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{-# OPTIONS --without-K #-}
open import Base
open import Homotopy.TruncatedHIT
open import Sets.Quotient
module Sets.QuotientUP {i j} (A : Set i) ⦃ A-set : is-set A
(R : AASet j) ⦃ R-prop : (x y : A) → is-prop (R x y) ⦄ where
-- [X →→ Y ~ R] is the set of functions [X → Y] respecting the relation [R]
_→→_~_ : {i j k}
(X : Set i) ⦃ X-set : is-set X ⦄
(Y : Set j) ⦃ Y-set : is-set Y ⦄
(R : X X Set k) ⦃ R-prop : (x x' : X) is-prop (R x x')⦄
Set _
X → Y ~ R = Σ (X Y) (λ f (x x' : X) (R x x' f x ≡ f x'))
module UP {k} (B : Set k) (B-set : is-set B) where
factor : ((A → B ~ R) (A / R B))
factor (f , p) = /-rec-nondep A R B f p B-set
extend : ((A / R B) (A → B ~ R))
extend f = ((f ◯ proj A R) , (λ x x' p₁ map f (eq A R x x' p₁)))
extend-factor : (f : A → B ~ R) extend (factor f) ≡ f
extend-factor (f , p) = map (λ x f , x)
(funext-dep (λ x
funext-dep (λ x'
funext-dep (λ p₁ π₁ (B-set _ _ _ _)))))
factor-extend : (f : A / R B) factor (extend f) ≡ f
factor-extend f =
funext-dep (/-rec A R (λ x factor (extend f) x ≡ f x)
(λ x refl _)
(λ x y p₁ π₁ (B-set _ _ _ _))
(λ x is-increasing-hlevel 1 _ (B-set _ _)))
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