module Cubical.Categories.DistLatticeSheaf.Diagram where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.HLevels
open import Cubical.Data.Nat
open import Cubical.Data.Nat.Order hiding (_<_)
open import Cubical.Data.Empty
open import Cubical.Data.Sigma
open import Cubical.Data.FinData
open import Cubical.Data.FinData.Order renaming (_<'Fin_ to _<_)
open import Cubical.Data.Sum
open import Cubical.Relation.Nullary
open import Cubical.Relation.Binary.Order.Poset
open import Cubical.Categories.Category
open import Cubical.Categories.Functor
open import Cubical.Categories.Limits.Limits
open import Cubical.Categories.Limits.Pullback
open import Cubical.Categories.Instances.DistLattice
open import Cubical.Algebra.DistLattice
open import Cubical.Algebra.Lattice
open import Cubical.Algebra.Semilattice
open import Cubical.Algebra.DistLattice.BigOps
private
variable
ℓ ℓ' ℓ'' : Level
module _ {ℓ : Level} where
data DLShfDiagOb (n : ℕ) : Type ℓ where
sing : Fin n → DLShfDiagOb n
pair : (i j : Fin n) → i < j → DLShfDiagOb n
data DLShfDiagHom (n : ℕ) : DLShfDiagOb n → DLShfDiagOb n → Type ℓ where
idAr : {x : DLShfDiagOb n} → DLShfDiagHom n x x
singPairL : {i j : Fin n} {i<j : i < j} → DLShfDiagHom n (sing i) (pair i j i<j)
singPairR : {i j : Fin n} {i<j : i < j}→ DLShfDiagHom n (sing j) (pair i j i<j)
module DLShfDiagHomPath where
variable
n : ℕ
Code : (x y : DLShfDiagOb n) → Type
Code (sing i) (sing j) = i ≡ j
Code (sing i) (pair j k j<k) =
(Σ[ p ∈ (i ≡ j) ] Σ[ i<k ∈ i < k ] PathP (λ ι → p ι < k) i<k j<k)
⊎ (Σ[ p ∈ (i ≡ k) ] Σ[ j<i ∈ j < i ] PathP (λ ι → j < p ι) j<i j<k)
Code (pair i j i<j) (sing k) = ⊥
Code (pair i j i<j) (pair k l k<l) =
Σ[ p ∈ (i ≡ k) × (j ≡ l) ] PathP (λ ι → fst p ι < snd p ι) i<j k<l
isSetCode : ∀ (x y : DLShfDiagOb n) → isSet (Code x y)
isSetCode (sing _) (sing _) = isProp→isSet (isSetFin _ _)
isSetCode (sing i) (pair j k j<k) =
isSet⊎
(isSetΣ (isProp→isSet (isSetFin _ _))
λ _ → isSetΣ (isProp→isSet (≤'FinIsPropValued _ _))
λ _ → isOfHLevelPathP 2 (isProp→isSet (≤'FinIsPropValued _ _)) _ _)
(isSetΣ (isProp→isSet (isSetFin _ _))
λ _ → isSetΣ (isProp→isSet (≤'FinIsPropValued _ _))
λ _ → isOfHLevelPathP 2 (isProp→isSet (≤'FinIsPropValued _ _)) _ _)
isSetCode (pair _ _ _) (sing _) = isProp→isSet isProp⊥
isSetCode (pair _ _ _) (pair _ _ _) =
isSetΣ
(isSet× (isProp→isSet (isSetFin _ _)) (isProp→isSet (isSetFin _ _)))
λ _ → isOfHLevelPathP 2 (isProp→isSet (≤'FinIsPropValued _ _)) _ _
encode : (x y : DLShfDiagOb n) → DLShfDiagHom n x y → Code x y
encode (sing i) (sing .i) idAr = refl
encode (sing i) (pair .i j i<j) singPairL = inl (refl , i<j , refl)
encode (sing j) (pair i .j i<j) singPairR = inr (refl , i<j , refl)
encode (pair i j i<j) (pair .i .j .i<j) idAr = (refl , refl) , refl
decode : (x y : DLShfDiagOb n) → Code x y → DLShfDiagHom n x y
decode (sing i) (sing j) p = subst (λ k → DLShfDiagHom _ (sing i) (sing k)) p idAr
decode (sing i) (pair j k j<k) (inl (p , i<k , q)) =
transport (λ ι → DLShfDiagHom _ (sing i) (pair (p ι) k (q ι))) singPairL
decode (sing i) (pair k j k<j) (inr (p , k<i , q)) =
transport (λ ι → DLShfDiagHom _ (sing i) (pair k (p ι) (q ι))) singPairR
decode (pair i j i<j) (pair k l k<l) (_ , p) =
transport (λ ι → DLShfDiagHom _ (pair _ _ i<j) (pair _ _ (p ι))) idAr
codeRetract : ∀ (x y : DLShfDiagOb n) (f : DLShfDiagHom n x y)
→ decode x y (encode x y f) ≡ f
codeRetract (sing i) (sing .i) idAr = transportRefl idAr
codeRetract (sing i) (pair .i k i<k) singPairL = transportRefl singPairL
codeRetract (sing i) (pair j .i j<i) singPairR = transportRefl singPairR
codeRetract (pair i j i<j) (pair .i .j .i<j) idAr = transportRefl idAr
isSetDLShfDiagHom : ∀ (x y : DLShfDiagOb n) → isSet (DLShfDiagHom n x y)
isSetDLShfDiagHom x y = isSetRetract (encode x y) (decode x y)
(codeRetract x y) (isSetCode x y)
open Category
DLShfDiag : ℕ → (ℓ : Level) → Category ℓ ℓ
ob (DLShfDiag n ℓ) = DLShfDiagOb n
Hom[_,_] (DLShfDiag n ℓ) = DLShfDiagHom n
id (DLShfDiag n ℓ) = idAr
_⋆_ (DLShfDiag n ℓ) idAr f = f
_⋆_ (DLShfDiag n ℓ) singPairL idAr = singPairL
_⋆_ (DLShfDiag n ℓ) singPairR idAr = singPairR
⋆IdL (DLShfDiag n ℓ) _ = refl
⋆IdR (DLShfDiag n ℓ) idAr = refl
⋆IdR (DLShfDiag n ℓ) singPairL = refl
⋆IdR (DLShfDiag n ℓ) singPairR = refl
⋆Assoc (DLShfDiag n ℓ) idAr _ _ = refl
⋆Assoc (DLShfDiag n ℓ) singPairL idAr _ = refl
⋆Assoc (DLShfDiag n ℓ) singPairR idAr _ = refl
isSetHom (DLShfDiag n ℓ) = let open DLShfDiagHomPath in (isSetDLShfDiagHom _ _)
module _ {C : Category ℓ ℓ'} {n : ℕ} {F : Functor (DLShfDiag n ℓ'') C} where
open Category
open Functor F
open Cone
isConeMorSingLemma : {c d : ob C} {f : C [ c , d ]}
(cc : Cone F c) (cd : Cone F d)
→ (∀ i → f ⋆⟨ C ⟩ coneOut cd (sing i) ≡ coneOut cc (sing i))
→ isConeMor cc cd f
isConeMorSingLemma cc cd singHyp (sing i) = singHyp i
isConeMorSingLemma {f = f} cc cd singHyp (pair i j i<j) =
f ⋆⟨ C ⟩ coneOut cd (pair i j i<j)
≡⟨ cong (λ x → f ⋆⟨ C ⟩ x) (sym (cd .coneOutCommutes singPairL)) ⟩
f ⋆⟨ C ⟩ (coneOut cd (sing i) ⋆⟨ C ⟩ F-hom singPairL)
≡⟨ sym (⋆Assoc C _ _ _) ⟩
(f ⋆⟨ C ⟩ coneOut cd (sing i)) ⋆⟨ C ⟩ F-hom singPairL
≡⟨ cong (λ x → x ⋆⟨ C ⟩ F-hom singPairL) (singHyp i) ⟩
coneOut cc (sing i) ⋆⟨ C ⟩ F-hom singPairL
≡⟨ cc .coneOutCommutes singPairL ⟩
coneOut cc (pair i j i<j) ∎
module _ (L' : DistLattice ℓ) where
private
L = fst L'
LCat = (DistLatticeCategory L') ^op
instance
_ = snd L'
open DistLatticeStr ⦃...⦄
open Join L'
open JoinSemilattice (Lattice→JoinSemilattice (DistLattice→Lattice L'))
open PosetStr (IndPoset .snd) hiding (_≤_)
open MeetSemilattice (Lattice→MeetSemilattice (DistLattice→Lattice L'))
using (∧≤RCancel ; ∧≤LCancel)
open Order (DistLattice→Lattice L')
open Category LCat
open Functor
open Cone
FinVec→Diag : {n : ℕ} → FinVec L n → Functor (DLShfDiag n ℓ) LCat
F-ob (FinVec→Diag α) (sing i) = α i
F-ob (FinVec→Diag α) (pair i j _) = α j ∧l α i
F-hom (FinVec→Diag α) idAr = is-refl _
F-hom (FinVec→Diag α) singPairL = ≤m→≤j _ _ (∧≤LCancel _ _)
F-hom (FinVec→Diag α) singPairR = ≤m→≤j _ _ (∧≤RCancel _ _)
F-id (FinVec→Diag α) = is-prop-valued _ _ _ _
F-seq (FinVec→Diag α) _ _ = is-prop-valued _ _ _ _
⋁Cone : {n : ℕ} (α : FinVec L n) → Cone (FinVec→Diag α) (⋁ α)
coneOut (⋁Cone α) (sing i) = ind≤⋁ α i
coneOut (⋁Cone α) (pair i _ _) = is-trans _ (α i) _ (≤m→≤j _ _ (∧≤LCancel _ _)) (ind≤⋁ α i)
coneOutCommutes (⋁Cone α) _ = is-prop-valued _ _ _ _
isLimCone⋁Cone : {n : ℕ} (α : FinVec L n) → isLimCone (FinVec→Diag α) (⋁ α) (⋁Cone α)
fst (fst (isLimCone⋁Cone α u uCone)) = ⋁IsMax α _ λ i → uCone .coneOut (sing i)
snd (fst (isLimCone⋁Cone α u uCone)) _ = is-prop-valued _ _ _ _
snd (isLimCone⋁Cone α _ uCone) _ = Σ≡Prop (λ _ → isPropIsConeMor uCone (⋁Cone α) _)
(is-prop-valued _ _ _ _)
module PullbacksAsDLShfDiags {ℓ'' : Level}
(C : Category ℓ ℓ')
(cspan : Cospan C)
(pback : Pullback C cspan) where
open Functor
open Cone
open Cospan ⦃...⦄
open Pullback ⦃...⦄
instance
_ = cspan
_ = pback
cospanAsDiag : Functor (DLShfDiag 2 ℓ'') C
F-ob cospanAsDiag (sing zero) = l
F-ob cospanAsDiag (sing one) = r
F-ob cospanAsDiag (pair _ _ _) = m
F-hom cospanAsDiag idAr = id C
F-hom cospanAsDiag {x = sing zero} singPairL = s₁
F-hom cospanAsDiag {x = sing one} singPairL = s₂
F-hom cospanAsDiag {x = sing zero} singPairR = s₁
F-hom cospanAsDiag {x = sing one} singPairR = s₂
F-id cospanAsDiag = refl
F-seq cospanAsDiag idAr idAr = sym (⋆IdL C _)
F-seq cospanAsDiag idAr singPairL = sym (⋆IdL C _)
F-seq cospanAsDiag idAr singPairR = sym (⋆IdL C _)
F-seq cospanAsDiag singPairL idAr = sym (⋆IdR C _)
F-seq cospanAsDiag singPairR idAr = sym (⋆IdR C _)
pbPrAsCone : Cone cospanAsDiag pbOb
coneOut pbPrAsCone (sing zero) = pbPr₁
coneOut pbPrAsCone (sing one) = pbPr₂
coneOut pbPrAsCone (pair _ _ _) = pbPr₁ ⋆⟨ C ⟩ s₁
coneOutCommutes pbPrAsCone idAr = ⋆IdR C _
coneOutCommutes pbPrAsCone (singPairL {zero}) = refl
coneOutCommutes pbPrAsCone (singPairL {suc zero}) = sym pbCommutes
coneOutCommutes pbPrAsCone (singPairR {zero} {zero}) = refl
coneOutCommutes pbPrAsCone (singPairR {zero} {suc zero}) = sym pbCommutes
coneOutCommutes pbPrAsCone (singPairR {suc zero} {zero}) = refl
coneOutCommutes pbPrAsCone (singPairR {suc zero} {suc zero}) = sym pbCommutes
pbAsLimit : isLimCone cospanAsDiag pbOb pbPrAsCone
pbAsLimit c cc = uniqueExists (fromPBUnivProp .fst .fst)
toConeMor
(λ _ → isPropIsConeMor cc pbPrAsCone _)
(λ f cf → cong fst (fromPBUnivProp .snd (f , fromConeMor cf)))
where
fromPBUnivProp : ∃![ hk ∈ C [ c , Pullback.pbOb pback ] ]
(coneOut cc (sing zero) ≡ hk ⋆⟨ C ⟩ pbPr₁) ×
(coneOut cc (sing one) ≡ hk ⋆⟨ C ⟩ pbPr₂)
fromPBUnivProp = univProp
(cc .coneOut (sing zero)) (cc .coneOut (sing one))
(cc .coneOutCommutes (singPairL {i<j = s≤s z≤}) ∙ sym (cc .coneOutCommutes singPairR))
toConeMor : isConeMor cc pbPrAsCone (fromPBUnivProp .fst .fst)
toConeMor (sing zero) = sym (fromPBUnivProp .fst .snd .fst)
toConeMor (sing one) = sym (fromPBUnivProp .fst .snd .snd)
toConeMor (pair zero j _) = path
where
path : fromPBUnivProp .fst .fst ⋆⟨ C ⟩ (pbPr₁ ⋆⟨ C ⟩ s₁) ≡ cc .coneOut (pair zero j _)
path = fromPBUnivProp .fst .fst ⋆⟨ C ⟩ (pbPr₁ ⋆⟨ C ⟩ s₁)
≡⟨ sym (⋆Assoc C _ _ _) ⟩
(fromPBUnivProp .fst .fst ⋆⟨ C ⟩ pbPr₁) ⋆⟨ C ⟩ s₁
≡⟨ cong (λ f → f ⋆⟨ C ⟩ s₁) (sym (fromPBUnivProp .fst .snd .fst)) ⟩
cc .coneOut (sing zero) ⋆⟨ C ⟩ s₁
≡⟨ cc .coneOutCommutes singPairL ⟩
cc .coneOut (pair zero j _) ∎
toConeMor (pair one j _) = path
where
path : fromPBUnivProp .fst .fst ⋆⟨ C ⟩ (pbPr₁ ⋆⟨ C ⟩ s₁)
≡ cc .coneOut (pair one j _)
path = fromPBUnivProp .fst .fst ⋆⟨ C ⟩ (pbPr₁ ⋆⟨ C ⟩ s₁)
≡⟨ cong (λ f → fromPBUnivProp .fst .fst ⋆⟨ C ⟩ f) pbCommutes ⟩
fromPBUnivProp .fst .fst ⋆⟨ C ⟩ (pbPr₂ ⋆⟨ C ⟩ s₂)
≡⟨ sym (⋆Assoc C _ _ _) ⟩
(fromPBUnivProp .fst .fst ⋆⟨ C ⟩ pbPr₂) ⋆⟨ C ⟩ s₂
≡⟨ cong (λ f → f ⋆⟨ C ⟩ s₂) (sym (fromPBUnivProp .fst .snd .snd)) ⟩
cc .coneOut (sing one) ⋆⟨ C ⟩ s₂
≡⟨ cc .coneOutCommutes singPairL ⟩
cc .coneOut (pair one j _) ∎
fromConeMor : {f : C [ c , pbOb ]}
→ isConeMor cc pbPrAsCone f
→ (coneOut cc (sing zero) ≡ f ⋆⟨ C ⟩ pbPr₁) ×
(coneOut cc (sing one) ≡ f ⋆⟨ C ⟩ pbPr₂)
fst (fromConeMor cf) = sym (cf (sing zero))
snd (fromConeMor cf) = sym (cf (sing one))
module DLShfDiagsAsPullbacks (C : Category ℓ ℓ')
(F : Functor (DLShfDiag 2 ℓ'') C)
(limF : LimCone F) where
open Cospan
open Pullback
open Functor ⦃...⦄
open Cone ⦃...⦄
open LimCone ⦃...⦄
instance
_ = F
_ = limF
_ = limF .limCone
DiagAsCospan : Cospan C
l DiagAsCospan = F-ob (sing zero)
m DiagAsCospan = F-ob (pair zero one (s≤s z≤))
r DiagAsCospan = F-ob (sing one)
s₁ DiagAsCospan = F-hom singPairL
s₂ DiagAsCospan = F-hom singPairR
LimAsPullback : Pullback C DiagAsCospan
pbOb LimAsPullback = lim
pbPr₁ LimAsPullback = coneOut (sing zero)
pbPr₂ LimAsPullback = coneOut (sing one)
pbCommutes LimAsPullback = coneOutCommutes singPairL ∙ sym (coneOutCommutes singPairR)
univProp LimAsPullback {d = d} f g cSq =
uniqueExists
(fromUnivProp .fst .fst)
(sym (fromUnivProp .fst .snd (sing zero)) , sym (fromUnivProp .fst .snd (sing one)))
(λ _ → isProp× (isSetHom C _ _) (isSetHom C _ _))
λ h' trs → cong fst (fromUnivProp .snd (h' , toConeMor h' trs))
where
theCone : Cone F d
Cone.coneOut theCone (sing zero) = f
Cone.coneOut theCone (sing one) = g
Cone.coneOut theCone (pair zero zero ())
Cone.coneOut theCone (pair zero one (s≤s z≤)) = f ⋆⟨ C ⟩ DiagAsCospan .s₁
Cone.coneOut theCone (pair one zero ())
Cone.coneOut theCone (pair one one (s≤s ()))
Cone.coneOutCommutes theCone {u} idAr = cong (seq' C (Cone.coneOut theCone u)) F-id
∙ ⋆IdR C (Cone.coneOut theCone u)
Cone.coneOutCommutes theCone {sing zero} {pair ._ one (s≤s z≤)} singPairL = refl
Cone.coneOutCommutes theCone {sing one} {pair ._ one (s≤s ())} singPairL
Cone.coneOutCommutes theCone {sing one} {pair zero ._ (s≤s z≤)} singPairR = sym cSq
Cone.coneOutCommutes theCone {sing one} {pair one ._ (s≤s ())} singPairR
fromUnivProp : ∃![ h ∈ C [ d , lim ] ] isConeMor theCone limCone h
fromUnivProp = LimCone.univProp limF d theCone
toConeMor : ∀ (h' : C [ d , lim ])
→ (f ≡ h' ⋆⟨ C ⟩ coneOut (sing zero)) × (g ≡ h' ⋆⟨ C ⟩ coneOut (sing one))
→ isConeMor theCone limCone h'
toConeMor h' (tr₁ , tr₂) (sing zero) = sym tr₁
toConeMor h' (tr₁ , tr₂) (sing one) = sym tr₂
toConeMor h' (tr₁ , tr₂) (pair zero one (s≤s z≤)) = path
where
path : h' ⋆⟨ C ⟩ coneOut (pair zero one (s≤s z≤))
≡ f ⋆⟨ C ⟩ F-hom singPairL
path = h' ⋆⟨ C ⟩ coneOut (pair zero one (s≤s z≤))
≡⟨ cong (seq' C h') (sym (coneOutCommutes singPairL)) ⟩
h' ⋆⟨ C ⟩ (coneOut (sing zero) ⋆⟨ C ⟩ F-hom singPairL)
≡⟨ sym (⋆Assoc C _ _ _) ⟩
(h' ⋆⟨ C ⟩ coneOut (sing zero)) ⋆⟨ C ⟩ F-hom singPairL
≡⟨ cong (λ x → seq' C x (F-hom singPairL)) (sym tr₁) ⟩
f ⋆⟨ C ⟩ F-hom singPairL ∎
toConeMor h' (tr₁ , tr₂) (pair one one (s≤s ()))