open import Level open import Category module CCCgraph where open import HomReasoning open import cat-utility open import Data.Product renaming (_×_ to _/\_ ) hiding ( <_,_> ) open import Category.Constructions.Product open import Relation.Binary.PropositionalEquality hiding ( [_] ) open import CCC open Functor -- ccc-1 : Hom A a 1 ≅ {*} -- ccc-2 : Hom A c (a × b) ≅ (Hom A c a ) × ( Hom A c b ) -- ccc-3 : Hom A a (c ^ b) ≅ Hom A (a × b) c open import Category.Sets -- Sets is a CCC postulate extensionality : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Relation.Binary.PropositionalEquality.Extensionality c₂ c₂ data One {c : Level } : Set c where OneObj : One -- () in Haskell ( or any one object set ) sets : {c : Level } → CCC (Sets {c}) sets = record { １ = One ; ○ = λ _ → λ _ → OneObj ; _∧_ = _∧_ ; <_,_> = <,> ; π = π ; π' = π' ; _<=_ = _<=_ ; _* = _* ; ε = ε ; isCCC = isCCC } where １ : Obj Sets １ = One ○ : (a : Obj Sets ) → Hom Sets a １ ○ a = λ _ → OneObj _∧_ : Obj Sets → Obj Sets → Obj Sets _∧_ a b = a /\ b <,> : {a b c : Obj Sets } → Hom Sets c a → Hom Sets c b → Hom Sets c ( a ∧ b) <,> f g = λ x → ( f x , g x ) π : {a b : Obj Sets } → Hom Sets (a ∧ b) a π {a} {b} = proj₁ π' : {a b : Obj Sets } → Hom Sets (a ∧ b) b π' {a} {b} = proj₂ _<=_ : (a b : Obj Sets ) → Obj Sets a <= b = b → a _* : {a b c : Obj Sets } → Hom Sets (a ∧ b) c → Hom Sets a (c <= b) f * = λ x → λ y → f ( x , y ) ε : {a b : Obj Sets } → Hom Sets ((a <= b ) ∧ b) a ε {a} {b} = λ x → ( proj₁ x ) ( proj₂ x ) isCCC : CCC.IsCCC Sets １ ○ _∧_ <,> π π' _<=_ _* ε isCCC = record { e2 = e2 ; e3a = λ {a} {b} {c} {f} {g} → e3a {a} {b} {c} {f} {g} ; e3b = λ {a} {b} {c} {f} {g} → e3b {a} {b} {c} {f} {g} ; e3c = e3c ; π-cong = π-cong ; e4a = e4a ; e4b = e4b ; *-cong = *-cong } where e2 : {a : Obj Sets} {f : Hom Sets a １} → Sets [ f ≈ ○ a ] e2 {a} {f} = extensionality Sets ( λ x → e20 x ) where e20 : (x : a ) → f x ≡ ○ a x e20 x with f x e20 x | OneObj = refl e3a : {a b c : Obj Sets} {f : Hom Sets c a} {g : Hom Sets c b} → Sets [ ( Sets [ π o ( <,> f g) ] ) ≈ f ] e3a = refl e3b : {a b c : Obj Sets} {f : Hom Sets c a} {g : Hom Sets c b} → Sets [ Sets [ π' o ( <,> f g ) ] ≈ g ] e3b = refl e3c : {a b c : Obj Sets} {h : Hom Sets c (a ∧ b)} → Sets [ <,> (Sets [ π o h ]) (Sets [ π' o h ]) ≈ h ] e3c = refl π-cong : {a b c : Obj Sets} {f f' : Hom Sets c a} {g g' : Hom Sets c b} → Sets [ f ≈ f' ] → Sets [ g ≈ g' ] → Sets [ <,> f g ≈ <,> f' g' ] π-cong refl refl = refl e4a : {a b c : Obj Sets} {h : Hom Sets (c ∧ b) a} → Sets [ Sets [ ε o <,> (Sets [ h * o π ]) π' ] ≈ h ] e4a = refl e4b : {a b c : Obj Sets} {k : Hom Sets c (a <= b)} → Sets [ (Sets [ ε o <,> (Sets [ k o π ]) π' ]) * ≈ k ] e4b = refl *-cong : {a b c : Obj Sets} {f f' : Hom Sets (a ∧ b) c} → Sets [ f ≈ f' ] → Sets [ f * ≈ f' * ] *-cong refl = refl open import graph module ccc-from-graph {c₁ c₂ : Level } (G : Graph {c₁} {c₂}) where open import Relation.Binary.PropositionalEquality renaming ( cong to ≡-cong ) hiding ( [_] ) open Graph data Objs : Set (suc c₁) where atom : (vertex G) → Objs ⊤ : Objs _∧_ : Objs → Objs → Objs _<=_ : Objs → Objs → Objs data Arrows : (b c : Objs ) → Set (suc c₁ ⊔ c₂) data Arrow : Objs → Objs → Set (suc c₁ ⊔ c₂) where --- case i arrow : {a b : vertex G} → (edge G) a b → Arrow (atom a) (atom b) π : {a b : Objs } → Arrow ( a ∧ b ) a π' : {a b : Objs } → Arrow ( a ∧ b ) b ε : {a b : Objs } → Arrow ((a <= b) ∧ b ) a _* : {a b c : Objs } → Arrows (c ∧ b ) a → Arrow c ( a <= b ) --- case v data Arrows where id : ( a : Objs ) → Arrows a a --- case i ○ : ( a : Objs ) → Arrows a ⊤ --- case i <_,_> : {a b c : Objs } → Arrows c a → Arrows c b → Arrows c (a ∧ b) -- case iii iv : {b c d : Objs } ( f : Arrow d c ) ( g : Arrows b d ) → Arrows b c -- cas iv _・_ : {a b c : Objs } (f : Arrows b c ) → (g : Arrows a b) → Arrows a c id a ・ g = g ○ a ・ g = ○ _ < f , g > ・ h = < f ・ h , g ・ h > iv f g ・ h = iv f ( g ・ h ) identityL : {A B : Objs} {f : Arrows A B} → (id B ・ f) ≡ f identityL = refl identityR : {A B : Objs} {f : Arrows A B} → (f ・ id A) ≡ f identityR {a} {a} {id a} = refl identityR {a} {⊤} {○ a} = refl identityR {a} {_} {< f , f₁ >} = cong₂ (λ j k → < j , k > ) identityR identityR identityR {a} {b} {iv f g} = cong (λ k → iv f k ) identityR assoc≡ : {a b c d : Objs} (f : Arrows c d) (g : Arrows b c) (h : Arrows a b) → (f ・ (g ・ h)) ≡ ((f ・ g) ・ h) assoc≡ (id a) g h = refl assoc≡ (○ a) g h = refl assoc≡ < f , f₁ > g h = cong₂ (λ j k → < j , k > ) (assoc≡ f g h) (assoc≡ f₁ g h) assoc≡ (iv f f1) g h = cong (λ k → iv f k ) ( assoc≡ f1 g h ) -- positive intutionistic calculus PL : Category (suc c₁) (suc c₁ ⊔ c₂) (suc c₁ ⊔ c₂) PL = record { Obj = Objs; Hom = λ a b → Arrows a b ; _o_ = λ{a} {b} {c} x y → x ・ y ; _≈_ = λ x y → x ≡ y ; Id = λ{a} → id a ; isCategory = record { 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≡ f g h } } where o-resp-≈ : {A B C : Objs} {f g : Arrows A B} {h i : Arrows B C} → f ≡ g → h ≡ i → (h ・ f) ≡ (i ・ g) o-resp-≈ refl refl = refl -------- -- -- Functor from Positive Logic to Sets -- -- open import Category.Sets -- postulate extensionality : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Relation.Binary.PropositionalEquality.Extensionalit y c₂ c₂ open import Data.List C = graphtocat.Chain G tr : {a b : vertex G} → edge G a b → ((y : vertex G) → C y a) → (y : vertex G) → C y b tr f x y = graphtocat.next f (x y) fobj : ( a : Objs ) → Set (c₁ ⊔ c₂) fobj (atom x) = ( y : vertex G ) → C y x fobj ⊤ = One fobj (a ∧ b) = ( fobj a /\ fobj b) fobj (a <= b) = fobj b → fobj a fmap : { a b : Objs } → Hom PL a b → fobj a → fobj b amap : { a b : Objs } → Arrow a b → fobj a → fobj b amap (arrow x) y = tr x y -- tr x amap π ( x , y ) = x amap π' ( x , y ) = y amap ε (f , x ) = f x amap (f *) x = λ y → fmap f ( x , y ) fmap (id a) x = x fmap (○ a) x = OneObj fmap < f , g > x = ( fmap f x , fmap g x ) fmap (iv x f) a = amap x ( fmap f a ) -- CS is a map from Positive logic to Sets -- Sets is CCC, so we have a cartesian closed category generated by a graph -- as a sub category of Sets CS : Functor PL (Sets {c₁ ⊔ c₂}) FObj CS a = fobj a FMap CS {a} {b} f = fmap {a} {b} f isFunctor CS = isf where _+_ = Category._o_ PL ++idR = IsCategory.identityR ( Category.isCategory PL ) distr : {a b c : Obj PL} { f : Hom PL a b } { g : Hom PL b c } → (z : fobj a ) → fmap (g + f) z ≡ fmap g (fmap f z) distr {a} {a₁} {a₁} {f} {id a₁} z = refl distr {a} {a₁} {⊤} {f} {○ a₁} z = refl distr {a} {b} {c ∧ d} {f} {< g , g₁ >} z = cong₂ (λ j k → j , k ) (distr {a} {b} {c} {f} {g} z) (distr {a} {b} {d} {f} {g₁} z) distr {a} {b} {c} {f} {iv {_} {_} {d} x g} z = adistr (distr {a} {b} {d} {f} {g} z) x where adistr : fmap (g + f) z ≡ fmap g (fmap f z) → ( x : Arrow d c ) → fmap ( iv x (g + f) ) z ≡ fmap ( iv x g ) (fmap f z ) adistr eq x = cong ( λ k → amap x k ) eq isf : IsFunctor PL Sets fobj fmap IsFunctor.identity isf = extensionality Sets ( λ x → refl ) IsFunctor.≈-cong isf refl = refl IsFunctor.distr isf {a} {b} {c} {g} {f} = extensionality Sets ( λ z → distr {a} {b} {c} {g} {f} z ) --- --- SubCategoy SC F A is a category with Obj = FObj F, Hom = FMap --- --- CCC ( SC (CS G)) Sets have to be proved --- SM can be eliminated if we have --- sobj (a : vertex g ) → {a} a set have only a --- smap (a b : vertex g ) → {a} → {b} record CCCObj {c₁ c₂ ℓ : Level} : Set (suc (ℓ ⊔ (c₂ ⊔ c₁))) where field cat : Category c₁ c₂ ℓ ≡←≈ : {a b : Obj cat } → { f g : Hom cat a b } → cat [ f ≈ g ] → f ≡ g ccc : CCC cat open CCCObj record CCCMap {c₁ c₂ ℓ c₁′ c₂′ ℓ′ : Level} (A : CCCObj {c₁} {c₂} {ℓ} ) (B : CCCObj {c₁′} {c₂′}{ℓ′} ) : Set (suc (ℓ′ ⊔ (c₂′ ⊔ c₁′) ⊔ ℓ ⊔ (c₂ ⊔ c₁))) where field cmap : Functor (cat A ) (cat B ) ccf : CCC (cat A) → CCC (cat B) open import Category.Cat open CCCMap open import Relation.Binary.Core Cart : {c₁ c₂ ℓ : Level} → Category (suc (c₁ ⊔ c₂ ⊔ ℓ)) (suc (ℓ ⊔ (c₂ ⊔ c₁))) (suc (ℓ ⊔ c₁ ⊔ c₂)) Cart {c₁} {c₂} {ℓ} = record { Obj = CCCObj {c₁} {c₂} {ℓ} ; Hom = CCCMap ; _o_ = λ {A} {B} {C} f g → record { cmap = (cmap f) ○ ( cmap g ) ; ccf = λ _ → ccf f ( ccf g (ccc A )) } ; _≈_ = λ {a} {b} f g → cmap f ≃ cmap g ; Id = λ {a} → record { cmap = identityFunctor ; ccf = λ x → x } ; isCategory = record { isEquivalence = λ {A} {B} → record { refl = λ {f} → let open ≈-Reasoning (CAT) in refl-hom {cat A} {cat B} {cmap f} ; sym = λ {f} {g} → let open ≈-Reasoning (CAT) in sym-hom {cat A} {cat B} {cmap f} {cmap g} ; trans = λ {f} {g} {h} → let open ≈-Reasoning (CAT) in trans-hom {cat A} {cat B} {cmap f} {cmap g} {cmap h} } ; identityL = λ {x} {y} {f} → let open ≈-Reasoning (CAT) in idL {cat x} {cat y} {cmap f} {_} {_} ; identityR = λ {x} {y} {f} → let open ≈-Reasoning (CAT) in idR {cat x} {cat y} {cmap f} ; o-resp-≈ = λ {x} {y} {z} {f} {g} {h} {i} → IsCategory.o-resp-≈ ( Category.isCategory CAT) {cat x}{cat y}{cat z} {cmap f} {cmap g} {cmap h} {cmap i} ; associative = λ {a} {b} {c} {d} {f} {g} {h} → let open ≈-Reasoning (CAT) in assoc {cat a} {cat b} {cat c} {cat d} {cmap f} {cmap g} {cmap h} }} open import graph open Graph record GMap {c₁ c₂ c₁' c₂' : Level} (x : Graph {c₁} {c₂} ) (y : Graph {c₁'} {c₂'} ) : Set (c₁ ⊔ c₂ ⊔ c₁' ⊔ c₂') where field vmap : vertex x → vertex y emap : {a b : vertex x} → edge x a b → edge y (vmap a) (vmap b) open GMap open import Relation.Binary.HeterogeneousEquality using (_≅_;refl ) renaming ( sym to ≅-sym ; trans to ≅-trans ; cong to ≅-cong ) data [_]_==_ {c₁ c₂ : Level} (C : Graph {c₁} {c₂}) {A B : vertex C} (f : edge C A B) : ∀{X Y : vertex C} → edge C X Y → Set (c₁ ⊔ c₂ ) where mrefl : {g : edge C A B} → (eqv : f ≡ g ) → [ C ] f == g _=m=_ : {c₁ c₂ c₁' c₂' : Level} {C : Graph {c₁} {c₂} } {D : Graph {c₁'} {c₂'} } → (F G : GMap C D) → Set (c₁ ⊔ c₂ ⊔ c₁' ⊔ c₂') _=m=_ {C = C} {D = D} F G = ∀{A B : vertex C} → (f : edge C A B) → [ D ] emap F f == emap G f _&_ : {c₁ c₂ c₁' c₂' c₁'' c₂'' : Level} {x : Graph {c₁} {c₂}} {y : Graph {c₁'} {c₂'}} {z : Graph {c₁''} {c₂''} } ( f : GMap y z ) ( g : GMap x y ) → GMap x z f & g = record { vmap = λ x → vmap f ( vmap g x ) ; emap = λ x → emap f ( emap g x ) } Grph : {c₁ c₂ : Level} → Category (suc (c₁ ⊔ c₂)) (c₁ ⊔ c₂) (c₁ ⊔ c₂) Grph {c₁} {c₂} = record { Obj = Graph {c₁} {c₂} ; Hom = GMap ; _o_ = _&_ ; _≈_ = _=m=_ ; Id = record { vmap = λ y → y ; emap = λ f → f } ; isCategory = record { isEquivalence = λ {A} {B} → ise ; identityL = λ e → mrefl refl ; identityR = λ e → mrefl refl ; o-resp-≈ = m--resp-≈ ; associative = λ e → mrefl refl }} where msym : {x y : Graph {c₁} {c₂} } { f g : GMap x y } → f =m= g → g =m= f msym {x} {y} f=g f = lemma ( f=g f ) where lemma : ∀{a b c d} {f : edge y a b} {g : edge y c d} → [ y ] f == g → [ y ] g == f lemma (mrefl Ff≈Gf) = mrefl (sym Ff≈Gf) mtrans : {x y : Graph {c₁} {c₂} } { f g h : GMap x y } → f =m= g → g =m= h → f =m= h mtrans {x} {y} f=g g=h f = lemma ( f=g f ) ( g=h f ) where lemma : ∀{a b c d e f} {p : edge y a b} {q : edge y c d} → {r : edge y e f} → [ y ] p == q → [ y ] q == r → [ y ] p == r lemma (mrefl eqv) (mrefl eqv₁) = mrefl ( trans eqv eqv₁ ) ise : {x y : Graph {c₁} {c₂} } → IsEquivalence {_} {c₁ ⊔ c₂} {_} ( _=m=_ {_} {_} {_} {_} {x} {y}) ise = record { refl = λ f → mrefl refl ; sym = msym ; trans = mtrans } m--resp-≈ : {A B C : Graph {c₁} {c₂} } {f g : GMap A B} {h i : GMap B C} → f =m= g → h =m= i → ( h & f ) =m= ( i & g ) m--resp-≈ {A} {B} {C} {f} {g} {h} {i} f=g h=i e = lemma (emap f e) (emap g e) (emap i (emap g e)) (f=g e) (h=i ( emap g e )) where lemma : {a b c d : vertex B } {z w : vertex C } (ϕ : edge B a b) (ψ : edge B c d) (π : edge C z w) → [ B ] ϕ == ψ → [ C ] (emap h ψ) == π → [ C ] (emap h ϕ) == π lemma _ _ _ (mrefl refl) (mrefl refl) = mrefl refl --- Forgetful functor module forgetful {c₁ c₂ : Level} where ≃-cong : {c₁ c₂ ℓ : Level} (B : Category c₁ c₂ ℓ ) → {a b a' b' : Obj B } → { f f' : Hom B a b } → { g g' : Hom B a' b' } → [_]_~_ B f g → B [ f ≈ f' ] → B [ g ≈ g' ] → [_]_~_ B f' g' ≃-cong B {a} {b} {a'} {b'} {f} {f'} {g} {g'} (refl {g2} eqv) f=f' g=g' = let open ≈-Reasoning B in refl {_} {_} {_} {B} {a'} {b'} {f'} {g'} ( begin f' ≈↑⟨ f=f' ⟩ f ≈⟨ eqv ⟩ g ≈⟨ g=g' ⟩ g' ∎ ) -- Grph does not allow morph on different level graphs -- simply assumes we have iso to the another level. This may means same axiom on CCCs results the same CCCs. postulate g21 : Graph {suc (c₁ ⊔ c₂)} {c₁ ⊔ c₂} → Graph {c₁} {c₂} m21 : (g : Graph {suc (c₁ ⊔ c₂)} {c₁ ⊔ c₂} ) → GMap {suc (c₁ ⊔ c₂)} {c₁ ⊔ c₂} {c₁} {c₂} g (g21 g) m12 : (g : Graph {suc (c₁ ⊔ c₂)} {c₁ ⊔ c₂} ) → GMap {c₁} {c₂} {suc (c₁ ⊔ c₂)} {c₁ ⊔ c₂} (g21 g) g giso→ : { g : Graph {suc (c₁ ⊔ c₂)} {c₁ ⊔ c₂} } → {a b : vertex g } → {e : edge g a b } → (m12 g & m21 g) =m= id1 Grph g giso← : { g : Graph {suc (c₁ ⊔ c₂)} {c₁ ⊔ c₂} } → {a b : vertex (g21 g) } → {e : edge (g21 g) a b } → (m21 g & m12 g ) =m= id1 Grph (g21 g) -- Grph [ Grph [ m21 g o m12 g ] ≈ id1 Grph (g21 g) ] fobj : Obj (Cart {suc (c₁ ⊔ c₂)} {c₁ ⊔ c₂} {c₁ ⊔ c₂}) → Obj Grph fobj a = record { vertex = Obj (cat a) ; edge = Hom (cat a) } fmap : {a b : Obj (Cart {suc (c₁ ⊔ c₂)} {c₁ ⊔ c₂} {c₁ ⊔ c₂} ) } → Hom (Cart ) a b → Hom (Grph {c₁} {c₂}) (g21 ( fobj a )) (g21 ( fobj b )) fmap {a} {b} f = record { vmap = λ e → vmap (m21 (fobj b)) (FObj (cmap f) (vmap (m12 (fobj a)) e )) ; emap = λ e → emap (m21 (fobj b)) (FMap (cmap f) (emap (m12 (fobj a)) e )) } UX : Functor (Cart {suc (c₁ ⊔ c₂)} {c₁ ⊔ c₂} {c₁ ⊔ c₂}) (Grph {c₁} {c₂}) FObj UX a = g21 (fobj a) FMap UX f = fmap f isFunctor UX = isf where isf : IsFunctor Cart Grph (λ z → g21 (fobj z)) fmap eff : (a : Obj Cart) (f : vertex (g21 (fobj a)) ) → edge (g21 (fobj a)) f f eff a f = {!!} IsFunctor.identity isf {a} {b} {f} = begin fmap (id1 Cart a) ≈⟨⟩ fmap {a} {a} (record { cmap = identityFunctor ; ccf = λ x → x }) ≈⟨⟩ record { vmap = λ e → vmap (m21 (fobj a)) (vmap (m12 (fobj a)) e ) ; emap = λ e → emap (m21 (fobj a)) (emap (m12 (fobj a)) e )} ≈⟨ giso← {fobj a} {f} {f} {eff a f } ⟩ record { vmap = λ y → y ; emap = λ f → f } ≈⟨⟩ id1 Grph (g21 (fobj a)) ∎ where open ≈-Reasoning Grph IsFunctor.distr isf {a} {b} {c} {f} {g} = begin fmap ( Cart [ g o f ] ) ≈⟨ {!!} ⟩ Grph [ fmap g o fmap f ] ∎ where open ≈-Reasoning Grph IsFunctor.≈-cong isf {a} {b} {f} {g} f=g e = {!!} where -- lemma ( (extensionality Sets ( λ z → lemma4 ( -- ≃-cong (cat b) (f=g (id1 (cat a) z)) (IsFunctor.identity (Functor.isFunctor (cmap f))) (IsFunctor.identity (Functor.isFunctor (cmap g))) -- )))) (f=g e) where lemma4 : {x y : vertex (fobj b)} → [_]_~_ (cat b) (id1 (cat b) x) (id1 (cat b) y) → x ≡ y lemma4 (refl eqv) = refl -- lemma : vmap (fmap f) ≡ vmap (fmap g) → [ cat b ] FMap (cmap f) e ~ FMap (cmap g) e → [ g21 (fobj b)] emap (fmap f) {!!} == emap (fmap g) {!!} -- lemma = {!!} -- refl (refl eqv) = mrefl (≡←≈ b eqv) open ccc-from-graph.Objs open ccc-from-graph.Arrow open ccc-from-graph.Arrows open graphtocat.Chain Sets0 : {c₂ : Level } → Category (suc c₂) c₂ c₂ Sets0 {c₂} = Sets {c₂} ccc-graph-univ : {c₁ c₂ : Level} → UniversalMapping (Grph {c₁} {c₂}) (Cart {suc (c₁ ⊔ c₂)} {c₁ ⊔ c₂} {c₁ ⊔ c₂}) forgetful.UX ccc-graph-univ {c₁} {c₂} = record { F = F ; η = η ; -- λ a → record { vmap = λ y → graphtocat.Chain {!!} {!!} {!!} ; emap = λ f x → {!!} } ; -- _* = solution ; isUniversalMapping = record { universalMapping = {!!} ; uniquness = {!!} } } where open forgetful open ccc-from-graph -- η : Hom Grph a (FObj UX (F a)) -- f : edge g x y -----------------------------------> m21 (record {vertex = fobj (atom x) ; edge = fmap h }) : Graph -- Graph g x ----------------------> y : vertex g ↑ -- arrow f : Hom (PL g) (atom x) (atom y) | -- PL g atom x ------------------> atom y : Obj (PL g) | UX : Functor Sets Graph -- | | -- | Functor (CS g) | -- ↓ | -- Sets ((z : vertx g) → C z x) ----> ((z : vertx g) → C z y) = h : Hom Sets (fobj (atom x)) (fobj (atom y)) -- cs : {c₁ c₂ : Level} → (g : Graph {c₁} {c₂} ) → Functor (ccc-from-graph.PL g) (Sets {_}) cs g = CS g F : Obj (Grph {c₁} {c₂}) → Obj (Cart {suc (c₁ ⊔ c₂)} {c₁ ⊔ c₂} {c₁ ⊔ c₂}) F g = record { cat = Sets {c₁ ⊔ c₂} ; ccc = sets ; ≡←≈ = λ eq → eq } η : (a : Obj (Grph {c₁} {c₂}) ) → Hom Grph a (FObj UX (F a)) η a = record { vmap = λ y → vm y ; emap = λ f → em f } where fo : Graph {suc (c₁ ⊔ c₂)} {c₁ ⊔ c₂} fo = forgetful.fobj {c₁} {c₂} (F a) vm : (y : vertex a ) → vertex (g21 fo) vm y = vmap (m21 fo) (ccc-from-graph.fobj a (atom y)) em : { x y : vertex a } (f : edge a x y ) → edge (FObj UX (F a)) (vm x) (vm y) em {x} {y} f = emap (m21 fo) (ccc-from-graph.fmap a (iv (arrow f) (id _))) -- k : ( y : vertex a) → Set (c₁ ⊔ c₂) -- k y = ( e : vertex a ) → graphtocat.Chain a e y -- mm : Graph {suc (c₁ ⊔ c₂)} {(c₁ ⊔ c₂)} -- mm = forgetful.fobj {c₁} {c₂} (F a) pl : {c₁ c₂ : Level} → (g : Graph {c₁} {c₂} ) → Category _ _ _ pl g = PL g cobj : {g : Obj (Grph {c₁} {c₂} ) } {c : Obj Cart} → Hom Grph g (FObj UX c) → Objs g → Obj (cat c) cobj {g} {c} f (atom x) = {!!} -- vmap f x cobj {g} {c} f ⊤ = CCC.１ (ccc c) cobj {g} {c} f (x ∧ y) = CCC._∧_ (ccc c) (cobj {g} {c} f x) (cobj {g} {c} f y) cobj {g} {c} f (b <= a) = CCC._<=_ (ccc c) (cobj {g} {c} f b) (cobj {g} {c} f a) c-map : {g : Obj (Grph )} {c : Obj Cart} {A B : Objs g} → (f : Hom Grph g (FObj UX c) ) → (p : Hom (pl g) A B) → Hom (cat c) (cobj {g} {c} f A) (cobj {g} {c} f B) c-map {g} {c} {atom a} {atom x} f y = {!!} c-map {g} {c} {⊤} {atom x} f (iv f1 y) = {!!} c-map {g} {c} {a ∧ b} {atom x} f (iv f1 y) = {!!} c-map {g} {c} {b <= a} {atom x} f y = {!!} c-map {g} {c} {a} {⊤} f x = CCC.○ (ccc c) (cobj f a) c-map {g} {c} {a} {x ∧ y} f z = CCC.<_,_> (ccc c) (c-map f {!!}) (c-map f {!!}) c-map {g} {c} {d} {b <= a} f x = CCC._* (ccc c) ( c-map f {!!}) solution : {g : Obj Grph } {c : Obj Cart } → Hom Grph g (FObj UX c) → Hom Cart (F g) c solution {g} {c} f = {!!} -- record { cmap = record { FObj = λ x → {!!} ; FMap = {!!} ; isFunctor = {!!} } ; ccf = {!!} }