Mercurial > hg > Members > kono > Proof > category
view limit-to.agda @ 440:ff36c500962e
fix limit
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 30 Aug 2016 14:22:47 +0900 |
| parents | 3fdf0aedc21d |
| children | 87600d338337 |
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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 -- 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 ) 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 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 -- 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 ; 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} } } 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 ) ∎ record Nil {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ) where field nil : {a b : Obj A } → Hom A a b nil-id : {a : Obj A } → A [ nil {a} {a} ≈ id1 A a ] -- too strong but we need this nil-idL : {a b c : Obj A } → { f : Hom A a b } → A [ A [ nil {b} {c} o f ] ≈ nil {a} {c} ] nil-idR : {a b c : Obj A } → { f : Hom A b c } → A [ A [ f o nil {a} {b} ] ≈ nil {a} {c} ] 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} A none a b f g = 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} → distr1 {a} {b} {c} {f} {g} ; ≈-cong = λ {a} {b} {c} {f} → ≈-cong {a} {b} {c} {f} } } where I = TwoCat {c₁} {c₂} {ℓ} 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 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 identity : {x : Obj I} → A [ fmap ( id1 I x ) ≈ id1 A (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} → 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 ) --- Equalizer --- f --- e -----→ --- c -----→ a b --- ^ / -----→ --- |k h g --- | / --- | / ^ --- | / | --- |/ | Γ --- d _ | --- |\ | --- \ K af --- \ -----→ --- \ t0 t1 --- -----→ --- ag --- --- open Limit I' : {c₁ c₂ ℓ : Level} → Category c₁ c₂ c₂ I' {c₁} {c₂} {ℓ} = TwoCat {c₁} {c₂} {ℓ} 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 } {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 none 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 open ≈-Reasoning A I : Category c₁ c₂ c₂ I = I' {c₁} {c₂} {ℓ} 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 ] 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 ∎ 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 ; isNTrans = record { commute = λ {x} {y} {f'} → commute1 {x} {y} {f'} d h fh=gh } } eqlim = lim Γ {c} {nat1 c e fe=ge } 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 eqlim d (nat1 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 = begin e o k h fh=gh ≈⟨ t0f=t eqlim {d} {nat1 d h fh=gh} {t0} ⟩ 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 ∎ 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 ) ⟩ k' ∎