The universal property of truncations

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

Created on 2022-03-08.
Last modified on 2025-11-10.

module foundation.universal-property-truncation where

open import foundation-core.universal-property-truncation public
Imports 
open import foundation.action-on-identifications-functions
open import foundation.contractible-types
open import foundation.dependent-pair-types
open import foundation.function-extensionality
open import foundation.identity-types
open import foundation.propositional-truncations
open import foundation.surjective-maps
open import foundation.truncations
open import foundation.type-arithmetic-dependent-function-types
open import foundation.universal-property-dependent-pair-types
open import foundation.universal-property-identity-types
open import foundation.universe-levels

open import foundation-core.contractible-maps
open import foundation-core.equivalences
open import foundation-core.fibers-of-maps
open import foundation-core.functoriality-dependent-function-types
open import foundation-core.functoriality-dependent-pair-types
open import foundation-core.torsorial-type-families
open import foundation-core.truncated-types
open import foundation-core.truncation-levels
open import foundation-core.type-theoretic-principle-of-choice

Properties

A map f : A → B is a k+1-truncation if and only if it is surjective and ap f : (x ＝ y) → (f x ＝ f y) is a k-truncation for all x y : A

module _
  {l1 l2 : Level} {k : 𝕋} {A : UU l1} (B : Truncated-Type l2 (succ-𝕋 k))
  {f : A → type-Truncated-Type B} (H : is-surjective f)
  ( K :
    (x y : A) → is-truncation (Id-Truncated-Type B (f x) (f y)) (ap f {x} {y}))
  where

  unique-extension-fiber-is-truncation-is-truncation-ap :
    {l : Level} (C : Truncated-Type l (succ-𝕋 k))
    (g : A → type-Truncated-Type C) (y : type-Truncated-Type B) →
    is-contr
      ( Σ ( type-Truncated-Type C)
          ( λ z → (t : fiber f y) → g (pr1 t) ＝ z))
  unique-extension-fiber-is-truncation-is-truncation-ap C g =
    apply-dependent-universal-property-surjection-is-surjective f H
      ( λ y → is-contr-Prop _)
      ( λ x →
        is-contr-equiv
          ( Σ (type-Truncated-Type C) (λ z → g x ＝ z))
          ( equiv-tot
            ( λ z →
              ( equiv-ev-refl' x) ∘e
              ( equiv-Π-equiv-family
                ( λ x' →
                  equiv-is-truncation
                    ( Id-Truncated-Type B (f x') (f x))
                    ( ap f)
                    ( K x' x)
                    ( Id-Truncated-Type C (g x') z))) ∘e
              ( equiv-ev-pair)))
          ( is-torsorial-Id (g x)))

  is-truncation-is-truncation-ap :
    is-truncation B f
  is-truncation-is-truncation-ap C =
    is-equiv-is-contr-map
      ( λ g →
        is-contr-equiv'
          ( (y : type-Truncated-Type B) →
            Σ ( type-Truncated-Type C)
              ( λ z → (t : fiber f y) → (g (pr1 t) ＝ z)))
          ( ( equiv-tot
              ( λ h →
                ( equiv-eq-htpy) ∘e
                ( equiv-Π-equiv-family
                  ( λ x → equiv-inv (g x) (h (f x)) ∘e equiv-ev-refl (f x))) ∘e
                ( equiv-swap-Π) ∘e
                ( equiv-Π-equiv-family (λ x → equiv-ev-pair)))) ∘e
            ( distributive-Π-Σ))
          ( is-contr-Π
            ( unique-extension-fiber-is-truncation-is-truncation-ap C g)))

module _
  {l1 l2 : Level} {k : 𝕋} {A : UU l1} (B : Truncated-Type l2 (succ-𝕋 k))
  {f : A → type-Truncated-Type B}
  where

  is-surjective-is-truncation :
    is-truncation B f → is-surjective f
  is-surjective-is-truncation H =
    map-inv-is-equiv
      ( dependent-universal-property-truncation-is-truncation B f H
        ( λ y → truncated-type-trunc-Prop k (fiber f y)))
      ( λ x → unit-trunc-Prop (x , refl))

is-surjective-unit-trunc-succ :
  {l : Level} {k : 𝕋} {A : UU l} → is-surjective (unit-trunc {k = succ-𝕋 k} {A})
is-surjective-unit-trunc-succ {k = k} {A} =
  is-surjective-is-truncation (trunc (succ-𝕋 k) A) is-truncation-trunc

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