Functoriality of dependent function types

Content created by Fredrik Bakke, Egbert Rijke, Jonathan Prieto-Cubides, Raymond Baker, Elisabeth Stenholm and Vojtěch Štěpančík.

Created on 2022-05-16.
Last modified on 2024-11-05.

module foundation-core.functoriality-dependent-function-types where
Imports 
open import foundation.dependent-pair-types
open import foundation.function-extensionality
open import foundation.implicit-function-types
open import foundation.universe-levels

open import foundation-core.contractible-maps
open import foundation-core.contractible-types
open import foundation-core.equivalences
open import foundation-core.families-of-equivalences
open import foundation-core.fibers-of-maps
open import foundation-core.function-types
open import foundation-core.functoriality-dependent-pair-types
open import foundation-core.homotopies
open import foundation-core.identity-types
open import foundation-core.injective-maps
open import foundation-core.retractions
open import foundation-core.type-theoretic-principle-of-choice

Properties

The operation map-Π preserves homotopies

htpy-map-Π :
  {l1 l2 l3 : Level} {I : UU l1} {A : I → UU l2} {B : I → UU l3}
  {f f' : (i : I) → A i → B i} (H : (i : I) → f i ~ f' i) →
  map-Π f ~ map-Π f'
htpy-map-Π H h = eq-htpy (λ i → H i (h i))

htpy-map-Π' :
  {l1 l2 l3 l4 : Level} {I : UU l1} {A : I → UU l2} {B : I → UU l3}
  {J : UU l4} (α : J → I) {f f' : (i : I) → A i → B i} →
  ((i : I) → (f i) ~ (f' i)) → map-Π' α f ~ map-Π' α f'
htpy-map-Π' α H = htpy-map-Π (H ∘ α)

The operation map-Π preserves composition

module _
  {l1 l2 l3 l4 : Level} {I : UU l1}
  {A : I → UU l2} {B : I → UU l3} {C : I → UU l4}
  where

  preserves-comp-map-Π :
    (g : (i : I) → B i → C i)
    (f : (i : I) → A i → B i) →
    map-Π (λ i → g i ∘ f i) ~ map-Π g ∘ map-Π f
  preserves-comp-map-Π g f = refl-htpy

The operation map-Π preserves identity functions

module _
  {l1 l2 : Level} {I : UU l1} {A : I → UU l2}
  where

  preserves-id-map-Π : map-Π (λ i → id {A = A i}) ~ id
  preserves-id-map-Π = refl-htpy

The fibers of map-Π

compute-fiber-map-Π :
  {l1 l2 l3 : Level} {I : UU l1} {A : I → UU l2} {B : I → UU l3}
  (f : (i : I) → A i → B i) (h : (i : I) → B i) →
  ((i : I) → fiber (f i) (h i)) ≃ fiber (map-Π f) h
compute-fiber-map-Π f h = equiv-tot (λ _ → equiv-eq-htpy) ∘e distributive-Π-Σ

compute-fiber-map-Π' :
  {l1 l2 l3 l4 : Level} {I : UU l1} {A : I → UU l2} {B : I → UU l3}
  {J : UU l4} (α : J → I) (f : (i : I) → A i → B i)
  (h : (j : J) → B (α j)) →
  ((j : J) → fiber (f (α j)) (h j)) ≃ fiber (map-Π' α f) h
compute-fiber-map-Π' α f = compute-fiber-map-Π (f ∘ α)

Families of equivalences induce equivalences of dependent function types

module _
  {l1 l2 l3 : Level} {I : UU l1} {A : I → UU l2} {B : I → UU l3}
  where

  abstract
    is-equiv-map-Π-is-fiberwise-equiv :
      {f : (i : I) → A i → B i} → is-fiberwise-equiv f → is-equiv (map-Π f)
    is-equiv-map-Π-is-fiberwise-equiv is-equiv-f =
      is-equiv-is-contr-map
        ( λ g →
          is-contr-equiv' _
            ( compute-fiber-map-Π _ g)
            ( is-contr-Π (λ i → is-contr-map-is-equiv (is-equiv-f i) (g i))))

  equiv-Π-equiv-family :
    (e : (i : I) → A i ≃ B i) → ((i : I) → A i) ≃ ((i : I) → B i)
  equiv-Π-equiv-family e =
    ( map-Π (λ i → map-equiv (e i)) ,
      is-equiv-map-Π-is-fiberwise-equiv (λ i → is-equiv-map-equiv (e i)))

Families of equivalences induce equivalences of implicit dependent function types

module _
  {l1 l2 l3 : Level} {I : UU l1} {A : I → UU l2} {B : I → UU l3}
  where

  is-equiv-map-implicit-Π-is-fiberwise-equiv :
      {f : (i : I) → A i → B i} → is-fiberwise-equiv f →
      is-equiv (map-implicit-Π f)
  is-equiv-map-implicit-Π-is-fiberwise-equiv is-equiv-f =
    is-equiv-comp _ _
      ( is-equiv-explicit-implicit-Π)
      ( is-equiv-comp _ _
        ( is-equiv-map-Π-is-fiberwise-equiv is-equiv-f)
        ( is-equiv-implicit-explicit-Π))

  equiv-implicit-Π-equiv-family :
    (e : (i : I) → (A i) ≃ (B i)) → ({i : I} → A i) ≃ ({i : I} → B i)
  equiv-implicit-Π-equiv-family e =
    ( equiv-implicit-explicit-Π) ∘e
    ( equiv-Π-equiv-family e) ∘e
    ( equiv-explicit-implicit-Π)
Computing the inverse of equiv-Π-equiv-family
module _
  {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3}
  where

  compute-inv-equiv-Π-equiv-family :
    (e : (x : A) → B x ≃ C x) →
    ( map-inv-equiv (equiv-Π-equiv-family e)) ~
    ( map-equiv (equiv-Π-equiv-family (λ x → inv-equiv (e x))))
  compute-inv-equiv-Π-equiv-family e f =
    is-injective-equiv
      ( equiv-Π-equiv-family e)
      ( ( is-section-map-inv-equiv (equiv-Π-equiv-family e) f) ∙
        ( eq-htpy (λ x → inv (is-section-map-inv-equiv (e x) (f x)))))

Families of retracts induce retracts of dependent function types

module _
  {l1 l2 l3 : Level} {I : UU l1} {A : I → UU l2} {B : I → UU l3}
  where

  abstract
    retraction-map-Π-fiberwise-retraction :
      {f : (i : I) → A i → B i} →
      ((i : I) → retraction (f i)) → retraction (map-Π f)
    retraction-map-Π-fiberwise-retraction {f} r =
      ( ( map-Π (λ i → map-retraction (f i) (r i))) ,
        ( λ h → eq-htpy (λ i → is-retraction-map-retraction (f i) (r i) (h i))))

See also

Recent changes