Set quotients of index 2

Content created by Fredrik Bakke, Egbert Rijke, Jonathan Prieto-Cubides, Julian KG, fernabnor and louismntnu.

Created on 2023-02-05.
Last modified on 2023-06-25.

{-# OPTIONS --lossy-unification #-}

module univalent-combinatorics.set-quotients-of-index-two where
Imports 
open import foundation.action-on-identifications-functions
open import foundation.commuting-squares-of-maps
open import foundation.contractible-types
open import foundation.coproduct-types
open import foundation.dependent-pair-types
open import foundation.embeddings
open import foundation.empty-types
open import foundation.equivalence-relations
open import foundation.equivalences
open import foundation.function-types
open import foundation.functoriality-set-quotients
open import foundation.identity-types
open import foundation.injective-maps
open import foundation.logical-equivalences
open import foundation.reflecting-maps-equivalence-relations
open import foundation.sets
open import foundation.universal-property-set-quotients
open import foundation.universe-levels

open import univalent-combinatorics.standard-finite-types

module _
  {l1 l2 l3 : Level}
  {A : UU l1} (R : Equivalence-Relation l2 A)
  (QR : Set l3) (f : reflecting-map-Equivalence-Relation R (type-Set QR))
  (Uf : {l : Level} → is-set-quotient l R QR f)
  (eA : type-Set QR ≃ Fin 2) (h : A → A)
  (H : {x y : A} →
    sim-Equivalence-Relation R x y ↔ sim-Equivalence-Relation R (h x) (h y))
  (h' : type-Set QR → type-Set QR)
  (x : A)
  (P :
    h' (map-reflecting-map-Equivalence-Relation R f x) ＝
    map-reflecting-map-Equivalence-Relation R f (h x))
  where

  cases-coherence-square-maps-eq-one-value-emb-is-set-quotient :
    is-emb h' →
    (y : A) (k k' k'' : Fin 2) →
    map-equiv eA (h' (map-reflecting-map-Equivalence-Relation R f x)) ＝ k →
    map-equiv eA (h' (map-reflecting-map-Equivalence-Relation R f y)) ＝ k' →
    map-equiv eA (map-reflecting-map-Equivalence-Relation R f (h y)) ＝ k'' →
    h' (map-reflecting-map-Equivalence-Relation R f y) ＝
    map-reflecting-map-Equivalence-Relation R f (h y)
  cases-coherence-square-maps-eq-one-value-emb-is-set-quotient H' y
    ( inl (inr star)) (inl (inr star)) k'' p q r =
    ( is-injective-map-equiv eA (q ∙ inv p)) ∙
      ( P ∙
        reflects-map-reflecting-map-Equivalence-Relation R f
          ( pr1 H
            ( map-equiv
              ( is-effective-is-set-quotient R QR f Uf x y)
              ( map-inv-is-equiv
                ( H'
                  ( map-reflecting-map-Equivalence-Relation R f x)
                  ( map-reflecting-map-Equivalence-Relation R f y))
                ( is-injective-map-equiv eA (p ∙ inv q))))))
  cases-coherence-square-maps-eq-one-value-emb-is-set-quotient H' y
    ( inl (inr star)) (inr star) (inl (inr star)) p q r =
    ex-falso
      ( neq-inl-inr
        ( inv p ∙
          ( ( ap
            ( map-equiv eA ∘ h')
            ( reflects-map-reflecting-map-Equivalence-Relation R f
              ( pr2 H
                (map-equiv
                  ( is-effective-is-set-quotient R QR f Uf (h x) (h y))
                  ( inv P ∙ is-injective-map-equiv eA (p ∙ inv r)))))) ∙
            ( q))))
  cases-coherence-square-maps-eq-one-value-emb-is-set-quotient H' y
    ( inl (inr star)) (inr star) (inr star) p q r =
    is-injective-map-equiv eA (q ∙ inv r)
  cases-coherence-square-maps-eq-one-value-emb-is-set-quotient H' y
    ( inr star) (inl (inr star)) (inl (inr star)) p q r =
    is-injective-map-equiv eA (q ∙ inv r)
  cases-coherence-square-maps-eq-one-value-emb-is-set-quotient H' y
    ( inr star) (inl (inr star)) (inr star) p q r =
    ex-falso
      ( neq-inr-inl
        ( inv p ∙
          ( ( ap
            ( map-equiv eA ∘ h')
            ( reflects-map-reflecting-map-Equivalence-Relation R f
              ( pr2 H
                (map-equiv
                  ( is-effective-is-set-quotient R QR f Uf (h x) (h y))
                  ( inv P ∙ is-injective-map-equiv eA (p ∙ inv r)))))) ∙
            ( q))))
  cases-coherence-square-maps-eq-one-value-emb-is-set-quotient H' y
    ( inr star) (inr star) k'' p q r =
    ( is-injective-map-equiv eA (q ∙ inv p)) ∙
      ( P ∙
        reflects-map-reflecting-map-Equivalence-Relation R f
          ( pr1 H
            ( map-equiv
              ( is-effective-is-set-quotient R QR f Uf x y)
              ( map-inv-is-equiv
                ( H'
                  ( map-reflecting-map-Equivalence-Relation R f x)
                  ( map-reflecting-map-Equivalence-Relation R f y))
                ( is-injective-map-equiv eA (p ∙ inv q))))))

  coherence-square-maps-eq-one-value-emb-is-set-quotient :
    is-emb h' →
    coherence-square-maps
      ( h)
      ( map-reflecting-map-Equivalence-Relation R f)
      ( map-reflecting-map-Equivalence-Relation R f)
      ( h')
  coherence-square-maps-eq-one-value-emb-is-set-quotient H' y =
    cases-coherence-square-maps-eq-one-value-emb-is-set-quotient H' y
      ( map-equiv eA (h' (map-reflecting-map-Equivalence-Relation R f x)))
      ( map-equiv eA (h' (map-reflecting-map-Equivalence-Relation R f y)))
      ( map-equiv eA (map-reflecting-map-Equivalence-Relation R f (h y)))
      ( refl)
      ( refl)
      ( refl)

  eq-equiv-eq-one-value-equiv-is-set-quotient :
    (P : is-equiv h) (Q : is-equiv h') →
    pair h' Q ＝ equiv-is-set-quotient R QR f R QR f Uf Uf ((h , P) , H)
  eq-equiv-eq-one-value-equiv-is-set-quotient P Q =
    ap pr1
      { x =
        pair
          ( pair h' Q)
          ( coherence-square-maps-eq-one-value-emb-is-set-quotient
            ( is-emb-is-equiv Q))}
      { y =
        center
          ( unique-equiv-is-set-quotient R QR f R QR f Uf Uf ((h , P) , H))}
      ( eq-is-contr
        ( unique-equiv-is-set-quotient R QR f R QR f Uf Uf ((h , P) , H)))

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