Equivalences of cospans

Content created by Egbert Rijke.

Created on 2024-01-28.
Last modified on 2024-01-28.

module foundation.equivalences-cospans where
Imports
open import foundation.cospans
open import foundation.dependent-pair-types
open import foundation.fundamental-theorem-of-identity-types
open import foundation.homotopy-induction
open import foundation.morphisms-cospans
open import foundation.structure-identity-principle
open import foundation.univalence
open import foundation.universe-levels

open import foundation-core.equivalences
open import foundation-core.homotopies
open import foundation-core.identity-types
open import foundation-core.torsorial-type-families

Idea

An equivalence of cospans from cospans (X , f , g) to (Y , h , k) between types A and B consists of an equivalence e : X ≃ Y such that the triangles

      e              e
  X ----> Y      X ----> Y
   \     /        \     /
  f \   / h      g \   / k
     V V            V V
      A              B

both commute.

Definitions

Equivalences of cospans

module _
  {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2}
  where

  equiv-cospan : cospan l3 A B  cospan l4 A B  UU (l1  l2  l3  l4)
  equiv-cospan c d =
    Σ ( codomain-cospan c  codomain-cospan d)
      ( λ e  coherence-hom-cospan c d (map-equiv e))

The identity equivalence of cospans

module _
  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2}
  where

  id-equiv-cospan : (c : cospan l3 A B)  equiv-cospan c c
  pr1 (id-equiv-cospan c) = id-equiv
  pr1 (pr2 (id-equiv-cospan c)) = refl-htpy
  pr2 (pr2 (id-equiv-cospan c)) = refl-htpy

Properties

Characterizing equality of cospans

module _
  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2}
  where

  equiv-eq-cospan : (c d : cospan l3 A B)  c  d  equiv-cospan c d
  equiv-eq-cospan c .c refl = id-equiv-cospan c

  is-torsorial-equiv-cospan :
    (c : cospan l3 A B)  is-torsorial (equiv-cospan c)
  is-torsorial-equiv-cospan c =
    is-torsorial-Eq-structure
      ( is-torsorial-equiv (pr1 c))
      ( codomain-cospan c , id-equiv)
      ( is-torsorial-Eq-structure
        ( is-torsorial-htpy' (left-map-cospan c))
        ( left-map-cospan c , refl-htpy)
        ( is-torsorial-htpy' (right-map-cospan c)))

  is-equiv-equiv-eq-cospan :
    (c d : cospan l3 A B)  is-equiv (equiv-eq-cospan c d)
  is-equiv-equiv-eq-cospan c =
    fundamental-theorem-id (is-torsorial-equiv-cospan c) (equiv-eq-cospan c)

  extensionality-cospan :
    (c d : cospan l3 A B)  (c  d)  (equiv-cospan c d)
  pr1 (extensionality-cospan c d) = equiv-eq-cospan c d
  pr2 (extensionality-cospan c d) = is-equiv-equiv-eq-cospan c d

  eq-equiv-cospan : (c d : cospan l3 A B)  equiv-cospan c d  c  d
  eq-equiv-cospan c d = map-inv-equiv (extensionality-cospan c d)

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