor-Precategory :
    is-prop is-pseudomonic-functor-Precategory
  is-prop-is-pseudomonic-functor-Precategory =
    is-prop-type-Prop is-pseudomonic-prop-functor-Precategory

  is-faithful-is-pseudomonic-functor-Precategory :
    is-pseudomonic-functor-Precategory →
    is-faithful-functor-Precategory C D F
  is-faithful-is-pseudomonic-functor-Precategory = pr1

  is-full-on-isos-is-pseudomonic-functor-Precategory :
    is-pseudomonic-functor-Precategory →
    is-full-on-isos-functor-Precategory C D F
  is-full-on-isos-is-pseudomonic-functor-Precategory = pr2

The type of pseudomonic functors between two precategories

pseudomonic-functor-Precategory :
  {l1 l2 l3 l4 : Level}
  (C : Precategory l1 l2) (D : Precategory l3 l4) →
  UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
pseudomonic-functor-Precategory C D =
  Σ (functor-Precategory C D) (is-pseudomonic-functor-Precategory C D)

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

  functor-pseudomonic-functor-Precategory : functor-Precategory C D
  functor-pseudomonic-functor-Precategory = pr1 F

  is-pseudomonic-pseudomonic-functor-Precategory :
    is-pseudomonic-functor-Precategory C D
      functor-pseudomonic-functor-Precategory
  is-pseudomonic-pseudomonic-functor-Precategory = pr2 F

  obj-pseudomonic-functor-Precategory : obj-Precategory C → obj-Precategory D
  obj-pseudomonic-functor-Precategory =
    obj-functor-Precategory C D functor-pseudomonic-functor-Precategory

  hom-pseudomonic-functor-Precategory :
    {x y : obj-Precategory C} →
    hom-Precategory C x y →
    hom-Precategory D
      ( obj-pseudomonic-functor-Precategory x)
      ( obj-pseudomonic-functor-Precategory y)
  hom-pseudomonic-functor-Precategory =
    hom-functor-Precategory C D functor-pseudomonic-functor-Precategory

  is-faithful-pseudomonic-functor-Precategory :
    is-faithful-functor-Precategory C D functor-pseudomonic-functor-Precategory
  is-faithful-pseudomonic-functor-Precategory =
    is-faithful-is-pseudomonic-functor-Precategory C D
      functor-pseudomonic-functor-Precategory
      is-pseudomonic-pseudomonic-functor-Precategory

  is-full-on-isos-pseudomonic-functor-Precategory :
    is-full-on-isos-functor-Precategory C D
      functor-pseudomonic-functor-Precategory
  is-full-on-isos-pseudomonic-functor-Precategory =
    is-full-on-isos-is-pseudomonic-functor-Precategory C D
      functor-pseudomonic-functor-Precategory
      is-pseudomonic-pseudomonic-functor-Precategory

Properties

Pseudomonic functors are equivalences on isomorphism-sets

module _
  {l1 l2 l3 l4 : Level}
  (C : Precategory l1 l2)
  (D : Precategory l3 l4)
  (F : functor-Precategory C D)
  (is-pseudomonic-F : is-pseudomonic-functor-Precategory C D F)
  {x y : obj-Precategory C}
  where

  is-equiv-preserves-iso-is-pseudomonic-functor-Precategory :
    is-equiv (preserves-iso-functor-Precategory C D F {x} {y})
  is-equiv-preserves-iso-is-pseudomonic-functor-Precategory =
    is-equiv-is-emb-is-surjective
      ( pr2 is-pseudomonic-F x y)
      ( is-faithful-on-isos-is-faithful-functor-Precategory
          C D F (pr1 is-pseudomonic-F) x y)

  equiv-iso-is-pseudomonic-functor-Precategory :
    iso-Precategory C x y ≃
    iso-Precategory D
      ( obj-functor-Precategory C D F x)
      ( obj-functor-Precategory C D F y)
  pr1 equiv-iso-is-pseudomonic-functor-Precategory =
    preserves-iso-functor-Precategory C D F
  pr2 equiv-iso-is-pseudomonic-functor-Precategory =
    is-equiv-preserves-iso-is-pseudomonic-functor-Precategory

  inv-equiv-iso-is-pseudomonic-functor-Precategory :
    iso-Precategory D
      ( obj-functor-Precategory C D F x)
      ( obj-functor-Precategory C D F y) ≃
    iso-Precategory C x y
  inv-equiv-iso-is-pseudomonic-functor-Precategory =
    inv-equiv equiv-iso-is-pseudomonic-functor-Precategory

  map-inv-iso-is-pseudomonic-functor-Precategory :
    iso-Precategory D
      ( obj-functor-Precategory C D F x)
      ( obj-functor-Precategory C D F y) →
    iso-Precategory C x y
  map-inv-iso-is-pseudomonic-functor-Precategory =
    map-equiv inv-equiv-iso-is-pseudomonic-functor-Precategory

The previous entry records what is also known as “essential injectivity”.

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

  equiv-iso-pseudomonic-functor-Precategory :
    iso-Precategory C x y ≃
    iso-Precategory D
      ( obj-pseudomonic-functor-Precategory C D F x)
      ( obj-pseudomonic-functor-Precategory C D F y)
  equiv-iso-pseudomonic-functor-Precategory =
    equiv-iso-is-pseudomonic-functor-Precategory C D
      ( functor-pseudomonic-functor-Precategory C D F)
      ( is-pseudomonic-pseudomonic-functor-Precategory C D F)

  inv-equiv-iso-pseudomonic-functor-Precategory :
    iso-Precategory D
      ( obj-pseudomonic-functor-Precategory C D F x)
      ( obj-pseudomonic-functor-Precategory C D F y) ≃
    iso-Precategory C x y
  inv-equiv-iso-pseudomonic-functor-Precategory =
    inv-equiv equiv-iso-pseudomonic-functor-Precategory

  map-inv-hom-pseudomonic-functor-Precategory :
    iso-Precategory D
      ( obj-pseudomonic-functor-Precategory C D F x)
      ( obj-pseudomonic-functor-Precategory C D F y) →
    iso-Precategory C x y
  map-inv-hom-pseudomonic-functor-Precategory =
    map-equiv inv-equiv-iso-pseudomonic-functor-Precategory

The previous entry records what is also known as “essential injectivity”.

Pseudomonic functors are conservative

module _
  {l1 l2 l3 l4 : Level}
  (C : Precategory l1 l2)
  (D : Precategory l3 l4)
  (F : functor-Precategory C D)
  (is-pseudomonic-F : is-pseudomonic-functor-Precategory C D F)
  {x y : obj-Precategory C}
  where

  is-conservative-is-pseudomonic-functor-Precategory :
    is-conservative-functor-Precategory C D F
  is-conservative-is-pseudomonic-functor-Precategory f is-iso-Ff =
    ind-trunc-Prop
      ( λ _ → is-iso-prop-Precategory C f)
      ( λ ((e , e' , l , r) , p) →
        ( e' ,
          ( inv
            ( ap
              ( λ g → comp-hom-Precategory C g e')
              ( is-injective-is-emb
                ( is-faithful-is-pseudomonic-functor-Precategory
                    ( C) D F is-pseudomonic-F _ _)
                ( ap pr1 p)))) ∙
          ( l) ,
          ( inv
            ( ap
              ( comp-hom-Precategory C e')
              ( is-injective-is-emb
                ( is-faithful-is-pseudomonic-functor-Precategory
                    ( C) D F is-pseudomonic-F _ _)
                ( ap pr1 p)))) ∙
          ( r)))
      ( pr2 is-pseudomonic-F _ _ (hom-functor-Precategory C D F f , is-iso-Ff))

See also

A wikidata identifier was not available for this concept.

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