Functoriality of function types

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

Created on 2022-02-08.
Last modified on 2023-09-13.

module foundation.functoriality-function-types where

open import foundation-core.functoriality-function-types public
Imports 
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.functoriality-dependent-function-types
open import foundation.unit-type
open import foundation.universe-levels

open import foundation-core.constant-maps
open import foundation-core.embeddings
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.propositional-maps
open import foundation-core.truncated-maps
open import foundation-core.truncation-levels

Properties

Equivalent types have equivalent function types

module _
  { l1 l2 l3 l4 : Level}
  { A' : UU l1} {B' : UU l2} {A : UU l3} (B : UU l4)
  ( e : A' ≃ A) (f : B' ≃ B)
  where

  map-equiv-function-type : (A' → B') → (A → B)
  map-equiv-function-type h = map-equiv f ∘ (h ∘ map-inv-equiv e)

  compute-map-equiv-function-type :
    (h : A' → B') (x : A') →
    map-equiv-function-type h (map-equiv e x) ＝ map-equiv f (h x)
  compute-map-equiv-function-type h x =
    ap (map-equiv f ∘ h) (is-retraction-map-inv-equiv e x)

  is-equiv-map-equiv-function-type : is-equiv map-equiv-function-type
  is-equiv-map-equiv-function-type =
    is-equiv-comp
      ( precomp (map-equiv (inv-equiv e)) B)
      ( postcomp A' (map-equiv f))
      ( is-equiv-postcomp-equiv f A')
      ( is-equiv-precomp-equiv (inv-equiv e) B)

  equiv-function-type : (A' → B') ≃ (A → B)
  pr1 equiv-function-type = map-equiv-function-type
  pr2 equiv-function-type = is-equiv-map-equiv-function-type

Two maps being homotopic is equivalent to them being homotopic after pre- or postcomposition by an equivalence

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

  equiv-htpy-precomp-htpy :
    ( f g : B → C) (e : A ≃ B) →
    ( (f ∘ map-equiv e) ~ (g ∘ map-equiv e)) ≃
    ( f ~ g)
  equiv-htpy-precomp-htpy f g e =
    equiv-Π
      ( eq-value f g)
      ( e)
      ( λ a → id-equiv)

  equiv-htpy-postcomp-htpy :
    ( e : B ≃ C) (f g : A → B) →
    ( f ~ g) ≃
    ( (map-equiv e ∘ f) ~ (map-equiv e ∘ g))
  equiv-htpy-postcomp-htpy e f g =
    equiv-Π-equiv-family
      ( λ a → equiv-ap e (f a) (g a))

A map is truncated iff postcomposition by it is truncated

module _
  {l1 l2 : Level} (k : 𝕋) {X : UU l1} {Y : UU l2} (f : X → Y)
  where

  is-trunc-map-postcomp-is-trunc-map :
    is-trunc-map k f →
    {l3 : Level} (A : UU l3) → is-trunc-map k (postcomp A f)
  is-trunc-map-postcomp-is-trunc-map is-trunc-f A =
    is-trunc-map-map-Π-is-trunc-map' k
      ( const A unit star)
      ( const unit (X → Y) f)
      ( const unit (is-trunc-map k f) is-trunc-f)

  is-trunc-map-is-trunc-map-postcomp :
    ({l3 : Level} (A : UU l3) → is-trunc-map k (postcomp A f)) →
    is-trunc-map k f
  is-trunc-map-is-trunc-map-postcomp is-trunc-postcomp-f =
    is-trunc-map-is-trunc-map-map-Π' k
      ( const unit (X → Y) f)
      ( λ {l} {J} α → is-trunc-postcomp-f {l} J)
      ( star)

module _
  {l1 l2 : Level} {X : UU l1} {Y : UU l2} (f : X → Y)
  where

  is-emb-postcomp-is-emb :
    is-emb f →
    {l3 : Level} (A : UU l3) → is-emb (postcomp A f)
  is-emb-postcomp-is-emb is-emb-f A =
    is-emb-is-prop-map
      ( is-trunc-map-postcomp-is-trunc-map neg-one-𝕋 f
        ( is-prop-map-is-emb is-emb-f)
        ( A))

  is-emb-is-emb-postcomp :
    ({l3 : Level} (A : UU l3) → is-emb (postcomp A f)) →
    is-emb f
  is-emb-is-emb-postcomp is-emb-postcomp-f =
    is-emb-is-prop-map
      ( is-trunc-map-is-trunc-map-postcomp neg-one-𝕋 f
        ( is-prop-map-is-emb ∘ is-emb-postcomp-f))

emb-postcomp :
  {l1 l2 l3 : Level} {X : UU l1} {Y : UU l2} (f : X ↪ Y) (A : UU l3) →
  (A → X) ↪ (A → Y)
pr1 (emb-postcomp f A) = postcomp A (map-emb f)
pr2 (emb-postcomp f A) = is-emb-postcomp-is-emb (map-emb f) (is-emb-map-emb f) A

See also

Recent changes