Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

fix merge conflict from ∙Susp → Susp∙ renaming #609

Merged
merged 1 commit into from
Oct 22, 2021
Merged

fix merge conflict from ∙Susp → Susp∙ renaming #609

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
10 changes: 5 additions & 5 deletions Cubical/HITs/Susp/LoopAdjunction.agda
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -27,15 +27,15 @@ private
Ω∙ (suc (suc n)) X = Ω∙ 1 (Ω∙ (suc n) X)

ΣX→∙YEquiv : ((X , x₀) : Pointed ℓ) ((Y , y₀) : Pointed ℓ')
→ (∙Susp X →∙ (Y , y₀)) ≃ (Σ[ y ∈ Y ] (X → (y₀ ≡ y)))
→ (Susp∙ X →∙ (Y , y₀)) ≃ (Σ[ y ∈ Y ] (X → (y₀ ≡ y)))
ΣX→∙YEquiv (X , x₀) (Y , y₀) =
isoToEquiv (iso left→right right→left right→left→right left→right→left)
where
left→right : (∙Susp X →∙ (Y , y₀)) → Σ[ y ∈ Y ] (X → (y₀ ≡ y))
left→right : (Susp∙ X →∙ (Y , y₀)) → Σ[ y ∈ Y ] (X → (y₀ ≡ y))
left→right (f , b) .fst = f south
left→right (f , b) .snd x = sym b ∙ cong f (merid x)

right→left : (Σ[ y ∈ Y ] (X → (y₀ ≡ y))) → (∙Susp X →∙ (Y , y₀))
right→left : (Σ[ y ∈ Y ] (X → (y₀ ≡ y))) → (Susp∙ X →∙ (Y , y₀))
right→left (y , g) .fst north = y₀
right→left (y , g) .fst south = y
right→left (y , g) .fst (merid x i) = g x i
Expand All @@ -45,7 +45,7 @@ private
right→left→right (y , g) i .fst = y
right→left→right (y , g) i .snd x = lUnit (g x) (~ i)

left→right→left : (f : ∙Susp X →∙ (Y , y₀)) → right→left (left→right f) ≡ f
left→right→left : (f : Susp∙ X →∙ (Y , y₀)) → right→left (left→right f) ≡ f
left→right→left (f , b) i .fst north = b (~ i)
left→right→left (f , b) i .fst south = f south
left→right→left (f , b) i .fst (merid x j) =
Expand Down Expand Up @@ -109,5 +109,5 @@ X→∙ΩYEquiv (X , x₀) (Y , y₀) =
(bottom (f , b) i j k)

{- The Main Theorem -}
ΣΩAdjunction : ((X , x₀) : Pointed ℓ) (Y : Pointed ℓ') → (∙Susp X →∙ Y) ≃ ((X , x₀) →∙ Ω∙ 1 Y)
ΣΩAdjunction : ((X , x₀) : Pointed ℓ) (Y : Pointed ℓ') → (Susp∙ X →∙ Y) ≃ ((X , x₀) →∙ Ω∙ 1 Y)
ΣΩAdjunction X Y = compEquiv (ΣX→∙YEquiv X Y) (invEquiv (X→∙ΩYEquiv X Y))