Path-split maps

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

Created on 2022-01-26.
Last modified on 2024-06-05.

module foundation.path-split-maps where

open import foundation-core.path-split-maps public
Imports 
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.iterated-dependent-product-types
open import foundation.logical-equivalences
open import foundation.universe-levels

open import foundation-core.contractible-types
open import foundation-core.propositions

Properties

Being path-split is a property

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

  abstract
    is-prop-is-path-split : (f : A → B) → is-prop (is-path-split f)
    is-prop-is-path-split f =
      is-prop-is-proof-irrelevant
        ( λ is-path-split-f →
          ( is-contr-product
            ( is-contr-section-is-equiv
              ( is-equiv-is-path-split f is-path-split-f))
            ( is-contr-iterated-Π 2
              ( λ x y →
                is-contr-section-is-equiv
                  ( is-emb-is-equiv
                    ( is-equiv-is-path-split f is-path-split-f) x y)))))

  abstract
    is-equiv-is-path-split-is-equiv :
      (f : A → B) → is-equiv (is-path-split-is-equiv f)
    is-equiv-is-path-split-is-equiv f =
      is-equiv-has-converse-is-prop
        ( is-property-is-equiv f)
        ( is-prop-is-path-split f)
        ( is-equiv-is-path-split f)

  equiv-is-path-split-is-equiv : (f : A → B) → is-equiv f ≃ is-path-split f
  equiv-is-path-split-is-equiv f =
    pair (is-path-split-is-equiv f) (is-equiv-is-path-split-is-equiv f)

  abstract
    is-equiv-is-equiv-is-path-split :
      (f : A → B) → is-equiv (is-equiv-is-path-split f)
    is-equiv-is-equiv-is-path-split f =
      is-equiv-has-converse-is-prop
        ( is-prop-is-path-split f)
        ( is-property-is-equiv f)
        ( is-path-split-is-equiv f)

  equiv-is-equiv-is-path-split : (f : A → B) → is-path-split f ≃ is-equiv f
  equiv-is-equiv-is-path-split f =
    ( is-equiv-is-path-split f , is-equiv-is-equiv-is-path-split f)

See also

References

[Shu14]
Mike Shulman. Universal properties without function extensionality. Blog post, 11 2014. URL: https://homotopytypetheory.org/2014/11/02/universal-properties-without-function-extensionality/.
[UF13]
The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study, 2013. URL: https://homotopytypetheory.org/book/, arXiv:1308.0729.

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