The universal property of the natural numbers

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

Created on 2022-02-08.
Last modified on 2024-02-06.

module elementary-number-theory.universal-property-natural-numbers where
open import elementary-number-theory.natural-numbers

open import foundation.action-on-identifications-functions
open import foundation.cartesian-product-types
open import foundation.contractible-types
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.function-types
open import foundation.fundamental-theorem-of-identity-types
open import foundation.homotopies
open import foundation.homotopy-induction
open import foundation.identity-types
open import foundation.structure-identity-principle
open import foundation.torsorial-type-families
open import foundation.universe-levels
open import foundation.whiskering-identifications-concatenation


The universal property of the natural numbers asserts that for any type X equipped with a point x : X and an endomorphism f : X → X, the type of structure preserving maps from to X is contractible.

module _
  {l : Level} {X : UU l} (x : X) (f : X  X)

  structure-preserving-map-ℕ : UU l
  structure-preserving-map-ℕ =
    Σ (  X)  h  (h zero-ℕ  x) × ((h  succ-ℕ) ~ (f  h)))

  htpy-structure-preserving-map-ℕ :
    (h k : structure-preserving-map-ℕ)  UU l
  htpy-structure-preserving-map-ℕ h k =
    Σ ( pr1 h ~ pr1 k)
      ( λ H 
        ( pr1 (pr2 h)  (H zero-ℕ  pr1 (pr2 k))) ×
        ( (n : ) 
          (pr2 (pr2 h) n  ap f (H n))  (H (succ-ℕ n)  pr2 (pr2 k) n)))

  refl-htpy-structure-preserving-map-ℕ :
    (h : structure-preserving-map-ℕ)  htpy-structure-preserving-map-ℕ h h
  refl-htpy-structure-preserving-map-ℕ h =
    triple refl-htpy refl  n  right-unit)

  htpy-eq-structure-preserving-map-ℕ :
    {h k : structure-preserving-map-ℕ}  h  k 
    htpy-structure-preserving-map-ℕ h k
  htpy-eq-structure-preserving-map-ℕ {h} refl =
    refl-htpy-structure-preserving-map-ℕ h

  is-torsorial-htpy-structure-preserving-map-ℕ :
    (h : structure-preserving-map-ℕ) 
    is-torsorial (htpy-structure-preserving-map-ℕ h)
  is-torsorial-htpy-structure-preserving-map-ℕ h =
      ( is-torsorial-htpy (pr1 h))
      ( pair (pr1 h) refl-htpy)
      ( is-torsorial-Eq-structure
        ( is-torsorial-Id (pr1 (pr2 h)))
        ( pair (pr1 (pr2 h)) refl)
        ( is-torsorial-htpy  n  (pr2 (pr2 h) n  refl))))

  is-equiv-htpy-eq-structure-preserving-map-ℕ :
    (h k : structure-preserving-map-ℕ) 
    is-equiv (htpy-eq-structure-preserving-map-ℕ {h} {k})
  is-equiv-htpy-eq-structure-preserving-map-ℕ h =
      ( is-torsorial-htpy-structure-preserving-map-ℕ h)
      ( λ k  htpy-eq-structure-preserving-map-ℕ {h} {k})

  eq-htpy-structure-preserving-map-ℕ :
    {h k : structure-preserving-map-ℕ} 
    htpy-structure-preserving-map-ℕ h k  h  k
  eq-htpy-structure-preserving-map-ℕ {h} {k} =
    map-inv-is-equiv (is-equiv-htpy-eq-structure-preserving-map-ℕ h k)

  center-structure-preserving-map-ℕ : structure-preserving-map-ℕ
  pr1 center-structure-preserving-map-ℕ =
    h :   X
    h zero-ℕ = x
    h (succ-ℕ n) = f (h n)
  pr1 (pr2 center-structure-preserving-map-ℕ) = refl
  pr2 (pr2 center-structure-preserving-map-ℕ) = refl-htpy

  contraction-structure-preserving-map-ℕ :
    (h : structure-preserving-map-ℕ)  center-structure-preserving-map-ℕ  h
  contraction-structure-preserving-map-ℕ h =
    eq-htpy-structure-preserving-map-ℕ (triple α β γ)
    α : pr1 center-structure-preserving-map-ℕ ~ pr1 h
    α zero-ℕ = inv (pr1 (pr2 h))
    α (succ-ℕ n) = ap f (α n)  inv (pr2 (pr2 h) n)
    β : pr1 (pr2 center-structure-preserving-map-ℕ)  (α zero-ℕ  pr1 (pr2 h))
    β = inv (left-inv (pr1 (pr2 h)))
    γ :
      (n : ) 
      ( pr2 (pr2 center-structure-preserving-map-ℕ) n  ap f (α n)) 
      ( α (succ-ℕ n)  pr2 (pr2 h) n)
    γ n = ( ( inv right-unit) 
            ( left-whisker-concat
              ( ap f (α n))
              ( inv (left-inv (pr2 (pr2 h) n))))) 
          ( inv
            ( assoc (ap f (α n)) (inv (pr2 (pr2 h) n)) (pr2 (pr2 h) n)))

  is-contr-structure-preserving-map-ℕ : is-contr structure-preserving-map-ℕ
  pr1 is-contr-structure-preserving-map-ℕ = center-structure-preserving-map-ℕ
  pr2 is-contr-structure-preserving-map-ℕ =

Recent changes