The action on equivalences of functions out of subuniverses

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-09-12.
Last modified on 2024-04-25.

module foundation.action-on-equivalences-functions-out-of-subuniverses where
open import foundation.action-on-higher-identifications-functions
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.equivalence-induction
open import foundation.subuniverses
open import foundation.universe-levels

open import foundation-core.contractible-types
open import foundation-core.equivalences
open import foundation-core.function-types
open import foundation-core.identity-types


Consider a subuniverse P of 𝒰 and a map f : P → B from the subuniverse P into some type B. Then f has an action on equivalences

  (X ≃ Y) → (f X = f Y)

which is uniquely determined by the identification action-equiv f id-equiv = refl. The action on equivalences fits in a commuting square of maps

                     ap f
       (X = Y) ---------------> (f X = f Y)
          |                          |
 equiv-eq |                          | id
          ∨                          ∨
       (X ≃ Y) ---------------> (f X = f Y)
                action-equiv f


module _
  {l1 l2 l3 : Level} (P : subuniverse l1 l2)
  {B : UU l3} (f : type-subuniverse P  B)

    unique-action-equiv-function-subuniverse :
      (X : type-subuniverse P) 
        ( Σ ( (Y : type-subuniverse P)  equiv-subuniverse P X Y  f X  f Y)
            ( λ h  h X id-equiv  refl))
    unique-action-equiv-function-subuniverse X =
      is-contr-map-ev-id-equiv-subuniverse P X
        ( λ Y e  f X  f Y)
        ( refl)

  action-equiv-function-subuniverse :
    (X Y : type-subuniverse P)  equiv-subuniverse P X Y  f X  f Y
  action-equiv-function-subuniverse X Y =
    ap f  eq-equiv-subuniverse P

  compute-action-equiv-function-subuniverse-id-equiv :
    (X : type-subuniverse P) 
    action-equiv-function-subuniverse X X id-equiv  refl
  compute-action-equiv-function-subuniverse-id-equiv X =
    ap² f (compute-eq-equiv-id-equiv-subuniverse P)

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