The structure identity principle

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

Created on 2022-01-29.
Last modified on 2024-01-11.

module foundation.structure-identity-principle where
open import foundation.dependent-pair-types
open import foundation.fundamental-theorem-of-identity-types
open import foundation.type-arithmetic-dependent-pair-types
open import foundation.universe-levels

open import foundation-core.contractible-types
open import foundation-core.equivalences
open import foundation-core.identity-types
open import foundation-core.torsorial-type-families


Structure is presented in type theory by dependent pair types. The structure identity principle is a way to characterize the identity type of a structure, using characterizations of the identity types of its components.


module _
  { l1 l2 l3 l4 : Level} {A : UU l1} {B : A  UU l2} {C : A  UU l3}
  { D : (x : A)  B x  C x  UU l4}

    is-torsorial-Eq-structure :
      (is-contr-AC : is-torsorial C) (t : Σ A C) 
      is-torsorial  y  D (pr1 t) y (pr2 t)) 
      is-torsorial  t  Σ (C (pr1 t)) (D (pr1 t) (pr2 t)))
    is-torsorial-Eq-structure is-contr-AC t is-contr-BD =
        ( Σ (Σ A C)  t  Σ (B (pr1 t))  y  D (pr1 t) y (pr2 t))))
        ( interchange-Σ-Σ D)
        ( is-contr-Σ is-contr-AC t is-contr-BD)


The structure identity principle

module _
  {l1 l2 l3 l4 : Level} { A : UU l1} {B : A  UU l2} {Eq-A : A  UU l3}
  (Eq-B : {x : A}  B x  Eq-A x  UU l4)
  {a : A} {b : B a} (refl-A : Eq-A a) (refl-B : Eq-B b refl-A)

    structure-identity-principle :
      {f : (x : A)  a  x  Eq-A x}
      {g : (y : B a)  b  y  Eq-B y refl-A} 
      (h : (z : Σ A B)  (pair a b)  z  Σ (Eq-A (pr1 z)) (Eq-B (pr2 z))) 
      ((x : A)  is-equiv (f x))  ((y : B a)  is-equiv (g y)) 
      (z : Σ A B)  is-equiv (h z)
    structure-identity-principle {f} {g} h H K =
        ( is-torsorial-Eq-structure
          ( fundamental-theorem-id' f H)
          ( pair a refl-A)
          ( fundamental-theorem-id' g K))
        ( h)

  map-extensionality-Σ :
    (f : (x : A)  (a  x)  Eq-A x)
    (g : (y : B a)  (b  y)  Eq-B y refl-A) 
    (z : Σ A B)  pair a b  z  Σ (Eq-A (pr1 z)) (Eq-B (pr2 z))
  pr1 (map-extensionality-Σ f g .(pair a b) refl) = refl-A
  pr2 (map-extensionality-Σ f g .(pair a b) refl) = refl-B

  extensionality-Σ :
    (f : (x : A)  (a  x)  Eq-A x)
    (g : (y : B a)  (b  y)  Eq-B y refl-A) 
    (z : Σ A B)  (pair a b  z)  Σ (Eq-A (pr1 z)) (Eq-B (pr2 z))
  pr1 (extensionality-Σ f g z) = map-extensionality-Σ f g z
  pr2 (extensionality-Σ f g z) =
      ( map-extensionality-Σ f g)
      ( λ x  is-equiv-map-equiv (f x))
      ( λ y  is-equiv-map-equiv (g y))
      ( z)

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