Weakly constant maps

Content created by Fredrik Bakke, Jonathan Prieto-Cubides and Egbert Rijke.

Created on 2022-02-09.
Last modified on 2023-06-08.

module foundation.weakly-constant-maps where
open import foundation.dependent-pair-types
open import foundation.universe-levels

open import foundation-core.identity-types
open import foundation-core.propositions
open import foundation-core.sets


A map f : A → B is said to be weakly constant if any two elements in A are mapped to identical elements in B.


is-weakly-constant-map :
  {l1 l2 : Level} {A : UU l1} {B : UU l2}  (A  B)  UU (l1  l2)
is-weakly-constant-map {A = A} f = (x y : A)  f x  f y

module _
  {l1 l2 : Level} {A : UU l1} (B : Set l2) (f : A  type-Set B)

    is-prop-is-weakly-constant-map-Set : is-prop (is-weakly-constant-map f)
    is-prop-is-weakly-constant-map-Set =
      is-prop-Π  x  is-prop-Π  y  is-set-type-Set B (f x) (f y)))

  is-weakly-constant-map-Prop : Prop (l1  l2)
  pr1 is-weakly-constant-map-Prop = is-weakly-constant-map f
  pr2 is-weakly-constant-map-Prop = is-prop-is-weakly-constant-map-Set

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