Constant type families

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-06-10.
Last modified on 2024-06-04.

module foundation.constant-type-families where
open import foundation.action-on-identifications-dependent-functions
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.transport-along-identifications
open import foundation.universe-levels

open import foundation-core.commuting-squares-of-identifications
open import foundation-core.dependent-identifications
open import foundation-core.equivalences


A type family B over A is said to be constant, if there is a type X equipped with a family of equivalences X ≃ B a indexed by a : A.

The standard constant type family over A with fiber B is the constant map const A 𝒰 B : A → 𝒰, where 𝒰 is a universe containing B.


The predicate of being a constant type family

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

  is-constant-type-family : UU (l1  lsuc l2)
  is-constant-type-family = Σ (UU l2)  X  (a : A)  X  B a)

  module _
    (H : is-constant-type-family)

    type-is-constant-type-family : UU l2
    type-is-constant-type-family = pr1 H

    equiv-is-constant-type-family : (a : A)  type-is-constant-type-family  B a
    equiv-is-constant-type-family = pr2 H

The (standard) constant type family

constant-type-family : {l1 l2 : Level} (A : UU l1) (B : UU l2)  A  UU l2
constant-type-family A B a = B

is-constant-type-family-constant-type-family :
  {l1 l2 : Level} (A : UU l1) (B : UU l2) 
  is-constant-type-family (constant-type-family A B)
pr1 (is-constant-type-family-constant-type-family A B) = B
pr2 (is-constant-type-family-constant-type-family A B) a = id-equiv


Transport in a standard constant type family

tr-constant-type-family :
  {l1 l2 : Level} {A : UU l1} {B : UU l2} {x y : A} (p : x  y) (b : B) 
  dependent-identification (constant-type-family A B) p b b
tr-constant-type-family refl b = refl

Computing dependent identifications in constant type families

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

  map-compute-dependent-identification-constant-type-family :
    {x y : A} (p : x  y) {x' y' : B} 
    x'  y'  dependent-identification  _  B) p x' y'
  map-compute-dependent-identification-constant-type-family p {x'} q =
    tr-constant-type-family p x'  q

  compute-dependent-identification-constant-type-family :
    {x y : A} (p : x  y) {x' y' : B} 
    (x'  y')  dependent-identification  _  B) p x' y'
  compute-dependent-identification-constant-type-family p {x'} {y'} =
    equiv-concat (tr-constant-type-family p x') y'

Dependent action on paths of sections of standard constant type families

apd-constant-type-family :
  {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A  B) {x y : A} (p : x  y) 
  apd f p  tr-constant-type-family p (f x)  ap f p
apd-constant-type-family f refl = refl

Naturality of transport in constant type families

For every equality p : x = x' in A and q : y = y' in B we have a commuting square of identifications

                    ap (tr (λ _ → B) p) q
          tr (λ _ → B) p y ------> tr (λ _ → B) p y'
                         |         |
  tr-constant-family p y |         | tr-constant-family p y'
                         ∨         ∨
                         y ------> y'.
module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2}

  naturality-tr-constant-type-family :
    {x x' : A} (p : x  x') {y y' : B} (q : y  y') 
      ( ap (tr  _  B) p) q)
      ( tr-constant-type-family p y)
      ( tr-constant-type-family p y')
      ( q)
  naturality-tr-constant-type-family p refl = right-unit

  naturality-inv-tr-constant-type-family :
    {x x' : A} (p : x  x') {y y' : B} (q : y  y') 
      ( q)
      ( inv (tr-constant-type-family p y))
      ( inv (tr-constant-type-family p y'))
      ( ap (tr  _  B) p) q)
  naturality-inv-tr-constant-type-family p refl = right-unit

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