cat/src/Cat/Category/Bij.agda

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{-# OPTIONS --cubical --allow-unsolved-metas #-}
module Cat.Category.Bij where
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open import Cubical.PathPrelude hiding ( Id )
open import Cubical.FromStdLib
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module _ {A : Set} {a : A} {P : A Set} where
Q : A Set
Q a = A
t : Σ[ a A ] P a Q a
t (a , Pa) = a
u : Q a Σ[ a A ] P a
u a = a , {!!}
tu-bij : (a : Q a) (t u) a a
tu-bij a = refl
v : P a Q a
v x = {!!}
w : Q a P a
w x = {!!}
vw-bij : (a : P a) (w v) a a
vw-bij a = ?
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-- tubij a with (t ∘ u) a
-- ... | q = {!!}
data Id {A : Set} (a : A) : Set where
id : A Id a
data Id' {A : Set} (a : A) : Set where
id' : A Id' a
T U : Set
T = Id a
U = Id' a
f : T U
f (id x) = id' x
g : U T
g (id' x) = id x
fg-bij : (x : U) (f g) x x
fg-bij (id' x) = {!!}