cat/src/Cat/Category/Pathy.agda

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{-# OPTIONS --cubical #-}
module Cat.Category.Pathy where
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open import Level
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open import Cubical
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{-
module _ { '} {A : Set } {x : A}
(P : y x y Set ') (d : P x ((λ i x))) where
pathJ' : (y : A) (p : x y) P y p
pathJ' _ p = transp (λ i uncurry P (contrSingl p i)) d
pathJprop' : pathJ' _ refl d
pathJprop' i
= primComp (λ _ P x refl) i (λ {j (i = i1) d}) d
module _ { '} {A : Set }
(P : (x y : A) x y Set ') (d : (x : A) P x x refl) where
pathJ'' : (x y : A) (p : x y) P x y p
pathJ'' _ _ p = transp (λ i
let
P' = uncurry P
q = (contrSingl p i)
in
{!uncurry (uncurry P)!} ) d
-}
module _ { '} {A : Set }
(C : (x y : A) x y Set ')
(c : (x : A) C x x refl) where
=-ind : (x y : A) (p : x y) C x y p
=-ind x y p = pathJ (C x) (c x) y p
module _ { ' : Level} {A : Set } {P : A Set } {x y : A} where
private
D : (x y : A) (x y) Set
D x y p = P x P y
id : { : Level} {A : Set } A A
id x = x
d : (x : A) D x x refl
d x = id {A = P x}
-- the p refers to the third argument
liftP : x y P x P y
liftP p = =-ind D d x y p
-- lift' : (u : P x) → (p : x ≡ y) → (x , u) ≡ (y , liftP p u)
-- lift' u p = {!!}