Functoriality of function types

Content created by Fredrik Bakke, Egbert Rijke, Jonathan Prieto-Cubides, Tom de Jong, Elisabeth Stenholm and Vojtěch Štěpančík.

Created on 2022-02-08.
Last modified on 2024-03-02.

module foundation.functoriality-function-types where
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.functoriality-dependent-function-types
open import foundation.postcomposition-functions
open import foundation.unit-type
open import foundation.universal-property-equivalences
open import foundation.universe-levels

open import foundation-core.embeddings
open import foundation-core.equivalences
open import foundation-core.function-types
open import foundation-core.identity-types
open import foundation-core.precomposition-functions
open import foundation-core.propositional-maps
open import foundation-core.truncated-maps
open import foundation-core.truncation-levels


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)

  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 =
      ( 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

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)

  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
      ( terminal-map A)
      ( point f)
      ( point 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
      ( point 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)

  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-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-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

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