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More stuff about opposite being an involution

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Frederik Hanghøj Iversen 2018-03-05 16:10:27 +01:00
parent b26ea18257
commit 7f4a8a65b8
1 changed files with 32 additions and 13 deletions
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43
src/Cat/Category.agda
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@ -301,23 +301,42 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
-- flipped. -- flipped.
module Opposite {ℓa ℓb : Level} where module Opposite {ℓa ℓb : Level} where
module _ (ℂ : Category ℓa ℓb) where module _ (ℂ : Category ℓa ℓb) where
open Category ℂ
private private
module ℂ = Category ℂ
opRaw : RawCategory ℓa ℓb opRaw : RawCategory ℓa ℓb
RawCategory.Object opRaw = Object RawCategory.Object opRaw = ℂ.Object
RawCategory.Arrow opRaw = Function.flip Arrow RawCategory.Arrow opRaw = Function.flip ℂ.Arrow
RawCategory.𝟙 opRaw = 𝟙 RawCategory.𝟙 opRaw = ℂ.𝟙
RawCategory._∘_ opRaw = Function.flip _∘_ RawCategory._∘_ opRaw = Function.flip ℂ._∘_
opIsCategory : IsCategory opRaw open RawCategory opRaw
IsCategory.isAssociative opIsCategory = sym isAssociative open Univalence opRaw
IsCategory.isIdentity opIsCategory = swap isIdentity
IsCategory.arrowsAreSets opIsCategory = arrowsAreSets isIdentity : IsIdentity 𝟙
IsCategory.univalent opIsCategory = {!!} isIdentity = swap ℂ.isIdentity
module _ {A B : ℂ.Object} where
univalent : isEquiv (A ≡ B) (A ≅ B)
(id-to-iso (swap ℂ.isIdentity) A B)
fst (univalent iso) = flipFiber (fst (ℂ.univalent (flipIso iso)))
where
flipIso : A ≅ B → B ℂ.≅ A
flipIso (f , f~ , iso) = f , f~ , swap iso
flipFiber
: fiber (ℂ.id-to-iso ℂ.isIdentity B A) (flipIso iso)
→ fiber ( id-to-iso isIdentity A B) iso
flipFiber (eq , eqIso) = sym eq , {!!}
snd (univalent iso) = {!!}
isCategory : IsCategory opRaw
IsCategory.isAssociative isCategory = sym ℂ.isAssociative
IsCategory.isIdentity isCategory = isIdentity
IsCategory.arrowsAreSets isCategory = ℂ.arrowsAreSets
IsCategory.univalent isCategory = univalent
opposite : Category ℓa ℓb opposite : Category ℓa ℓb
raw opposite = opRaw Category.raw opposite = opRaw
Category.isCategory opposite = opIsCategory Category.isCategory opposite = isCategory
-- As demonstrated here a side-effect of having no-eta-equality on constructors -- As demonstrated here a side-effect of having no-eta-equality on constructors
-- means that we need to pick things apart to show that things are indeed -- means that we need to pick things apart to show that things are indeed