The action on identifications of functions

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-06-10.
Last modified on 2023-09-11.

module foundation.action-on-identifications-functions where
Imports 
open import foundation.universe-levels

open import foundation-core.constant-maps
open import foundation-core.function-types
open import foundation-core.identity-types
open import foundation-core.transport-along-identifications

Idea

Any function f : A → B preserves identifications, in the sense that it maps identifications p : x ＝ y in A to an identification ap f p : f x ＝ f y in B. This action on identifications can be thought of as the functoriality of identity types.

Definition

The functorial action of functions on identity types

ap :
  {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) {x y : A} →
  x ＝ y → (f x) ＝ (f y)
ap f refl = refl

Properties

The identity function acts trivially on identifications

ap-id :
  {l : Level} {A : UU l} {x y : A} (p : x ＝ y) → (ap id p) ＝ p
ap-id refl = refl

The action on identifications of a composite function is the composite of the actions

ap-comp :
  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} (g : B → C)
  (f : A → B) {x y : A} (p : x ＝ y) → (ap (g ∘ f) p) ＝ ((ap g ∘ ap f) p)
ap-comp g f refl = refl

The action on identifications of any map preserves refl

ap-refl :
  {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) (x : A) →
  (ap f (refl {x = x})) ＝ refl
ap-refl f x = refl

The action on identifications of any map preserves concatenation of identifications

ap-concat :
  {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) {x y z : A}
  (p : x ＝ y) (q : y ＝ z) → (ap f (p ∙ q)) ＝ ((ap f p) ∙ (ap f q))
ap-concat f refl q = refl

ap-concat-eq :
  {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) {x y z : A}
  (p : x ＝ y) (q : y ＝ z) (r : x ＝ z)
  (H : r ＝ (p ∙ q)) → (ap f r) ＝ ((ap f p) ∙ (ap f q))
ap-concat-eq f p q .(p ∙ q) refl = ap-concat f p q

The action on identifications of any map preserves inverses

ap-inv :
  {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) {x y : A}
  (p : x ＝ y) → ap f (inv p) ＝ inv (ap f p)
ap-inv f refl = refl

The action on identifications of a constant map is constant

ap-const :
  {l1 l2 : Level} {A : UU l1} {B : UU l2} (b : B) {x y : A}
  (p : x ＝ y) → (ap (const A B b) p) ＝ refl
ap-const b refl = refl

Transporting along the action on identifications of a function

tr-ap :
  {l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} {C : UU l3} {D : C → UU l4}
  (f : A → C) (g : (x : A) → B x → D (f x))
  {x y : A} (p : x ＝ y) (z : B x) →
  tr D (ap f p) (g x z) ＝ g y (tr B p z)
tr-ap f g refl z = refl

The action on identifications of concatenating by refl on the right

Note that _∙ refl is only homotopic to the identity function. Therefore we will compute here the action on identifications of the map _∙ refl.

inv-ap-refl-concat :
  {l : Level} {A : UU l} {x y : A} {p q : x ＝ y} (r : p ＝ q) →
  (right-unit ∙ (r ∙ inv right-unit)) ＝ (ap (_∙ refl) r)
inv-ap-refl-concat refl = right-inv right-unit

ap-refl-concat :
  {l : Level} {A : UU l} {x y : A} {p q : x ＝ y} (r : p ＝ q) →
  (ap (_∙ refl) r) ＝ (right-unit ∙ (r ∙ inv right-unit))
ap-refl-concat = inv ∘ inv-ap-refl-concat

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