Mercurial > hg > Members > kono > Proof > category
view limit-to.agda @ 421:06ffcad985ac
fix free-monoid
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 24 Mar 2016 12:07:32 +0900 |
| parents | 3e44951b9bdb |
| children | 3a4a99a8edbe |
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open import Category -- https://github.com/konn/category-agda open import Level module limit-to where 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 --- ------> --- 0 1 --- ------> --- g data TwoObject {c₁ : Level} : Set c₁ where t0 : TwoObject t1 : TwoObject 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 RawHom : Maybe ( Arrow {c₁} {c₂} t0 t1 a b ) open TwoHom hom : ∀{ c₁ c₂ } { a b : TwoObject {c₁} } -> ∀ (f : TwoHom {c₁} {c₂ } a b ) → Maybe ( Arrow {c₁} {c₂} t0 t1 a b ) hom {_} {_} {a} {b} f with RawHom f hom {_} {_} {t0} {t0} _ | just id-t0 = just id-t0 hom {_} {_} {t1} {t1} _ | just id-t1 = just id-t1 hom {_} {_} {t0} {t1} _ | just arrow-f = just arrow-f hom {_} {_} {t0} {t1} _ | just arrow-g = just arrow-g hom {_} {_} {t1} {t0} _ | just inv-f = just inv-f hom {_} {_} {_ } {_ } _ | _ = nothing open TwoHom -- arrow composition _×_ : ∀ {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 with hom f | hom g _×_ {_} {_} {_} {_} {_} f g | nothing | _ = record { RawHom = nothing } _×_ {_} {_} {_} {_} {_} f g | just _ | nothing = record { RawHom = nothing } _×_ {_} {_} {t0} {t1} {t1} f g | just id-t1 | just arrow-f = record { RawHom = just arrow-f } _×_ {_} {_} {t0} {t1} {t1} f g | just id-t1 | just arrow-g = record { RawHom = just arrow-g } _×_ {_} {_} {t1} {t1} {t1} f g | just id-t1 | just id-t1 = record { RawHom = just id-t1 } _×_ {_} {_} {t1} {t1} {t0} f g | just inv-f | just id-t1 = record { RawHom = just inv-f } _×_ {_} {_} {t0} {t0} {t1} f g | just arrow-f | just id-t0 = record { RawHom = just arrow-f } _×_ {_} {_} {t0} {t0} {t1} f g | just arrow-g | just id-t0 = record { RawHom = just arrow-g } _×_ {_} {_} {t0} {t0} {t0} f g | just id-t0 | just id-t0 = record { RawHom = just id-t0 } _×_ {_} {_} {t1} {t0} {t0} f g | just id-t0 | just inv-f = record { RawHom = just inv-f } _×_ {_} {_} {_} {_} {_} f g | just _ | just _ = record { RawHom = nothing } _==_ : ∀{ 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 { RawHom = just id-t1 } map2hom {_} {_} {t0} {t1} (just arrow-f) = record { RawHom = just arrow-f } map2hom {_} {_} {t0} {t1} (just arrow-g) = record { RawHom = just arrow-g } map2hom {_} {_} {t0} {t0} (just id-t0) = record { RawHom = just id-t0 } map2hom {_} {_} {_} {_} _ = record { RawHom = 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 ) ==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 } -> ∀ {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 a b ) ) -> x ≡ y → f x == f y ==cong f 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 -- 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 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 TwoId : {c₁ c₂ : Level } (a : TwoObject {c₁} ) -> (TwoHom {c₁} {c₂ } a a ) TwoId {_} {_} t0 = record { RawHom = just id-t0 } TwoId {_} {_} t1 = record { RawHom = just id-t1 } open import maybeCat -- 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_ = \{a} {b} {c} x y -> _×_ {c₁ } { c₂} {a} {b} {c} x y ; _≈_ = \x y -> hom x == hom y ; Id = \{a} -> TwoId {c₁ } { c₂} a ; isCategory = record { isEquivalence = record {refl = ==refl ; trans = ==trans ; sym = ==sym } ; identityL = \{a b f} -> identityL {c₁} {c₂ } {a} {b} {f} ; identityR = \{a b f} -> identityR {c₁} {c₂ } {a} {b} {f} ; o-resp-≈ = \{a b c f g h i} -> o-resp-≈ {c₁} {c₂ } {a} {b} {c} {f} {g} {h} {i} ; associative = \{a b c d f g h } -> assoc-× {c₁} {c₂} {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 with hom f | hom g | hom h | hom i o-resp-≈ {c₁} {c₂} {t0} {t0} {t0} {f} {g} {h} {i} nothing nothing | nothing | nothing | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {t0} {t0} {t1} {f} {g} {h} {i} nothing nothing | nothing | nothing | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {t0} {t1} {t0} {f} {g} {h} {i} nothing nothing | nothing | nothing | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {t0} {t1} {t1} {f} {g} {h} {i} nothing nothing | nothing | nothing | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {t1} {t0} {t0} {f} {g} {h} {i} nothing nothing | nothing | nothing | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {t1} {t0} {t1} {f} {g} {h} {i} nothing nothing | nothing | nothing | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {t1} {t1} {t0} {f} {g} {h} {i} nothing nothing | nothing | nothing | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {t1} {t1} {t1} {f} {g} {h} {i} nothing nothing | nothing | nothing | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {t0} {t0} {t0} {f} {g} {h} {i} f≡g nothing | just _ | just _ | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {t0} {t0} {t1} {f} {g} {h} {i} f≡g nothing | just _ | just _ | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {t0} {t1} {t0} {f} {g} {h} {i} f≡g nothing | just _ | just _ | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {t0} {t1} {t1} {f} {g} {h} {i} f≡g nothing | just _ | just _ | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {t1} {t0} {t0} {f} {g} {h} {i} f≡g nothing | just _ | just _ | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {t1} {t0} {t1} {f} {g} {h} {i} f≡g nothing | just _ | just _ | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {t1} {t1} {t0} {f} {g} {h} {i} f≡g nothing | just _ | just _ | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {t1} {t1} {t1} {f} {g} {h} {i} f≡g nothing | just _ | just _ | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {t0} {t0} {t0} {f} {g} {h} {i} nothing h≡i | nothing | nothing | just _ | just _ = nothing o-resp-≈ {c₁} {c₂} {t0} {t0} {t1} {f} {g} {h} {i} nothing h≡i | nothing | nothing | just _ | just _ = nothing o-resp-≈ {c₁} {c₂} {t0} {t1} {t0} {f} {g} {h} {i} nothing h≡i | nothing | nothing | just _ | just _ = nothing o-resp-≈ {c₁} {c₂} {t0} {t1} {t1} {f} {g} {h} {i} nothing h≡i | nothing | nothing | just _ | just _ = nothing o-resp-≈ {c₁} {c₂} {t1} {t0} {t0} {f} {g} {h} {i} nothing h≡i | nothing | nothing | just _ | just _ = nothing o-resp-≈ {c₁} {c₂} {t1} {t0} {t1} {f} {g} {h} {i} nothing h≡i | nothing | nothing | just _ | just _ = nothing o-resp-≈ {c₁} {c₂} {t1} {t1} {t0} {f} {g} {h} {i} nothing h≡i | nothing | nothing | just _ | just _ = nothing o-resp-≈ {c₁} {c₂} {t1} {t1} {t1} {f} {g} {h} {i} nothing h≡i | nothing | nothing | just _ | just _ = nothing o-resp-≈ {c₁} {c₂} {_} {_} {_} {f} {g} {h} {i} f≡g (just refl) | nothing | just _ | just jh | just .jh = {!!} o-resp-≈ {c₁} {c₂} {_} {_} {_} {f} {g} {h} {i} f≡g (just refl) | just _ | nothing | just jh | just .jh = {!!} o-resp-≈ {c₁} {c₂} {_} {_} {_} {f} {g} {h} {i} (just refl) h≡i | just jf | just .jf | nothing | just _ = {!!} o-resp-≈ {c₁} {c₂} {_} {_} {_} {f} {g} {h} {i} (just refl) h≡i | just jf | just .jf | just _ | nothing = {!!} o-resp-≈ {c₁} {c₂} {t0} {t0} {t0} {f} {g} {h} {i} f≡g h≡i | just id-t0 | nothing | just id-t0 | nothing = ? o-resp-≈ {c₁} {c₂} {_} {_} {_} {f} {g} {h} {i} f≡g h≡i | just _ | nothing | just _ | nothing = {!!} o-resp-≈ {c₁} {c₂} {_} {_} {_} {f} {g} {h} {i} f≡g h≡i | nothing | just _ | just _ | nothing = {!!} o-resp-≈ {c₁} {c₂} {_} {_} {_} {f} {g} {h} {i} f≡g h≡i | nothing | just _ | nothing | just _ = {!!} o-resp-≈ {c₁} {c₂} {_} {_} {_} {f} {g} {h} {i} f≡g h≡i | just _ | nothing | nothing | just _ = {!!} o-resp-≈ {c₁} {c₂} {t0} {t0} {t0} {f} {g} {h} {i} nothing h≡i | nothing | nothing | nothing | just _ = nothing o-resp-≈ {c₁} {c₂} {t0} {t0} {t1} {f} {g} {h} {i} nothing h≡i | nothing | nothing | nothing | just _ = nothing o-resp-≈ {c₁} {c₂} {t0} {t1} {t0} {f} {g} {h} {i} nothing h≡i | nothing | nothing | nothing | just _ = nothing o-resp-≈ {c₁} {c₂} {t0} {t1} {t1} {f} {g} {h} {i} nothing h≡i | nothing | nothing | nothing | just _ = nothing o-resp-≈ {c₁} {c₂} {t1} {t0} {t0} {f} {g} {h} {i} nothing h≡i | nothing | nothing | nothing | just _ = nothing o-resp-≈ {c₁} {c₂} {t1} {t0} {t1} {f} {g} {h} {i} nothing h≡i | nothing | nothing | nothing | just _ = nothing o-resp-≈ {c₁} {c₂} {t1} {t1} {t0} {f} {g} {h} {i} nothing h≡i | nothing | nothing | nothing | just _ = nothing o-resp-≈ {c₁} {c₂} {t1} {t1} {t1} {f} {g} {h} {i} nothing h≡i | nothing | nothing | nothing | just _ = nothing o-resp-≈ {c₁} {c₂} {t0} {t0} {t0} {f} {g} {h} {i} nothing h≡i | nothing | nothing | just _ | nothing = nothing o-resp-≈ {c₁} {c₂} {t0} {t0} {t1} {f} {g} {h} {i} nothing h≡i | nothing | nothing | just _ | nothing = nothing o-resp-≈ {c₁} {c₂} {t0} {t1} {t0} {f} {g} {h} {i} nothing h≡i | nothing | nothing | just _ | nothing = nothing o-resp-≈ {c₁} {c₂} {t0} {t1} {t1} {f} {g} {h} {i} nothing h≡i | nothing | nothing | just _ | nothing = nothing o-resp-≈ {c₁} {c₂} {t1} {t0} {t0} {f} {g} {h} {i} nothing h≡i | nothing | nothing | just _ | nothing = nothing o-resp-≈ {c₁} {c₂} {t1} {t0} {t1} {f} {g} {h} {i} nothing h≡i | nothing | nothing | just _ | nothing = nothing o-resp-≈ {c₁} {c₂} {t1} {t1} {t0} {f} {g} {h} {i} nothing h≡i | nothing | nothing | just _ | nothing = nothing o-resp-≈ {c₁} {c₂} {t1} {t1} {t1} {f} {g} {h} {i} nothing h≡i | nothing | nothing | just _ | nothing = nothing o-resp-≈ {c₁} {c₂} {t0} {t0} {t0} {f} {g} {h} {i} f≡g nothing | nothing | just _ | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {_} {_} {_} {f} {g} {h} {i} f≡g h≡i | nothing | just _ | nothing | nothing = {!!} o-resp-≈ {c₁} {c₂} {t0} {t0} {t0} {f} {g} {h} {i} f≡g nothing | just _ | nothing | nothing | nothing = nothing o-resp-≈ {c₁} {c₂} {_} {_} {_} {f} {g} {h} {i} f≡g h≡i | just _ | nothing | nothing | nothing = {!!} o-resp-≈ {c₁} {c₂} {t0} {t0} {t0} {f} {g} {h} {i} (just refl) (just refl) | just id-t0 | just id-t0 | just id-t0 | just id-t0 = ==refl o-resp-≈ {c₁} {c₂} {t1} {t1} {t1} {f} {g} {h} {i} (just refl) (just refl) | just id-t1 | just id-t1 | just id-t1 | just id-t1 = ==refl o-resp-≈ {c₁} {c₂} {t0} {t1} {t1} {f} {g} {h} {i} (just refl) (just refl) | just arrow-f | just arrow-f | just id-t1 | just id-t1 = ==refl o-resp-≈ {c₁} {c₂} {t0} {t1} {t1} {f} {g} {h} {i} (just refl) (just refl) | just arrow-g | just arrow-g | just id-t1 | just id-t1 = ==refl o-resp-≈ {c₁} {c₂} {t0} {t0} {t1} {f} {g} {h} {i} (just refl) (just refl) | just id-t0 | just id-t0 | just arrow-f | just arrow-f = ==refl o-resp-≈ {c₁} {c₂} {t0} {t0} {t1} {f} {g} {h} {i} (just refl) (just refl) | just id-t0 | just id-t0 | just arrow-g | just arrow-g = ==refl o-resp-≈ {c₁} {c₂} {t1} {t1} {t0} {f} {g} {h} {i} (just refl) (just refl) | just id-t1 | just id-t1 | just inv-f | just inv-f = ==refl o-resp-≈ {c₁} {c₂} {t1} {t0} {t0} {f} {g} {h} {i} (just refl) (just refl) | just inv-f | just inv-f | just id-t0 | just id-t0 = ==refl o-resp-≈ {c₁} {c₂} {t1} {t0} {t1} {f} {g} {h} {i} (just refl) (just refl) | just inv-f | just inv-f | just arrow-f | just arrow-f = ==refl o-resp-≈ {c₁} {c₂} {t1} {t0} {t1} {f} {g} {h} {i} (just refl) (just refl) | just inv-f | just inv-f | just arrow-g | just arrow-g = ==refl o-resp-≈ {c₁} {c₂} {t0} {t1} {t0} {f} {g} {h} {i} (just refl) (just refl) | just arrow-f | just arrow-f | just inv-f | just inv-f = ==refl o-resp-≈ {c₁} {c₂} {t0} {t1} {t0} {f} {g} {h} {i} (just refl) (just refl) | just arrow-g | just arrow-g | just inv-f | just inv-f = ==refl -- o-resp-≈ {c₁} {c₂} {a} {b} {c} {f} {g} {h} {i} (just refl) (just refl) | just jf | just .jf | just jh | just .jh = -- let open ==-Reasoning {c₁} {c₂ } in begin -- {!!} -- ==⟨ {!!} ⟩ -- {!!} -- ∎ indexFunctor : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( a b : Obj (MaybeCat A )) ( f g : Hom A a b ) -> Functor (TwoCat {c₁} {c₂} {c₂} ) (MaybeCat A ) indexFunctor {c₁} {c₂} {ℓ} A a b f g = record { FObj = λ a → fobj a ; FMap = λ {a} {b} f → fmap {a} {b} f ; isFunctor = record { identity = \{x} -> identity {x} ; distr = \ {a} {b} {c} {f} {g} -> distr1 {a} {b} {c} {f} {g} ; ≈-cong = \ {a} {b} {c} {f} -> ≈-cong {a} {b} {c} {f} } } where I = TwoCat {c₁} {c₂} {ℓ} MA = MaybeCat A open ≈-Reasoning (MA) fobj : Obj I -> Obj A fobj t0 = a fobj t1 = b fmap : {x y : Obj I } -> (TwoHom {c₁} {c₂} x y ) -> Hom MA (fobj x) (fobj y) fmap {x} {y} h with hom h fmap {t0} {t0} h | just id-t0 = id1 MA a fmap {t1} {t1} h | just id-t1 = id1 MA b fmap {t0} {t1} h | just arrow-f = record { hom = just f } fmap {t0} {t1} h | just arrow-g = record { hom = just g } fmap {_} {_} h | _ = record { hom = nothing } identity : {x : Obj I} → MA [ fmap ( id1 I x ) ≈ id1 MA (fobj x) ] identity {t0} = refl-hom identity {t1} = refl-hom distr1 : {a₁ : Obj I} {b₁ : Obj I} {c : Obj I} {f₁ : Hom I a₁ b₁} {g₁ : Hom I b₁ c} → MA [ fmap (I [ g₁ o f₁ ]) ≈ MA [ fmap g₁ o fmap f₁ ] ] distr1 {a1} {b1} {c1} {f1} {g1} with hom g1 | hom f1 distr1 {t0} {t0} {t0} {f1} {g1} | nothing | nothing = nothing distr1 {t0} {t0} {t1} {f1} {g1} | nothing | nothing = nothing distr1 {t0} {t1} {t0} {f1} {g1} | nothing | nothing = nothing distr1 {t0} {t1} {t1} {f1} {g1} | nothing | nothing = nothing distr1 {t1} {t0} {t0} {f1} {g1} | nothing | nothing = nothing distr1 {t1} {t0} {t1} {f1} {g1} | nothing | nothing = nothing distr1 {t1} {t1} {t0} {f1} {g1} | nothing | nothing = nothing distr1 {t1} {t1} {t1} {f1} {g1} | nothing | nothing = nothing distr1 {t0} {t0} {t0} {f1} {g1} | nothing | just id-t0 = nothing distr1 {t0} {t0} {t1} {f1} {g1} | nothing | just id-t0 = nothing distr1 {t1} {t1} {t0} {f1} {g1} | nothing | just id-t1 = nothing distr1 {t1} {t1} {t1} {f1} {g1} | nothing | just id-t1 = nothing distr1 {t0} {t1} {t1} {f1} {g1} | nothing | just arrow-f = nothing distr1 {t0} {t1} {t0} {f1} {g1} | nothing | just arrow-f = nothing distr1 {t0} {t1} {t1} {f1} {g1} | nothing | just arrow-g = nothing distr1 {t0} {t1} {t0} {f1} {g1} | nothing | just arrow-g = nothing distr1 {t1} {t0} {t0} {f1} {g1} | nothing | just inv-f = nothing distr1 {t1} {t0} {t1} {f1} {g1} | nothing | just inv-f = nothing distr1 {t0} {t0} {t0} {f1} {g1} | just id-t0 | nothing = nothing distr1 {t1} {t0} {t0} {f1} {g1} | just id-t0 | nothing = nothing distr1 {t0} {t1} {t1} {f1} {g1} | just id-t1 | nothing = nothing distr1 {t1} {t1} {t1} {f1} {g1} | just id-t1 | nothing = nothing distr1 {t0} {t0} {t1} {f1} {g1} | just arrow-f | nothing = nothing distr1 {t1} {t0} {t1} {f1} {g1} | just arrow-f | nothing = nothing distr1 {t0} {t0} {t1} {f1} {g1} | just arrow-g | nothing = nothing distr1 {t1} {t0} {t1} {f1} {g1} | just arrow-g | nothing = nothing distr1 {t0} {t1} {t0} {f1} {g1} | just inv-f | nothing = nothing distr1 {t1} {t1} {t0} {f1} {g1} | just inv-f | nothing = nothing 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 idL distr1 {t0} {t1} {t0} {_} {_} | (just inv-f) | (just _) = nothing distr1 {t1} {t0} {t1} {_} {_} | (just arrow-f) | (just _) = nothing distr1 {t1} {t0} {t1} {_} {_} | (just arrow-g) | (just _) = nothing ≈-cong : {a : Obj I} {b : Obj I} {f g : Hom I a b} → I [ f ≈ g ] → MA [ fmap f ≈ fmap g ] ≈-cong {_} {_} {f1} {g1} f≈g with hom f1 | hom g1 ≈-cong {t0} {t0} {f1} {g1} f≈g | nothing | nothing = nothing ≈-cong {t0} {t1} {f1} {g1} f≈g | nothing | nothing = nothing ≈-cong {t1} {t0} {f1} {g1} f≈g | nothing | nothing = nothing ≈-cong {t1} {t1} {f1} {g1} f≈g | nothing | nothing = nothing ≈-cong {t0} {t0} {f1} {g1} f≈g | nothing | just id-t0 = {!!} ≈-cong {t0} {t1} {f1} {g1} f≈g | nothing | just arrow-f = {!!} ≈-cong {t0} {t1} {f1} {g1} f≈g | nothing | just arrow-g = {!!} ≈-cong {t1} {t0} {f1} {g1} f≈g | nothing | just inv-f = nothing ≈-cong {t1} {t1} {f1} {g1} f≈g | nothing | just id-t1 = {!!} ≈-cong {t0} {t0} {f1} {g1} f≈g | just id-t0 | nothing = {!!} ≈-cong {t1} {t1} {f1} {g1} f≈g | just id-t1 | nothing = {!!} ≈-cong {t0} {t1} {f1} {g1} f≈g | just arrow-f | nothing = {!!} ≈-cong {t0} {t1} {f1} {g1} f≈g | just arrow-g | nothing = {!!} ≈-cong {t1} {t0} {f1} {g1} f≈g | just inv-f | nothing = nothing ≈-cong {t0} {t0} {f1} {g1} f≈g | just id-t0 | just id-t0 = refl-hom ≈-cong {t1} {t1} {f1} {g1} f≈g | just id-t1 | just id-t1 = refl-hom ≈-cong {t0} {t1} {f1} {g1} f≈g | just arrow-f | just arrow-f = refl-hom ≈-cong {t0} {t1} {f1} {g1} f≈g | just arrow-g | just arrow-g = refl-hom ≈-cong {t0} {t1} {f1} {g1} f≈g | just arrow-g | just arrow-f = {!!} ≈-cong {t1} {t0} {f1} {g1} f≈g | just inv-f | just inv-f = refl-hom ≈-cong {t0} {t1} {f1} {g1} f≈g | just arrow-f | just arrow-g = begin {!!} ≈⟨ {!!} ⟩ {!!} ∎ --- Equalizer --- f --- e ------> --- c ------> a b --- ^ / ------> --- |k h g --- | / --- | / --- | / --- |/ --- d open Limit lim-to-equ : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) -> (lim : (I : Category c₁ c₂ ℓ) ( Γ : Functor I 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 ] ] ) → Equalizer A e f g lim-to-equ {c₁} {c₂} {ℓ } A lim {a} {b} {c} f g e fe=ge = record { fe=ge = fe=ge ; k = λ {d} h fh=gh → k {d} h fh=gh ; ek=h = λ {d} {h} {fh=gh} → ek=h d h fh=gh ; uniqueness = λ {d} {h} {fh=gh} {k'} → uniquness d h fh=gh k' } where I = TwoCat {c₁} {c₂} {ℓ } Γ = {!!} 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 = {!!} 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 = {!!} nat : (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 nat d h fh=gh = record { TMap = λ x → nmap x d h ; isNTrans = record { commute = λ {x} {y} {f'} -> commute1 {x} {y} {f'} d h fh=gh } } k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c k {d} h fh=gh = {!!} -- limit (lim I Γ {c} {nat c e fe=ge }) d (nat d h fh=gh ) ek=h : (d : Obj A ) (h : Hom A d a ) -> ( fh=gh : A [ A [ f o h ] ≈ A [ g o h ] ] ) -> A [ A [ e o k h fh=gh ] ≈ h ] ek=h d h fh=gh = {!!} 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 = {!!}