Mercurial > hg > Members > kono > Proof > category
changeset 444:59e47e75188f
complete connection for finite category
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 14 Oct 2016 19:12:01 +0900 |
| parents | f526f4b68565 |
| children | 00cf84846d81 |
| files | limit-to.agda |
| diffstat | 1 files changed, 59 insertions(+), 468 deletions(-) [+] |
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--- a/limit-to.agda Sun Sep 04 21:05:39 2016 +0900 +++ b/limit-to.agda Fri Oct 14 19:12:01 2016 +0900 @@ -6,288 +6,75 @@ open import cat-utility open import HomReasoning open import Relation.Binary.Core -open import Data.Maybe open Functor - -- If we have limit then we have equalizer --- two objects category --- ---- f +--- t0 f t1 --- -----→ ---- 0 1 +--- --- -----→ ---- g +--- t2 g t3 -data TwoObject {c₁ : Level} : Set c₁ where - t0 : TwoObject - t1 : TwoObject +data Object4 {c₁ : Level} : Set c₁ where + t0 : Object4 + t1 : Object4 + t2 : Object4 + t4 : Object4 --- constrainted arrow --- we need inverse of f to complete cases -data Arrow {c₁ c₂ : Level } ( t00 t11 : TwoObject {c₁} ) : TwoObject {c₁} → TwoObject {c₁} → Set c₂ where - id-t0 : Arrow t00 t11 t00 t00 - id-t1 : Arrow t00 t11 t11 t11 - arrow-f : Arrow t00 t11 t00 t11 - arrow-g : Arrow t00 t11 t00 t11 - inv-f : Arrow t00 t11 t11 t00 - -record TwoHom {c₁ c₂ : Level} (a b : TwoObject {c₁} ) : Set c₂ where - field - hom : Maybe ( Arrow {c₁} {c₂} t0 t1 a b ) +record TwoHom {c₁ : Level} {Object : Set c₁} (a b : Object) : Set c₁ where open TwoHom -- arrow composition -comp : ∀ {c₁ c₂} → {a b c : TwoObject {c₁}} → Maybe ( Arrow {c₁} {c₂} t0 t1 b c ) → Maybe ( Arrow {c₁} {c₂} t0 t1 a b ) → Maybe ( Arrow {c₁} {c₂} t0 t1 a c ) -comp {_} {_} {_} {_} {_} nothing _ = nothing -comp {_} {_} {_} {_} {_} (just _ ) nothing = nothing -comp {_} {_} {t0} {t1} {t1} (just id-t1 ) ( just arrow-f ) = just arrow-f -comp {_} {_} {t0} {t1} {t1} (just id-t1 ) ( just arrow-g ) = just arrow-g -comp {_} {_} {t1} {t1} {t1} (just id-t1 ) ( just id-t1 ) = just id-t1 -comp {_} {_} {t1} {t1} {t0} (just inv-f ) ( just id-t1 ) = just inv-f -comp {_} {_} {t0} {t0} {t1} (just arrow-f ) ( just id-t0 ) = just arrow-f -comp {_} {_} {t0} {t0} {t1} (just arrow-g ) ( just id-t0 ) = just arrow-g -comp {_} {_} {t0} {t0} {t0} (just id-t0 ) ( just id-t0 ) = just id-t0 -comp {_} {_} {t1} {t0} {t0} (just id-t0 ) ( just inv-f ) = just inv-f -comp {_} {_} {_} {_} {_} (just _ ) ( just _ ) = nothing - - -_×_ : ∀ {c₁ c₂} → {a b c : TwoObject {c₁}} → ( TwoHom {c₁} {c₂} b c ) → ( TwoHom {c₁} {c₂} a b ) → ( TwoHom {c₁} {c₂} a c ) -_×_ {c₁} {c₂} {a} {b} {c} f g = record { hom = comp {c₁} {c₂} {a} {b} {c} (hom f) (hom g ) } - - -_==_ : ∀{ c₁ c₂ a b } → Rel (Maybe (Arrow {c₁} {c₂} t0 t1 a b )) (c₂) -_==_ = Eq _≡_ - -map2hom : ∀{ c₁ c₂ } → {a b : TwoObject {c₁}} → Maybe ( Arrow {c₁} {c₂} t0 t1 a b ) → TwoHom {c₁} {c₂ } a b -map2hom {_} {_} {t1} {t1} (just id-t1) = record { hom = just id-t1 } -map2hom {_} {_} {t0} {t1} (just arrow-f) = record { hom = just arrow-f } -map2hom {_} {_} {t0} {t1} (just arrow-g) = record { hom = just arrow-g } -map2hom {_} {_} {t0} {t0} (just id-t0) = record { hom = just id-t0 } -map2hom {_} {_} {_} {_} _ = record { hom = nothing } - -record TwoHom1 {c₁ c₂ : Level} (a : TwoObject {c₁} ) (b : TwoObject {c₁} ) : Set c₂ where - field - Map : TwoHom {c₁} {c₂ } a b - iso-Map : Map ≡ map2hom ( hom Map ) - -open TwoHom1 - -==refl : ∀{ c₁ c₂ a b } → ∀ {x : Maybe (Arrow {c₁} {c₂} t0 t1 a b )} → x == x -==refl {_} {_} {_} {_} {just x} = just refl -==refl {_} {_} {_} {_} {nothing} = nothing - -==sym : ∀{ c₁ c₂ a b } → ∀ {x y : Maybe (Arrow {c₁} {c₂} t0 t1 a b )} → x == y → y == x -==sym (just x≈y) = just (≡-sym x≈y) -==sym nothing = nothing - -==trans : ∀{ c₁ c₂ a b } → ∀ {x y z : Maybe (Arrow {c₁} {c₂} t0 t1 a b ) } → - x == y → y == z → x == z -==trans (just x≈y) (just y≈z) = just (≡-trans x≈y y≈z) -==trans nothing nothing = nothing - -==cong : ∀{ c₁ c₂ a b c d } → ∀ {x y : Maybe (Arrow {c₁} {c₂} t0 t1 a b )} → - (f : Maybe (Arrow {c₁} {c₂} t0 t1 a b ) → Maybe (Arrow {c₁} {c₂} t0 t1 c d ) ) → x == y → f x == f y -==cong { c₁} {c₂} {a} {b } {c} {d} {nothing} {nothing} f nothing = ==refl -==cong { c₁} {c₂} {a} {b } {c} {d} {just x} {just .x} f (just refl) = ==refl - - -module ==-Reasoning {c₁ c₂ : Level} where - - infixr 2 _∎ - infixr 2 _==⟨_⟩_ _==⟨⟩_ - infix 1 begin_ - - - data _IsRelatedTo_ {c₁ c₂ : Level} {a b : TwoObject {c₁} } (x y : (Maybe (Arrow {c₁} {c₂} t0 t1 a b ))) : - Set c₂ where - relTo : (x≈y : x == y ) → x IsRelatedTo y - - begin_ : { a b : TwoObject {c₁} } {x : Maybe (Arrow {c₁} {c₂} t0 t1 a b ) } {y : Maybe (Arrow {c₁} {c₂} t0 t1 a b )} → - x IsRelatedTo y → x == y - begin relTo x≈y = x≈y - - _==⟨_⟩_ : { a b : TwoObject {c₁} } (x : Maybe (Arrow {c₁} {c₂} t0 t1 a b )) {y z : Maybe (Arrow {c₁} {c₂} t0 t1 a b ) } → - x == y → y IsRelatedTo z → x IsRelatedTo z - _ ==⟨ x≈y ⟩ relTo y≈z = relTo (==trans x≈y y≈z) - - - _==⟨⟩_ : { a b : TwoObject {c₁} }(x : Maybe (Arrow {c₁} {c₂} t0 t1 a b )) {y : Maybe (Arrow {c₁} {c₂} t0 t1 a b )} - → x IsRelatedTo y → x IsRelatedTo y - _ ==⟨⟩ x≈y = x≈y - - _∎ : { a b : TwoObject {c₁} }(x : Maybe (Arrow {c₁} {c₂} t0 t1 a b )) → x IsRelatedTo x - _∎ _ = relTo ==refl - - --- TwoHom1Eq : {c₁ c₂ : Level } {a b : TwoObject {c₁} } {f g : ( TwoHom1 {c₁} {c₂ } a b)} → --- hom (Map f) == hom (Map g) → f == g --- TwoHom1Eq = ? - - --- f g h --- d <- c <- b <- a --- --- It can be proved without Arrow constraints - -assoc-× : {c₁ c₂ : Level } {a b c d : TwoObject {c₁} } - {f : (TwoHom {c₁} {c₂ } c d )} → - {g : (TwoHom {c₁} {c₂ } b c )} → - {h : (TwoHom {c₁} {c₂ } a b )} → - hom ( f × (g × h)) == hom ((f × g) × h ) -assoc-× {c₁} {c₂} {a} {b} {c} {d} {f} {g} {h} with hom f | hom g | hom h -assoc-× {c₁} {c₂} {t0} {t0} {t0} {t0} {f} {g} {h} | nothing | _ | _ = nothing -assoc-× {c₁} {c₂} {t0} {t0} {t0} {t1} {f} {g} {h} | nothing | _ | _ = nothing -assoc-× {c₁} {c₂} {t0} {t0} {t1} {t0} {f} {g} {h} | nothing | _ | _ = nothing -assoc-× {c₁} {c₂} {t0} {t0} {t1} {t1} {f} {g} {h} | nothing | _ | _ = nothing -assoc-× {c₁} {c₂} {t0} {t1} {t0} {t0} {f} {g} {h} | nothing | _ | _ = nothing -assoc-× {c₁} {c₂} {t0} {t1} {t0} {t1} {f} {g} {h} | nothing | _ | _ = nothing -assoc-× {c₁} {c₂} {t0} {t1} {t1} {t0} {f} {g} {h} | nothing | _ | _ = nothing -assoc-× {c₁} {c₂} {t0} {t1} {t1} {t1} {f} {g} {h} | nothing | _ | _ = nothing -assoc-× {c₁} {c₂} {t1} {t0} {t0} {t0} {f} {g} {h} | nothing | _ | _ = nothing -assoc-× {c₁} {c₂} {t1} {t0} {t0} {t1} {f} {g} {h} | nothing | _ | _ = nothing -assoc-× {c₁} {c₂} {t1} {t0} {t1} {t0} {f} {g} {h} | nothing | _ | _ = nothing -assoc-× {c₁} {c₂} {t1} {t0} {t1} {t1} {f} {g} {h} | nothing | _ | _ = nothing -assoc-× {c₁} {c₂} {t1} {t1} {t0} {t0} {f} {g} {h} | nothing | _ | _ = nothing -assoc-× {c₁} {c₂} {t1} {t1} {t0} {t1} {f} {g} {h} | nothing | _ | _ = nothing -assoc-× {c₁} {c₂} {t1} {t1} {t1} {t0} {f} {g} {h} | nothing | _ | _ = nothing -assoc-× {c₁} {c₂} {t1} {t1} {t1} {t1} {f} {g} {h} | nothing | _ | _ = nothing -assoc-× {c₁} {c₂} {t0} {t0} {t0} {t0} {f} {g} {h} | just _ | nothing | _ = nothing -assoc-× {c₁} {c₂} {t0} {t0} {t0} {t1} {f} {g} {h} | just _ | nothing | _ = nothing -assoc-× {c₁} {c₂} {t0} {t0} {t1} {t0} {f} {g} {h} | just _ | nothing | _ = nothing -assoc-× {c₁} {c₂} {t0} {t0} {t1} {t1} {f} {g} {h} | just _ | nothing | _ = nothing -assoc-× {c₁} {c₂} {t0} {t1} {t0} {t0} {f} {g} {h} | just _ | nothing | _ = nothing -assoc-× {c₁} {c₂} {t0} {t1} {t0} {t1} {f} {g} {h} | just _ | nothing | _ = nothing -assoc-× {c₁} {c₂} {t0} {t1} {t1} {t0} {f} {g} {h} | just _ | nothing | _ = nothing -assoc-× {c₁} {c₂} {t0} {t1} {t1} {t1} {f} {g} {h} | just _ | nothing | _ = nothing -assoc-× {c₁} {c₂} {t1} {t0} {t0} {t0} {f} {g} {h} | just _ | nothing | _ = nothing -assoc-× {c₁} {c₂} {t1} {t0} {t0} {t1} {f} {g} {h} | just _ | nothing | _ = nothing -assoc-× {c₁} {c₂} {t1} {t0} {t1} {t0} {f} {g} {h} | just _ | nothing | _ = nothing -assoc-× {c₁} {c₂} {t1} {t0} {t1} {t1} {f} {g} {h} | just _ | nothing | _ = nothing -assoc-× {c₁} {c₂} {t1} {t1} {t0} {t0} {f} {g} {h} | just _ | nothing | _ = nothing -assoc-× {c₁} {c₂} {t1} {t1} {t0} {t1} {f} {g} {h} | just _ | nothing | _ = nothing -assoc-× {c₁} {c₂} {t1} {t1} {t1} {t0} {f} {g} {h} | just _ | nothing | _ = nothing -assoc-× {c₁} {c₂} {t1} {t1} {t1} {t1} {f} {g} {h} | just _ | nothing | _ = nothing -assoc-× {c₁} {c₂} {t0} {t0} {t0} {t0} {f} {g} {h} | just id-t0 | just id-t0 | just id-t0 = ==refl -assoc-× {c₁} {c₂} {t0} {t0} {t0} {t1} {f} {g} {h} | just arrow-f | just id-t0 | just id-t0 = ==refl -assoc-× {c₁} {c₂} {t0} {t0} {t0} {t1} {f} {g} {h} | just arrow-g | just id-t0 | just id-t0 = ==refl -assoc-× {c₁} {c₂} {t0} {t0} {t1} {t1} {f} {g} {h} | just id-t1 | just arrow-f | just id-t0 = ==refl -assoc-× {c₁} {c₂} {t0} {t0} {t1} {t1} {f} {g} {h} | just id-t1 | just arrow-g | just id-t0 = ==refl -assoc-× {c₁} {c₂} {t0} {t1} {t1} {t1} {f} {g} {h} | just id-t1 | just id-t1 | just arrow-f = ==refl -assoc-× {c₁} {c₂} {t0} {t1} {t1} {t1} {f} {g} {h} | just id-t1 | just id-t1 | just arrow-g = ==refl -assoc-× {c₁} {c₂} {t1} {t1} {t1} {t1} {f} {g} {h} | just id-t1 | just id-t1 | just id-t1 = ==refl --- remaining all failure case (except inf-f case ) -assoc-× {c₁} {c₂} {t0} {t0} {t0} {t0} {f} {g} {h} | just id-t0 | just id-t0 | nothing = nothing -assoc-× {c₁} {c₂} {t1} {t0} {t0} {t0} {f} {g} {h} | just id-t0 | just id-t0 | nothing = nothing -assoc-× {c₁} {c₂} {t0} {t1} {t1} {t1} {f} {g} {h} | just id-t1 | just id-t1 | nothing = nothing -assoc-× {c₁} {c₂} {t1} {t1} {t1} {t1} {f} {g} {h} | just id-t1 | just id-t1 | nothing = nothing -assoc-× {c₁} {c₂} {t0} {t1} {t0} {t0} {f} {g} {h} | just id-t0 | just inv-f | nothing = nothing -assoc-× {c₁} {c₂} {t1} {t1} {t0} {t0} {f} {g} {h} | just id-t0 | just inv-f | nothing = nothing -assoc-× {c₁} {c₂} {t1} {t0} {t0} {t1} {f} {g} {h} | just arrow-f | just id-t0 | nothing = nothing -assoc-× {c₁} {c₂} {t1} {t0} {t0} {t1} {f} {g} {h} | just arrow-g | just id-t0 | nothing = nothing -assoc-× {c₁} {c₂} {t0} {t0} {t0} {t1} {f} {g} {h} | just arrow-f | just id-t0 | nothing = nothing -assoc-× {c₁} {c₂} {t0} {t0} {t0} {t1} {f} {g} {h} | just arrow-g | just id-t0 | nothing = nothing -assoc-× {c₁} {c₂} {t0} {t0} {t1} {t1} {f} {g} {h} | just id-t1 | just arrow-f | nothing = nothing -assoc-× {c₁} {c₂} {t0} {t0} {t1} {t1} {f} {g} {h} | just id-t1 | just arrow-g | nothing = nothing -assoc-× {c₁} {c₂} {t1} {t0} {t1} {t1} {f} {g} {h} | just id-t1 | just arrow-f | nothing = nothing -assoc-× {c₁} {c₂} {t1} {t0} {t1} {t1} {f} {g} {h} | just id-t1 | just arrow-g | nothing = nothing -assoc-× {c₁} {c₂} {t0} {t1} {t1} {t0} {f} {g} {h} | just inv-f | just id-t1 | nothing = nothing -assoc-× {c₁} {c₂} {t1} {t1} {t1} {t0} {f} {g} {h} | just inv-f | just id-t1 | nothing = nothing -assoc-× {_} {_} {t0} {t0} {t1} {t0} {_} {_} {_} | (just _) | (just _) | nothing = nothing -assoc-× {_} {_} {t0} {t1} {t0} {t1} {_} {_} {_} | (just _) | (just _) | nothing = nothing -assoc-× {_} {_} {t1} {t0} {t1} {t0} {_} {_} {_} | (just _) | (just _) | nothing = nothing -assoc-× {_} {_} {t1} {t1} {t0} {t1} {_} {_} {_} | (just _) | (just _) | nothing = nothing -assoc-× {_} {_} {t0} {t0} {t1} {t0} {_} {_} {_} | (just _) | (just arrow-f) | (just id-t0) = nothing -assoc-× {_} {_} {t0} {t0} {t1} {t0} {_} {_} {_} | (just _) | (just arrow-g) | (just id-t0) = nothing -assoc-× {_} {_} {t0} {t1} {t0} {t0} {_} {_} {_} | (just id-t0) | (just inv-f) | (just _) = nothing -assoc-× {_} {_} {t0} {t1} {t0} {t1} {_} {_} {_} | (just _) | (just _) | (just _) = nothing -assoc-× {_} {_} {t0} {t1} {t1} {t0} {_} {_} {_} | (just inv-f) | (just id-t1) | (just arrow-f) = nothing -assoc-× {_} {_} {t0} {t1} {t1} {t0} {_} {_} {_} | (just inv-f) | (just id-t1) | (just arrow-g) = nothing -assoc-× {_} {_} {t1} {t0} {t0} {t0} {_} {_} {_} | (just id-t0) | (just id-t0) | (just inv-f) = ==refl -assoc-× {_} {_} {t1} {t0} {t0} {t1} {_} {_} {_} | (just arrow-f) | (just id-t0) | (just inv-f) = nothing -assoc-× {_} {_} {t1} {t0} {t0} {t1} {_} {_} {_} | (just arrow-g) | (just id-t0) | (just inv-f) = nothing -assoc-× {_} {_} {t1} {t0} {t1} {t0} {_} {_} {_} | (just _) | (just _) | (just _) = nothing -assoc-× {_} {_} {t1} {t0} {t1} {t1} {_} {_} {_} | (just id-t1) | (just arrow-f) | (just _) = nothing -assoc-× {_} {_} {t1} {t0} {t1} {t1} {_} {_} {_} | (just id-t1) | (just arrow-g) | (just _) = nothing -assoc-× {_} {_} {t1} {t1} {t0} {t0} {_} {_} {_} | (just id-t0) | (just inv-f) | (just id-t1) = ==refl -assoc-× {_} {_} {t1} {t1} {t0} {t1} {_} {_} {_} | (just arrow-f) | (just inv-f) | (just id-t1) = ==refl -assoc-× {_} {_} {t1} {t1} {t0} {t1} {_} {_} {_} | (just arrow-g) | (just inv-f) | (just id-t1) = ==refl -assoc-× {_} {_} {t1} {t1} {t1} {t0} {_} {_} {_} | (just inv-f) | (just id-t1) | (just id-t1) = ==refl +_×_ : ∀ {c₁ } → { Object : Set c₁} → {a b c : Object } → ( TwoHom {c₁} {Object} b c ) → ( TwoHom {c₁} {Object} a b ) + → ( TwoHom {c₁} {Object} a c ) +_×_ {c₁} {Object} {a} {b} {c} f g = record { } -TwoId : {c₁ c₂ : Level } (a : TwoObject {c₁} ) → (TwoHom {c₁} {c₂ } a a ) -TwoId {_} {_} t0 = record { hom = just id-t0 } -TwoId {_} {_} t1 = record { hom = just id-t1 } - -open import maybeCat +TwoId : {c₁ : Level } {Object : Set c₁} (a : Object ) → (TwoHom {c₁} {Object} a a ) +TwoId {_} a = record { } --- identityL {c₁} {c₂} {_} {b} {nothing} = let open ==-Reasoning {c₁} {c₂} in --- begin --- (TwoId b × nothing) --- ==⟨ {!!} ⟩ --- nothing --- ∎ open import Relation.Binary -TwoCat : {c₁ c₂ ℓ : Level } → Category c₁ c₂ c₂ -TwoCat {c₁} {c₂} {ℓ} = record { - Obj = TwoObject {c₁} ; - Hom = λ a b → ( TwoHom {c₁} {c₂ } a b ) ; - _o_ = λ{a} {b} {c} x y → _×_ {c₁ } { c₂} {a} {b} {c} x y ; - _≈_ = λ x y → hom x == hom y ; - Id = λ{a} → TwoId {c₁ } { c₂} a ; +TwoCat : {c₁ ℓ : Level } → (Object : Set c₁) → Category c₁ c₁ c₁ +TwoCat {c₁} {ℓ} Object = record { + Obj = Object ; + Hom = λ a b → ( TwoHom {c₁} {Object} a b ) ; + _o_ = λ{a} {b} {c} x y → _×_ {c₁ } {Object} {a} {b} {c} x y ; + _≈_ = λ x y → x ≡ y ; + Id = λ{a} → TwoId {c₁ } {Object} a ; isCategory = record { - isEquivalence = record {refl = ==refl ; trans = ==trans ; sym = ==sym } ; - identityL = λ{a b f} → identityL {c₁} {c₂ } {a} {b} {f} ; - identityR = λ{a b f} → identityR {c₁} {c₂ } {a} {b} {f} ; - o-resp-≈ = λ{a b c f g h i} → o-resp-≈ {c₁} {c₂ } {a} {b} {c} {f} {g} {h} {i} ; - associative = λ{a b c d f g h } → assoc-× {c₁} {c₂} {a} {b} {c} {d} {f} {g} {h} + isEquivalence = record {refl = refl ; trans = ≡-trans ; sym = ≡-sym} ; + identityL = λ{a b f} → identityL {a} {b} {f} ; + identityR = λ{a b f} → identityR {a} {b} {f} ; + o-resp-≈ = λ{a b c f g h i} → o-resp-≈ {a} {b} {c} {f} {g} {h} {i} ; + associative = λ{a b c d f g h } → assoc-× {a} {b} {c} {d} {f} {g} {h} } } where - identityL : {c₁ c₂ : Level } {A B : TwoObject {c₁}} {f : ( TwoHom {c₁} {c₂ } A B) } → hom ((TwoId B) × f) == hom f - identityL {c₁} {c₂} {_} {_} {f} with hom f - identityL {c₁} {c₂} {t0} {t0} {_} | nothing = nothing - identityL {c₁} {c₂} {t0} {t1} {_} | nothing = nothing - identityL {c₁} {c₂} {t1} {t0} {_} | nothing = nothing - identityL {c₁} {c₂} {t1} {t1} {_} | nothing = nothing - identityL {c₁} {c₂} {t1} {t0} {_} | just inv-f = ==refl - identityL {c₁} {c₂} {t1} {t1} {_} | just id-t1 = ==refl - identityL {c₁} {c₂} {t0} {t0} {_} | just id-t0 = ==refl - identityL {c₁} {c₂} {t0} {t1} {_} | just arrow-f = ==refl - identityL {c₁} {c₂} {t0} {t1} {_} | just arrow-g = ==refl - identityR : {c₁ c₂ : Level } {A B : TwoObject {c₁}} {f : ( TwoHom {c₁} {c₂ } A B) } → hom ( f × TwoId A ) == hom f - identityR {c₁} {c₂} {_} {_} {f} with hom f - identityR {c₁} {c₂} {t0} {t0} {_} | nothing = nothing - identityR {c₁} {c₂} {t0} {t1} {_} | nothing = nothing - identityR {c₁} {c₂} {t1} {t0} {_} | nothing = nothing - identityR {c₁} {c₂} {t1} {t1} {_} | nothing = nothing - identityR {c₁} {c₂} {t1} {t0} {_} | just inv-f = ==refl - identityR {c₁} {c₂} {t1} {t1} {_} | just id-t1 = ==refl - identityR {c₁} {c₂} {t0} {t0} {_} | just id-t0 = ==refl - identityR {c₁} {c₂} {t0} {t1} {_} | just arrow-f = ==refl - identityR {c₁} {c₂} {t0} {t1} {_} | just arrow-g = ==refl - o-resp-≈ : {c₁ c₂ : Level } {A B C : TwoObject {c₁} } {f g : ( TwoHom {c₁} {c₂ } A B)} {h i : ( TwoHom B C)} → - hom f == hom g → hom h == hom i → hom ( h × f ) == hom ( i × g ) - o-resp-≈ {c₁} {c₂} {a} {b} {c} {f} {g} {h} {i} f==g h==i = - let open ==-Reasoning {c₁} {c₂ } in begin - hom ( h × f ) - ==⟨⟩ - comp (hom h) (hom f) - ==⟨ ==cong (λ x → comp ( hom h ) x ) f==g ⟩ - comp (hom h) (hom g) - ==⟨ ==cong (λ x → comp x ( hom g ) ) h==i ⟩ - comp (hom i) (hom g) - ==⟨⟩ - hom ( i × g ) - ∎ + identityL : {A B : Object } {f : ( TwoHom {c₁} {Object} A B) } → ((TwoId {_} {Object} B) × f) ≡ f + identityL = {!!} + identityR : {A B : Object } {f : ( TwoHom {_} {Object} A B) } → ( f × TwoId A ) ≡ f + identityR = {!!} + o-resp-≈ : {A B C : Object } {f g : ( TwoHom {c₁} {Object} A B)} {h i : ( TwoHom B C)} → + f ≡ g → h ≡ i → ( h × f ) ≡ ( i × g ) + o-resp-≈ {a} {b} {c} {f} {g} {h} {i} f==g h==i = {!!} + assoc-× : {a b c d : Object } + {f : (TwoHom {c₁} {Object} c d )} → + {g : (TwoHom {c₁} {Object} b c )} → + {h : (TwoHom {c₁} {Object} a b )} → + ( f × (g × h)) ≡ ((f × g) × h ) + assoc-× {a} {b} {c} {d} {f} {g} {h} = {!!} + + record Nil {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ) where field @@ -298,47 +85,9 @@ open Nil -justFunctor : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) → Nil A → Functor (MaybeCat A ) A -justFunctor{c₁} {c₂} {ℓ} A none = record { - FObj = λ a → fobj a - ; FMap = λ {a} {b} f → fmap {a} {b} f - ; isFunctor = record { - identity = λ{x} → identity {x} - ; distr = λ {a} {b} {c} {f} {g} → distr2 {a} {b} {c} {f} {g} - ; ≈-cong = λ {a} {b} {c} {f} → ≈-cong {a} {b} {c} {f} - } - } where - MA = MaybeCat A - fobj : Obj MA → Obj A - fobj = λ x → x - fmap : {x y : Obj MA } → Hom MA x y → Hom A x y - fmap {x} {y} f with MaybeHom.hom f - fmap {x} {y} _ | nothing = nil none - fmap {x} {y} _ | (just f) = f - identity : {x : Obj MA} → A [ fmap (id1 MA x) ≈ id1 A (fobj x) ] - identity = let open ≈-Reasoning (A) in refl-hom - distr2 : {a : Obj MA} {b : Obj MA} {c : Obj MA} {f : Hom MA a b} {g : Hom MA b c} → A [ fmap (MA [ g o f ]) ≈ A [ fmap g o fmap f ] ] - distr2 {a} {b} {c} {f} {g} with MaybeHom.hom f | MaybeHom.hom g - distr2 | nothing | nothing = let open ≈-Reasoning (A) in sym ( nil-idR none ) - distr2 | nothing | just ga = let open ≈-Reasoning (A) in sym ( nil-idR none ) - distr2 | just fa | nothing = let open ≈-Reasoning (A) in sym ( nil-idL none ) - distr2 | just f | just g = let open ≈-Reasoning (A) in refl-hom - ≈-cong : {a : Obj MA} {b : Obj MA} {f g : Hom MA a b} → MA [ f ≈ g ] → A [ fmap f ≈ fmap g ] - ≈-cong {a} {b} {f} {g} eq with MaybeHom.hom f | MaybeHom.hom g - ≈-cong {a} {b} {f} {g} nothing | nothing | nothing = let open ≈-Reasoning (A) in refl-hom - ≈-cong {a} {b} {f} {g} (just eq) | just _ | just _ = eq - --- indexFunctor' : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (none : Nil A) ( a b : Obj A) ( f g : Hom A a b ) --- → ( obj→ : Obj A → TwoObject {c₁} ) --- → ( hom→ : { a b : Obj A } → Hom A a b → Arrow t0 t0 (obj→ a) (obj→ b) ) --- → ( { x y : Obj A } { h i : Hom A x y } → A [ h ≈ i ] → hom→ h ≡ hom→ i ) --- → Functor A A --- this one does not work on fmap ( g o f ) ≈ ( fmap g o fmap f ) --- because g o f can be arrow-f when g is arrow-g --- ideneity and ≈-cong are easy -indexFunctor : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (none : Nil A) ( a b : Obj A) ( f g : Hom A a b ) → Functor (TwoCat {c₁} {c₂} {c₂} ) A +indexFunctor : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (none : Nil A) ( a b : Obj A) ( f g : Hom A a b ) → Functor (TwoCat Object4 ) A indexFunctor {c₁} {c₂} {ℓ} A none a b f g = record { FObj = λ a → fobj a ; FMap = λ {a} {b} f → fmap {a} {b} f @@ -348,76 +97,19 @@ ; ≈-cong = λ {a} {b} {c} {f} → ≈-cong {a} {b} {c} {f} } } where - I = TwoCat {c₁} {c₂} {ℓ} + I = TwoCat Object4 fobj : Obj I → Obj A - fobj t0 = a - fobj t1 = b - fmap : {x y : Obj I } → (TwoHom {c₁} {c₂} x y ) → Hom A (fobj x) (fobj y) - fmap {x} {y} h with hom h - fmap {t0} {t0} h | just id-t0 = id1 A a - fmap {t1} {t1} h | just id-t1 = id1 A b - fmap {t0} {t1} h | just arrow-f = f - fmap {t0} {t1} h | just arrow-g = g - fmap {_} {_} h | _ = nil none + fobj = {!!} + fmap : {x y : Obj I } → (TwoHom {c₁} {Object4} x y ) → Hom A (fobj x) (fobj y) + fmap {_} {_} h = {!!} open ≈-Reasoning A ≈-cong : {a : Obj I} {b : Obj I} {f g : Hom I a b} → I [ f ≈ g ] → A [ fmap f ≈ fmap g ] - ≈-cong {a} {b} {f1} {g1} f≈g with hom f1 | hom g1 - ≈-cong {t0} {t0} {f1} {g1} nothing | nothing | nothing = refl-hom - ≈-cong {t0} {t1} {f1} {g1} nothing | nothing | nothing = refl-hom - ≈-cong {t1} {t0} {f1} {g1} nothing | nothing | nothing = refl-hom - ≈-cong {t1} {t1} {f1} {g1} nothing | nothing | nothing = refl-hom - ≈-cong {t0} {t0} {f1} {g1} (just refl) | just id-t0 | just id-t0 = refl-hom - ≈-cong {t1} {t1} {f1} {g1} (just refl) | just id-t1 | just id-t1 = refl-hom - ≈-cong {t0} {t1} {f1} {g1} (just refl) | just arrow-f | just arrow-f = refl-hom - ≈-cong {t0} {t1} {f1} {g1} (just refl) | just arrow-g | just arrow-g = refl-hom - ≈-cong {t1} {t0} {f1} {g1} (just refl) | just inv-f | just inv-f = refl-hom + ≈-cong {a} {b} {f1} {g1} f≈g = {!!} identity : {x : Obj I} → A [ fmap ( id1 I x ) ≈ id1 A (fobj x) ] - identity {t0} = refl-hom - identity {t1} = refl-hom + identity = {!!} distr1 : {a₁ : Obj I} {b₁ : Obj I} {c : Obj I} {f₁ : Hom I a₁ b₁} {g₁ : Hom I b₁ c} → A [ fmap (I [ g₁ o f₁ ]) ≈ A [ fmap g₁ o fmap f₁ ] ] - distr1 {a1} {b1} {c1} {f1} {g1} with hom g1 | hom f1 - distr1 {t0} {t0} {t0} {f1} {g1} | nothing | nothing = sym ( nil-idR none ) - distr1 {t0} {t0} {t1} {f1} {g1} | nothing | nothing = sym ( nil-idR none ) - distr1 {t0} {t1} {t0} {f1} {g1} | nothing | nothing = sym ( nil-idR none ) - distr1 {t0} {t1} {t1} {f1} {g1} | nothing | nothing = sym ( nil-idR none ) - distr1 {t1} {t0} {t0} {f1} {g1} | nothing | nothing = sym ( nil-idR none ) - distr1 {t1} {t0} {t1} {f1} {g1} | nothing | nothing = sym ( nil-idR none ) - distr1 {t1} {t1} {t0} {f1} {g1} | nothing | nothing = sym ( nil-idR none ) - distr1 {t1} {t1} {t1} {f1} {g1} | nothing | nothing = sym ( nil-idR none ) - distr1 {t0} {t0} {t0} {f1} {g1} | nothing | just id-t0 = sym ( nil-idL none ) - distr1 {t0} {t0} {t1} {f1} {g1} | nothing | just id-t0 = sym ( nil-idL none ) - distr1 {t1} {t1} {t0} {f1} {g1} | nothing | just id-t1 = sym ( nil-idL none ) - distr1 {t1} {t1} {t1} {f1} {g1} | nothing | just id-t1 = sym ( nil-idL none ) - distr1 {t0} {t1} {t1} {f1} {g1} | nothing | just arrow-f = sym ( nil-idL none ) - distr1 {t0} {t1} {t0} {f1} {g1} | nothing | just arrow-f = sym ( nil-idL none ) - distr1 {t0} {t1} {t1} {f1} {g1} | nothing | just arrow-g = sym ( nil-idL none ) - distr1 {t0} {t1} {t0} {f1} {g1} | nothing | just arrow-g = sym ( nil-idL none ) - distr1 {t1} {t0} {t0} {f1} {g1} | nothing | just inv-f = sym ( nil-idL none ) - distr1 {t1} {t0} {t1} {f1} {g1} | nothing | just inv-f = sym ( nil-idL none ) - distr1 {t0} {t0} {t0} {f1} {g1} | just id-t0 | nothing = sym ( nil-idR none ) - distr1 {t1} {t0} {t0} {f1} {g1} | just id-t0 | nothing = sym ( nil-idR none ) - distr1 {t0} {t1} {t1} {f1} {g1} | just id-t1 | nothing = sym ( nil-idR none ) - distr1 {t1} {t1} {t1} {f1} {g1} | just id-t1 | nothing = sym ( nil-idR none ) - distr1 {t0} {t0} {t1} {f1} {g1} | just arrow-f | nothing = sym ( nil-idR none ) - distr1 {t1} {t0} {t1} {f1} {g1} | just arrow-f | nothing = sym ( nil-idR none ) - distr1 {t0} {t0} {t1} {f1} {g1} | just arrow-g | nothing = sym ( nil-idR none ) - distr1 {t1} {t0} {t1} {f1} {g1} | just arrow-g | nothing = sym ( nil-idR none ) - distr1 {t0} {t1} {t0} {f1} {g1} | just inv-f | nothing = sym ( nil-idR none ) - distr1 {t1} {t1} {t0} {f1} {g1} | just inv-f | nothing = sym ( nil-idR none ) - distr1 {t0} {t0} {t0} {f1} {g1} | just id-t0 | just id-t0 = sym idL - distr1 {t1} {t0} {t0} {f1} {g1} | just id-t0 | just inv-f = sym idL - distr1 {t0} {t0} {t1} {f1} {g1} | just arrow-f | just id-t0 = sym idR - distr1 {t0} {t0} {t1} {f1} {g1} | just arrow-g | just id-t0 = sym idR - distr1 {t1} {t1} {t1} {f1} {g1} | just id-t1 | just id-t1 = sym idL - distr1 {t0} {t1} {t1} {f1} {g1} | just id-t1 | just arrow-f = sym idL - distr1 {t0} {t1} {t1} {f1} {g1} | just id-t1 | just arrow-g = sym idL - distr1 {t1} {t1} {t0} {f1} {g1} | just inv-f | just id-t1 = sym ( nil-idL none ) - distr1 {t0} {t1} {t0} {_} {_} | (just inv-f) | (just _) = sym ( nil-idL none ) - distr1 {t1} {t0} {t1} {_} {_} | (just arrow-f) | (just _) = sym ( nil-idR none ) - distr1 {t1} {t0} {t1} {_} {_} | (just arrow-g) | (just _) = sym ( nil-idR none ) - - + distr1 {a1} {b1} {c1} {f1} {g1} = {!!} --- Equalizer --- f @@ -441,11 +133,11 @@ open Limit -I' : {c₁ c₂ ℓ : Level} → Category c₁ c₂ c₂ -I' {c₁} {c₂} {ℓ} = TwoCat {c₁} {c₂} {ℓ} +I' : {c₁ : Level} → Category c₁ c₁ c₁ +I' {c₁} = TwoCat {c₁} Object4 lim-to-equ : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( none : Nil A ) - (lim : ( Γ : Functor (I' {c₁} {c₂} {ℓ}) A ) → {a0 : Obj A } + (lim : ( Γ : Functor (I' {c₁} ) A ) → {a0 : Obj A } {u : NTrans I' A ( K A I' a0 ) Γ } → Limit A I' Γ a0 u ) -- completeness → {a b c : Obj A} (f g : Hom A a b ) → (e : Hom A c a ) → (fe=ge : A [ A [ f o e ] ≈ A [ g o e ] ] ) → IsEqualizer A e f g @@ -456,98 +148,15 @@ ; uniqueness = λ {d} {h} {fh=gh} {k'} → uniquness d h fh=gh k' } where open ≈-Reasoning A - I : Category c₁ c₂ c₂ - I = I' {c₁} {c₂} {ℓ} + I : Category c₁ c₁ c₁ + I = I' {c₁} Γ : Functor I A Γ = indexFunctor {c₁} {c₂} {ℓ} A none a b f g nmap : (x : Obj I ) ( d : Obj (A) ) (h : Hom A d a ) → Hom A (FObj (K A I d) x) (FObj Γ x) - nmap x d h with x - ... | t0 = h - ... | t1 = A [ f o h ] + nmap x d h = {!!} commute1 : {x y : Obj I} {f' : Hom I x y} (d : Obj A) (h : Hom A d a ) → A [ A [ f o h ] ≈ A [ g o h ] ] → A [ A [ FMap Γ f' o nmap x d h ] ≈ A [ nmap y d h o FMap (K A I d) f' ] ] - commute1 {x} {y} {f'} d h fh=gh with hom f' - commute1 {t0} {t0} {f'} d h fh=gh | nothing = begin - nil none o h - ≈⟨ car ( nil-id none ) ⟩ - id1 A a o h - ≈⟨ idL ⟩ - h - ≈↑⟨ idR ⟩ - h o id1 A d - ∎ - commute1 {t0} {t1} {f'} d h fh=gh | nothing = begin - nil none o h - ≈↑⟨ car ( nil-idL none ) ⟩ - (nil none o f ) o h - ≈⟨ car ( car ( nil-id none ) ) ⟩ - (id1 A b o f ) o h - ≈⟨ car idL ⟩ - f o h - ≈↑⟨ idR ⟩ - (f o h ) o id1 A d - ∎ - commute1 {t1} {t0} {f'} d h fh=gh | nothing = begin - nil none o ( f o h ) - ≈⟨ assoc ⟩ - ( nil none o f ) o h - ≈⟨ car ( nil-idL none ) ⟩ - nil none o h - ≈⟨ car ( nil-id none ) ⟩ -- nil-id is need here - id1 A a o h - ≈⟨ idL ⟩ - h - ≈↑⟨ idR ⟩ - h o id1 A d - ∎ - commute1 {t1} {t1} {f'} d h fh=gh | nothing = begin - nil none o ( f o h ) - ≈⟨ car ( nil-id none ) ⟩ - id1 A b o ( f o h ) - ≈⟨ idL ⟩ - f o h - ≈↑⟨ idR ⟩ - ( f o h ) o id1 A d - ∎ - commute1 {t1} {t0} {f'} d h fh=gh | just inv-f = begin - nil none o ( f o h ) - ≈⟨ assoc ⟩ - ( nil none o f ) o h - ≈⟨ car ( nil-idL none ) ⟩ - nil none o h - ≈⟨ car ( nil-id none ) ⟩ - id1 A a o h - ≈⟨ idL ⟩ - h - ≈↑⟨ idR ⟩ - h o id1 A d - ∎ - commute1 {t0} {t1} {f'} d h fh=gh | just arrow-f = begin - f o h - ≈↑⟨ idR ⟩ - (f o h ) o id1 A d - ∎ - commute1 {t0} {t1} {f'} d h fh=gh | just arrow-g = begin - g o h - ≈↑⟨ fh=gh ⟩ - f o h - ≈↑⟨ idR ⟩ - (f o h ) o id1 A d - ∎ - commute1 {t0} {t0} {f'} d h fh=gh | just id-t0 = begin - id1 A a o h - ≈⟨ idL ⟩ - h - ≈↑⟨ idR ⟩ - h o id1 A d - ∎ - commute1 {t1} {t1} {f'} d h fh=gh | just id-t1 = begin - id1 A b o ( f o h ) - ≈⟨ idL ⟩ - f o h - ≈↑⟨ idR ⟩ - ( f o h ) o id1 A d - ∎ + commute1 = {!!} nat1 : (d : Obj A) → (h : Hom A d a ) → A [ A [ f o h ] ≈ A [ g o h ] ] → NTrans I A (K A I d) Γ nat1 d h fh=gh = record { TMap = λ x → nmap x d h ; @@ -565,32 +174,14 @@ h ∎ uniq-nat : {i : Obj I} → (d : Obj A ) (h : Hom A d a ) ( k' : Hom A d c ) → A [ A [ e o k' ] ≈ h ] → A [ A [ nmap i c e o k' ] ≈ nmap i d h ] - uniq-nat {t0} d h k' ek'=h = begin - nmap t0 c e o k' - ≈⟨⟩ - e o k' - ≈⟨ ek'=h ⟩ - h - ≈⟨⟩ - nmap t0 d h - ∎ - uniq-nat {t1} d h k' ek'=h = begin - nmap t1 c e o k' - ≈⟨⟩ - (f o e ) o k' - ≈↑⟨ assoc ⟩ - f o ( e o k' ) - ≈⟨ cdr ek'=h ⟩ - f o h - ≈⟨⟩ - nmap t1 d h - ∎ + uniq-nat d h k' ek'=h = {!!} uniquness : (d : Obj A ) (h : Hom A d a ) → ( fh=gh : A [ A [ f o h ] ≈ A [ g o h ] ] ) → ( k' : Hom A d c ) → A [ A [ e o k' ] ≈ h ] → A [ k h fh=gh ≈ k' ] uniquness d h fh=gh k' ek'=h = begin k h fh=gh - ≈⟨ limit-uniqueness eqlim k' ( λ{i} → uniq-nat {i} d h k' ek'=h ) ⟩ + -- ≈⟨ limit-uniqueness eqlim k' ( λ{i} → uniq-nat {i} d h k' ek'=h ) ⟩ + ≈⟨ {!!} ⟩ k' ∎