Equality of dependent pair types

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

Created on 2022-02-07.
Last modified on 2024-02-06.

module foundation-core.equality-dependent-pair-types where
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.universe-levels

open import foundation-core.dependent-identifications
open import foundation-core.equivalences
open import foundation-core.function-types
open import foundation-core.homotopies
open import foundation-core.identity-types
open import foundation-core.transport-along-identifications


An identification (pair x y) = (pair x' y') in a dependent pair type Σ A B is equivalently described as a pair pair α β consisting of an identification α : x = x' and an identification β : (tr B α y) = y'.


module _
  {l1 l2 : Level} {A : UU l1} {B : A  UU l2}

  Eq-Σ : (s t : Σ A B)  UU (l1  l2)
  Eq-Σ s t =
    Σ (pr1 s  pr1 t)  α  dependent-identification B α (pr2 s) (pr2 t))


The type Id s t is equivalent to Eq-Σ s t for any s t : Σ A B

  refl-Eq-Σ : (s : Σ A B)  Eq-Σ s s
  pr1 (refl-Eq-Σ (pair a b)) = refl
  pr2 (refl-Eq-Σ (pair a b)) = refl

  pair-eq-Σ : {s t : Σ A B}  s  t  Eq-Σ s t
  pair-eq-Σ {s} refl = refl-Eq-Σ s

  eq-pair-eq-base :
    {x y : A} {s : B x} (p : x  y)  (x , s)  (y , tr B p s)
  eq-pair-eq-base refl = refl

  eq-pair-eq-base' :
    {x y : A} {t : B y} (p : x  y)  (x , tr B (inv p) t)  (y , t)
  eq-pair-eq-base' refl = refl

  eq-pair-eq-fiber :
    {x : A} {s t : B x}  s  t  (x , s)  (x , t)
  eq-pair-eq-fiber {x} = ap {B = Σ A B} (pair x)

  eq-pair-Σ :
    {s t : Σ A B}
    (α : pr1 s  pr1 t) 
    dependent-identification B α (pr2 s) (pr2 t)  s  t
  eq-pair-Σ refl = eq-pair-eq-fiber

  eq-pair-Σ' : {s t : Σ A B}  Eq-Σ s t  s  t
  eq-pair-Σ' p = eq-pair-Σ (pr1 p) (pr2 p)

  ap-pr1-eq-pair-eq-fiber :
    {x : A} {s t : B x} (p : s  t)  ap pr1 (eq-pair-eq-fiber p)  refl
  ap-pr1-eq-pair-eq-fiber refl = refl

  is-retraction-pair-eq-Σ :
    (s t : Σ A B)  pair-eq-Σ {s} {t}  eq-pair-Σ' {s} {t} ~ id {A = Eq-Σ s t}
  is-retraction-pair-eq-Σ (pair x y) (pair .x .y) (pair refl refl) = refl

  is-section-pair-eq-Σ :
    (s t : Σ A B)  ((eq-pair-Σ' {s} {t})  (pair-eq-Σ {s} {t})) ~ id
  is-section-pair-eq-Σ (pair x y) .(pair x y) refl = refl

    is-equiv-eq-pair-Σ : (s t : Σ A B)  is-equiv (eq-pair-Σ' {s} {t})
    is-equiv-eq-pair-Σ s t =
        ( pair-eq-Σ)
        ( is-section-pair-eq-Σ s t)
        ( is-retraction-pair-eq-Σ s t)

  equiv-eq-pair-Σ : (s t : Σ A B)  Eq-Σ s t  (s  t)
  pr1 (equiv-eq-pair-Σ s t) = eq-pair-Σ'
  pr2 (equiv-eq-pair-Σ s t) = is-equiv-eq-pair-Σ s t

    is-equiv-pair-eq-Σ : (s t : Σ A B)  is-equiv (pair-eq-Σ {s} {t})
    is-equiv-pair-eq-Σ s t =
        ( eq-pair-Σ')
        ( is-retraction-pair-eq-Σ s t)
        ( is-section-pair-eq-Σ s t)

  equiv-pair-eq-Σ : (s t : Σ A B)  (s  t)  Eq-Σ s t
  pr1 (equiv-pair-eq-Σ s t) = pair-eq-Σ
  pr2 (equiv-pair-eq-Σ s t) = is-equiv-pair-eq-Σ s t

  η-pair : (t : Σ A B)  (pair (pr1 t) (pr2 t))  t
  η-pair t = refl

  eq-base-eq-pair-Σ : {s t : Σ A B}  (s  t)  (pr1 s  pr1 t)
  eq-base-eq-pair-Σ p = pr1 (pair-eq-Σ p)

  dependent-eq-family-eq-pair-Σ :
    {s t : Σ A B}  (p : s  t) 
    dependent-identification B (eq-base-eq-pair-Σ p) (pr2 s) (pr2 t)
  dependent-eq-family-eq-pair-Σ p = pr2 (pair-eq-Σ p)

Lifting equality to the total space

module _
  {l1 l2 : Level} {A : UU l1} {B : A  UU l2}

  lift-eq-Σ :
    {x y : A} (p : x  y) (b : B x)  (pair x b)  (pair y (tr B p b))
  lift-eq-Σ refl b = refl

Transport in a family of dependent pair types

tr-Σ :
  {l1 l2 l3 : Level} {A : UU l1} {a0 a1 : A} {B : A  UU l2}
  (C : (x : A)  B x  UU l3) (p : a0  a1) (z : Σ (B a0)  x  C a0 x)) 
  tr  a  (Σ (B a) (C a))) p z 
  pair (tr B p (pr1 z)) (tr (ind-Σ C) (eq-pair-Σ p refl) (pr2 z))
tr-Σ C refl z = refl

Transport in a family over a dependent pair type

tr-eq-pair-Σ :
  {l1 l2 l3 : Level} {A : UU l1} {a0 a1 : A}
  {B : A  UU l2} {b0 : B a0} {b1 : B a1} (C : (Σ A B)  UU l3)
  (p : a0  a1) (q : dependent-identification B p b0 b1) (u : C (a0 , b0)) 
  tr C (eq-pair-Σ p q) u 
  tr  x  C (a1 , x)) q (tr C (eq-pair-Σ p refl) u)
tr-eq-pair-Σ C refl refl u = refl

See also

Recent changes