Split surjective maps

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

Created on 2022-01-26.
Last modified on 2023-09-06.

module foundation.split-surjective-maps where
open import foundation.dependent-pair-types
open import foundation.type-theoretic-principle-of-choice
open import foundation.universe-levels

open import foundation-core.cartesian-product-types
open import foundation-core.equivalences
open import foundation-core.fibers-of-maps
open import foundation-core.function-types
open import foundation-core.injective-maps
open import foundation-core.retractions
open import foundation-core.sections


A map f : A → B is split surjective if we can construct for every b : B an element in the fiber of b, meaning an element a : A equipped with an identification f a = b.


Note that split-surjectiveness is the Curry-Howard interpretation of surjectiveness. However, this is not a property, and the split surjective maps don't fit in a factorization system along with the injective maps.


Split surjective maps

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2}

  is-split-surjective : (A  B)  UU (l1  l2)
  is-split-surjective f = (b : B)  fiber f b

  split-surjection : UU (l1  l2)
  split-surjection = Σ (A  B) is-split-surjective

  map-split-surjection : split-surjection  (A  B)
  map-split-surjection = pr1

  is-split-surjective-split-surjection :
    (f : split-surjection)  is-split-surjective (map-split-surjection f)
  is-split-surjective-split-surjection = pr2


Split surjections are equivalent to maps equipped with a section

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

  section-is-split-surjective : is-split-surjective f  section f
  pr1 (section-is-split-surjective s) = pr1  s
  pr2 (section-is-split-surjective s) = pr2  s

  is-split-surjective-section : section f  is-split-surjective f
  pr1 (is-split-surjective-section s b) = pr1 s b
  pr2 (is-split-surjective-section s b) = pr2 s b

  equiv-section-is-split-surjective : is-split-surjective f  section f
  equiv-section-is-split-surjective = distributive-Π-Σ

  equiv-is-split-surjective-section : section f  is-split-surjective f
  equiv-is-split-surjective-section = inv-distributive-Π-Σ

A map is an equivalence if and only if it is injective and split surjective

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

  retraction-is-split-surjective-is-injective :
    is-injective f  is-split-surjective f  retraction f
  pr1 (retraction-is-split-surjective-is-injective l s) = pr1  s
  pr2 (retraction-is-split-surjective-is-injective l s) = l  (pr2  (s  f))

  is-equiv-is-split-surjective-is-injective :
    is-injective f  is-split-surjective f  is-equiv f
  pr1 (is-equiv-is-split-surjective-is-injective l s) =
    section-is-split-surjective f s
  pr2 (is-equiv-is-split-surjective-is-injective l s) =
    retraction-is-split-surjective-is-injective l s

  is-split-surjective-is-equiv : is-equiv f  is-split-surjective f
  is-split-surjective-is-equiv = is-split-surjective-section f  pr1

  is-split-surjective-is-injective-is-equiv :
    is-equiv f  is-injective f × is-split-surjective f
  pr1 (is-split-surjective-is-injective-is-equiv is-equiv-f) =
    is-injective-is-equiv is-equiv-f
  pr2 (is-split-surjective-is-injective-is-equiv is-equiv-f) =
    is-split-surjective-is-equiv is-equiv-f

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