Functoriality of propositional truncations

Content created by Fredrik Bakke, Egbert Rijke, Jonathan Prieto-Cubides and Elisabeth Stenholm.

Created on 2022-02-08.
Last modified on 2024-04-11.

module foundation.functoriality-propositional-truncation where
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.function-extensionality
open import foundation.logical-equivalences
open import foundation.propositional-truncations
open import foundation.universe-levels
open import foundation.whiskering-homotopies-composition

open import foundation-core.contractible-types
open import foundation-core.equivalences
open import foundation-core.function-types
open import foundation-core.homotopies
open import foundation-core.propositions


The universal property of propositional truncations can be used to define the functorial action of propositional truncations.


  unique-map-trunc-Prop :
    {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A  B) 
      ( Σ ( type-hom-Prop (trunc-Prop A) (trunc-Prop B))
          ( λ h  (h  unit-trunc-Prop) ~ (unit-trunc-Prop  f)))
  unique-map-trunc-Prop {l1} {l2} {A} {B} f =
    universal-property-trunc-Prop A (trunc-Prop B) (unit-trunc-Prop  f)

  map-trunc-Prop :
    {l1 l2 : Level} {A : UU l1} {B : UU l2} 
    (A  B)  type-hom-Prop (trunc-Prop A) (trunc-Prop B)
  map-trunc-Prop f =
    pr1 (center (unique-map-trunc-Prop f))


Propositional truncations of homotopic maps are homotopic

  htpy-map-trunc-Prop :
    { l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A  B) 
    ( (map-trunc-Prop f)  unit-trunc-Prop) ~ (unit-trunc-Prop  f)
  htpy-map-trunc-Prop f =
    pr2 (center (unique-map-trunc-Prop f))

  htpy-uniqueness-map-trunc-Prop :
    { l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A  B) 
    ( h : type-hom-Prop (trunc-Prop A) (trunc-Prop B)) 
    ( ( h  unit-trunc-Prop) ~ (unit-trunc-Prop  f)) 
    (map-trunc-Prop f) ~ h
  htpy-uniqueness-map-trunc-Prop f h H =
    htpy-eq (ap pr1 (contraction (unique-map-trunc-Prop f) (pair h H)))

The propositional truncation of the identity map is the identity map

  id-map-trunc-Prop :
    { l1 : Level} {A : UU l1}  map-trunc-Prop (id {A = A}) ~ id
  id-map-trunc-Prop {l1} {A} =
    htpy-uniqueness-map-trunc-Prop id id refl-htpy

The propositional truncation of a composite is the composite of the propositional truncations

  preserves-comp-map-trunc-Prop :
    { l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3}
    ( g : B  C) (f : A  B) 
    ( map-trunc-Prop (g  f)) ~
    ( (map-trunc-Prop g)  (map-trunc-Prop f))
  preserves-comp-map-trunc-Prop g f =
      ( g  f)
      ( (map-trunc-Prop g)  (map-trunc-Prop f))
      ( ( (map-trunc-Prop g) ·l (htpy-map-trunc-Prop f)) ∙h
        ( ( htpy-map-trunc-Prop g) ·r f))

The functorial action of propositional truncations preserves equivalences

  map-equiv-trunc-Prop :
    {l1 l2 : Level} {A : UU l1} {B : UU l2} 
    (A  B)  type-trunc-Prop A  type-trunc-Prop B
  map-equiv-trunc-Prop e = map-trunc-Prop (map-equiv e)

  map-inv-equiv-trunc-Prop :
    {l1 l2 : Level} {A : UU l1} {B : UU l2} 
    (A  B)  type-trunc-Prop B  type-trunc-Prop A
  map-inv-equiv-trunc-Prop e = map-equiv-trunc-Prop (inv-equiv e)

  equiv-trunc-Prop :
    {l1 l2 : Level} {A : UU l1} {B : UU l2} 
    (A  B)  (type-trunc-Prop A  type-trunc-Prop B)
  pr1 (equiv-trunc-Prop e) = map-equiv-trunc-Prop e
  pr2 (equiv-trunc-Prop e) =
      ( is-prop-type-trunc-Prop)
      ( is-prop-type-trunc-Prop)
      ( map-inv-equiv-trunc-Prop e)

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