------------------------------------------------------------------------
-- Partiality algebra categories
------------------------------------------------------------------------
{-# OPTIONS --cubical --safe #-}
module Partiality-algebra.Category where
open import Equality.Propositional.Cubical
open import Logical-equivalence using (_⇔_)
open import Prelude hiding (T)
open import Bijection equality-with-J as Bijection using (_↔_)
open import Category equality-with-J as Category
open import Equality.Path.Isomorphisms.Univalence equality-with-paths
open import Equivalence equality-with-J as Eq using (_≃_)
open import Function-universe equality-with-J as F hiding (id; _∘_)
open import H-level equality-with-J
open import H-level.Closure equality-with-J
open import Structure-identity-principle equality-with-J
open import Univalence-axiom equality-with-J
open import Partiality-algebra as PA hiding (id; _∘_)
------------------------------------------------------------------------
-- Equality characterisation lemmas for Partiality-algebra-with
abstract
-- An equality characterisation lemma for Partiality-algebra-with.
equality-characterisation-Partiality-algebra-with₁ :
∀ {a p q} {A : Type a} {T : Type p}
{P₁ P₂ : Partiality-algebra-with T q A} →
let module P₁ = Partiality-algebra-with P₁
module P₂ = Partiality-algebra-with P₂
in
(∃ λ (⊑≡⊑ : ∀ x y → (x P₁.⊑ y) ≡ (x P₂.⊑ y)) →
P₁.never ≡ P₂.never
×
(∀ x → P₁.now x ≡ P₂.now x)
×
∀ s → P₁.⨆ s ≡ P₂.⨆ (Σ-map id (≡⇒→ (⊑≡⊑ _ _) ∘_) s))
↔
P₁ ≡ P₂
equality-characterisation-Partiality-algebra-with₁
{q = q} {A} {T} {P₁} {P₂} =
(∃ λ (⊑≡⊑ : ∀ x y → (x P₁.⊑ y) ≡ (x P₂.⊑ y)) →
P₁.never ≡ P₂.never
×
(∀ x → P₁.now x ≡ P₂.now x)
×
∀ s → P₁.⨆ s ≡ P₂.⨆ (Σ-map id (≡⇒→ (⊑≡⊑ _ _) ∘_) s)) ↝⟨ ∃-cong (λ ⊑≡⊑ → ∃-cong λ _ → ∃-cong λ _ → ∀-cong ext λ s →
≡⇒↝ _ $ cong (λ (p : {f : ℕ → T} → _) →
P₁.⨆ s ≡ P₂.⨆ (Σ-map id p s)) $
(implicit-Π-closure ext 1 λ _ →
Π-closure ext 1 λ _ →
Π-closure ext 1 λ _ →
P₂.⊑-propositional)
(≡⇒→ (⊑≡⊑ _ _) ∘_)
(λ {f} → ≡⇒→ (cong (λ _⊑_ → ∀ n → f n ⊑ f (suc n))
(⟨ext⟩ (⟨ext⟩ ∘ ⊑≡⊑))))) ⟩
(∃ λ (⊑≡⊑ : ∀ x y → (x P₁.⊑ y) ≡ (x P₂.⊑ y)) →
P₁.never ≡ P₂.never
×
(∀ x → P₁.now x ≡ P₂.now x)
×
∀ s → P₁.⨆ s ≡
P₂.⨆ (Σ-map id
(λ {f} →
≡⇒→ (cong (λ _⊑_ → ∀ n → f n ⊑ f (suc n))
(⟨ext⟩ (⟨ext⟩ ∘ ⊑≡⊑))))
s)) ↝⟨ Σ-cong (∀-cong ext λ _ →
Eq.extensionality-isomorphism ext) (λ _ → F.id) ⟩
(∃ λ (⊑≡⊑ : ∀ x → P₁._⊑_ x ≡ P₂._⊑_ x) →
P₁.never ≡ P₂.never
×
(∀ x → P₁.now x ≡ P₂.now x)
×
∀ s → P₁.⨆ s ≡
P₂.⨆ (Σ-map id
(λ {f} →
≡⇒→ (cong (λ _⊑_ → ∀ n → f n ⊑ f (suc n))
(⟨ext⟩ ⊑≡⊑)))
s)) ↝⟨ Σ-cong (Eq.extensionality-isomorphism ext) (λ _ → F.id) ⟩
(∃ λ (⊑≡⊑ : P₁._⊑_ ≡ P₂._⊑_) →
P₁.never ≡ P₂.never
×
(∀ x → P₁.now x ≡ P₂.now x)
×
∀ s → P₁.⨆ s ≡
P₂.⨆ (Σ-map id
(λ {f} →
≡⇒→ (cong (λ _⊑_ → ∀ n → f n ⊑ f (suc n))
⊑≡⊑))
s)) ↔⟨⟩
(∃ λ (⊑≡⊑ : P₁._⊑_ ≡ P₂._⊑_) →
P₁.never ≡ P₂.never
×
(∀ x → P₁.now x ≡ P₂.now x)
×
∀ s → P₁.⨆ s ≡
P₂.⨆ ( proj₁ s
, ≡⇒→ (cong (λ _⊑_ →
∀ n → proj₁ s n ⊑ proj₁ s (suc n))
⊑≡⊑)
(proj₂ s)
)) ↝⟨ ∃-cong (λ ⊑≡⊑ → ∃-cong λ _ → ∃-cong λ _ → ∀-cong ext λ _ →
≡⇒↝ _ $ cong (λ eq → _ ≡ P₂.⨆ (_ , ≡⇒→ eq _)) $ sym $
cong-∘ _ _ ⊑≡⊑) ⟩
(∃ λ (⊑≡⊑ : P₁._⊑_ ≡ P₂._⊑_) →
P₁.never ≡ P₂.never
×
(∀ x → P₁.now x ≡ P₂.now x)
×
∀ s → P₁.⨆ s ≡
P₂.⨆ ( proj₁ s
, ≡⇒→ (cong (uncurry λ _⊑_ (f : ℕ → T) →
∀ n → f n ⊑ f (suc n))
(cong (_, _) ⊑≡⊑))
(proj₂ s)
)) ↝⟨ ∃-cong (λ ⊑≡⊑ → ∃-cong λ _ → ∃-cong λ _ → ∀-cong ext λ _ →
≡⇒↝ _ $ cong (λ inc → _ ≡ P₂.⨆ (_ , inc)) $ sym $
subst-in-terms-of-≡⇒↝ equivalence (cong (_, _) ⊑≡⊑) _ _) ⟩
(∃ λ (⊑≡⊑ : P₁._⊑_ ≡ P₂._⊑_) →
P₁.never ≡ P₂.never
×
(∀ x → P₁.now x ≡ P₂.now x)
×
∀ s → P₁.⨆ s ≡
P₂.⨆ ( proj₁ s
, subst (uncurry λ _⊑_ (f : ℕ → T) →
∀ n → f n ⊑ f (suc n))
(cong (_, _) ⊑≡⊑)
(proj₂ s)
)) ↝⟨ ∃-cong (λ ⊑≡⊑ → ∃-cong λ _ → ∃-cong λ _ → ∀-cong ext λ _ →
≡⇒↝ _ $ cong (λ eq → _ ≡
P₂.⨆ (_ , subst (uncurry λ _⊑_ (f : ℕ → T) →
∀ n → f n ⊑ f (suc n))
eq _)) $ sym $
Σ-≡,≡→≡-subst-const ⊑≡⊑ refl) ⟩
(∃ λ (⊑≡⊑ : P₁._⊑_ ≡ P₂._⊑_) →
P₁.never ≡ P₂.never
×
(∀ x → P₁.now x ≡ P₂.now x)
×
∀ s → P₁.⨆ s ≡
P₂.⨆ ( proj₁ s
, subst (uncurry λ _⊑_ (f : ℕ → T) →
∀ n → f n ⊑ f (suc n))
(Σ-≡,≡→≡ ⊑≡⊑ (subst-const ⊑≡⊑))
(proj₂ s)
)) ↝⟨ ∃-cong (λ ⊑≡⊑ → ∃-cong λ _ → ∃-cong λ _ → ∀-cong ext λ _ →
≡⇒↝ _ $ cong (λ s → _ ≡ P₂.⨆ s) $ sym $
push-subst-pair′ {y≡z = ⊑≡⊑} _ _ (subst-const ⊑≡⊑)) ⟩
(∃ λ (⊑≡⊑ : P₁._⊑_ ≡ P₂._⊑_) →
P₁.never ≡ P₂.never
×
(∀ x → P₁.now x ≡ P₂.now x)
×
∀ s → P₁.⨆ s ≡
P₂.⨆ (subst (λ _⊑_ → ∃ λ f → ∀ n → f n ⊑ f (suc n))
⊑≡⊑ s)) ↝⟨ ∃-cong (λ ⊑≡⊑ → ∃-cong λ _ → ∃-cong λ _ → ∀-cong ext λ _ →
≡⇒↝ _ $ cong (λ eq → _ ≡
P₂.⨆ (subst (λ _⊑_ → ∃ λ f → ∀ n →
f n ⊑ f (suc n)) eq _)) $
sym $ sym-sym ⊑≡⊑) ⟩
(∃ λ (⊑≡⊑ : P₁._⊑_ ≡ P₂._⊑_) →
P₁.never ≡ P₂.never
×
(∀ x → P₁.now x ≡ P₂.now x)
×
∀ s → P₁.⨆ s ≡
P₂.⨆ (subst (λ _⊑_ → ∃ λ f → ∀ n → f n ⊑ f (suc n))
(sym (sym ⊑≡⊑)) s)) ↝⟨ ∃-cong (λ ⊑≡⊑ → ∃-cong λ _ → ∃-cong λ _ → ∀-cong ext λ _ →
≡⇒↝ _ $ cong (_ ≡_) $ sym $
subst-→-domain _ (sym ⊑≡⊑)) ⟩
(∃ λ (⊑≡⊑ : P₁._⊑_ ≡ P₂._⊑_) →
P₁.never ≡ P₂.never
×
(∀ x → P₁.now x ≡ P₂.now x)
×
∀ s → P₁.⨆ s ≡
subst (λ _⊑_ → ∃ (λ f → ∀ n → f n ⊑ f (suc n)) → T)
(sym ⊑≡⊑) P₂.⨆ s) ↔⟨ ∃-cong (λ _ → ∃-cong λ _ →
Eq.extensionality-isomorphism ext
×-cong
Eq.extensionality-isomorphism ext) ⟩
(∃ λ (⊑≡⊑ : P₁._⊑_ ≡ P₂._⊑_) →
P₁.never ≡ P₂.never
×
P₁.now ≡ P₂.now
×
P₁.⨆ ≡ subst _ (sym ⊑≡⊑) P₂.⨆) ↝⟨ ∃-cong (λ ⊑≡⊑ → ∃-cong λ _ → ∃-cong λ _ →
≡⇒↝ _ $ elim (λ ⊑≡⊑ → ∀ {⨆₁ ⨆₂} →
(⨆₁ ≡ subst _ (sym ⊑≡⊑) ⨆₂) ≡
(subst _ ⊑≡⊑ ⨆₁ ≡ ⨆₂))
(λ _ → refl) ⊑≡⊑) ⟩
(∃ λ (⊑≡⊑ : P₁._⊑_ ≡ P₂._⊑_) →
P₁.never ≡ P₂.never
×
P₁.now ≡ P₂.now
×
subst _ ⊑≡⊑ P₁.⨆ ≡ P₂.⨆) ↝⟨ ∃-cong (λ ⊑≡⊑ →
≡⇒↝ _ (cong (_≡ _) $ sym $ subst-const ⊑≡⊑)
×-cong
≡⇒↝ _ (cong (_≡ _) $ sym $ subst-const ⊑≡⊑)
×-cong
F.id) ⟩
(∃ λ (⊑≡⊑ : P₁._⊑_ ≡ P₂._⊑_) →
subst _ ⊑≡⊑ P₁.never ≡ P₂.never
×
subst _ ⊑≡⊑ P₁.now ≡ P₂.now
×
subst _ ⊑≡⊑ P₁.⨆ ≡ P₂.⨆) ↝⟨ ∃-cong (λ _ → ∃-cong λ _ → ≡×≡↔≡) ⟩
(∃ λ (⊑≡⊑ : P₁._⊑_ ≡ P₂._⊑_) →
subst _ ⊑≡⊑ P₁.never ≡ P₂.never
×
(subst _ ⊑≡⊑ P₁.now , subst _ ⊑≡⊑ P₁.⨆) ≡
(P₂.now , P₂.⨆)) ↝⟨ ∃-cong (λ _ → ≡×≡↔≡) ⟩
(∃ λ (⊑≡⊑ : P₁._⊑_ ≡ P₂._⊑_) →
(subst _ ⊑≡⊑ P₁.never , subst _ ⊑≡⊑ P₁.now , subst _ ⊑≡⊑ P₁.⨆) ≡
(P₂.never , P₂.now , P₂.⨆)) ↝⟨ ∃-cong (λ ⊑≡⊑ → ≡⇒↝ _ $ cong (λ x → (subst _ ⊑≡⊑ P₁.never , x) ≡ _) $
sym $ push-subst-, {y≡z = ⊑≡⊑} _
(λ _⊑_ → (∃ λ f → ∀ n → f n ⊑ f (suc n)) → _)) ⟩
(∃ λ (⊑≡⊑ : P₁._⊑_ ≡ P₂._⊑_) →
(subst _ ⊑≡⊑ P₁.never , subst _ ⊑≡⊑ (P₁.now , P₁.⨆)) ≡
(P₂.never , P₂.now , P₂.⨆)) ↝⟨ ∃-cong (λ ⊑≡⊑ → ≡⇒↝ _ $ cong (_≡ _) $ sym $
push-subst-, {y≡z = ⊑≡⊑} _ _) ⟩
(∃ λ (⊑≡⊑ : P₁._⊑_ ≡ P₂._⊑_) →
subst _ ⊑≡⊑ (P₁.never , P₁.now , P₁.⨆) ≡
(P₂.never , P₂.now , P₂.⨆)) ↔⟨⟩
(∃ λ (⊑≡⊑ : P₁._⊑_ ≡ P₂._⊑_) →
subst _ ⊑≡⊑ (proj₂ (proj₁ (_↔_.to rearrange P₁))) ≡
proj₂ (proj₁ (_↔_.to rearrange P₂))) ↝⟨ Bijection.Σ-≡,≡↔≡ ⟩
proj₁ (_↔_.to rearrange P₁) ≡ proj₁ (_↔_.to rearrange P₂) ↝⟨ ignore-propositional-component
(×-closure 1 (implicit-Π-closure ext 1 λ _ →
implicit-Π-closure ext 1 λ _ →
Π-closure ext 1 λ _ →
Π-closure ext 1 λ _ →
P₂.T-is-set) $
×-closure 1 (H-level-propositional ext 2) $
×-closure 1 (Π-closure ext 1 λ _ →
P₂.⊑-propositional) $
×-closure 1 (implicit-Π-closure ext 1 λ _ →
implicit-Π-closure ext 1 λ _ →
implicit-Π-closure ext 1 λ _ →
Π-closure ext 1 λ _ →
Π-closure ext 1 λ _ →
P₂.⊑-propositional) $
×-closure 1 (Π-closure ext 1 λ _ →
P₂.⊑-propositional) $
×-closure 1 (Π-closure ext 1 λ _ →
Π-closure ext 1 λ _ →
P₂.⊑-propositional) $
×-closure 1 (Π-closure ext 1 λ _ →
Π-closure ext 1 λ _ →
Π-closure ext 1 λ _ →
P₂.⊑-propositional)
(implicit-Π-closure ext 1 λ _ →
implicit-Π-closure ext 1 λ _ →
Is-proposition-propositional ext)) ⟩
_↔_.to rearrange P₁ ≡ _↔_.to rearrange P₂ ↔⟨ Eq.≃-≡ (Eq.↔⇒≃ rearrange) ⟩□
P₁ ≡ P₂ □
where
module P₁ = Partiality-algebra-with P₁
module P₂ = Partiality-algebra-with P₂
rearrange :
Partiality-algebra-with T q A
↔
∃ λ ((_⊑_ , never , _ , ⨆) :
∃ λ (_⊑_ : T → T → Type q) →
T ×
(A → T) ×
((∃ λ (f : ℕ → T) → ∀ n → f n ⊑ f (suc n)) → T)) →
(∀ {x y} → x ⊑ y → y ⊑ x → x ≡ y)
×
Is-set T
×
(∀ x → x ⊑ x)
×
(∀ {x y z} → x ⊑ y → y ⊑ z → x ⊑ z)
×
(∀ x → never ⊑ x)
×
(∀ s → ∀ n → proj₁ s n ⊑ ⨆ s)
×
(∀ s ub → (∀ n → proj₁ s n ⊑ ub) → ⨆ s ⊑ ub)
×
(∀ {x y} → Is-proposition (x ⊑ y))
rearrange = record
{ surjection = record
{ logical-equivalence = record
{ to = λ P → let open Partiality-algebra-with P in
( _⊑_
, never
, now
, ⨆
)
, antisymmetry
, T-is-set-unused
, ⊑-refl
, ⊑-trans
, never⊑
, upper-bound
, least-upper-bound
, ⊑-propositional
; from = λ where
((LE , ne , n , l) , a , u , r , t , le , ub , lub , p) →
record
{ _⊑_ = LE
; never = ne
; now = n
; ⨆ = l
; antisymmetry = a
; T-is-set-unused = u
; ⊑-refl = r
; ⊑-trans = t
; never⊑ = le
; upper-bound = ub
; least-upper-bound = lub
; ⊑-propositional = p
}
}
; right-inverse-of = λ _ → refl
}
; left-inverse-of = λ _ → refl
}
-- Another equality characterisation lemma for
-- Partiality-algebra-with.
equality-characterisation-Partiality-algebra-with₂ :
∀ {a p q} {A : Type a} {T : Type p}
{P₁ P₂ : Partiality-algebra-with T q A} →
let module P₁ = Partiality-algebra-with P₁
module P₂ = Partiality-algebra-with P₂
in
(∃ λ (⊑⇔⊑ : ∀ x y → x P₁.⊑ y ⇔ x P₂.⊑ y) →
P₁.never ≡ P₂.never
×
(∀ x → P₁.now x ≡ P₂.now x)
×
∀ s → P₁.⨆ s ≡ P₂.⨆ (Σ-map id (_⇔_.to (⊑⇔⊑ _ _) ∘_) s))
↔
P₁ ≡ P₂
equality-characterisation-Partiality-algebra-with₂
{q = q} {A} {T} {P₁} {P₂} =
(∃ λ (⊑⇔⊑ : ∀ x y → x P₁.⊑ y ⇔ x P₂.⊑ y) →
P₁.never ≡ P₂.never
×
(∀ x → P₁.now x ≡ P₂.now x)
×
∀ s → P₁.⨆ s ≡ P₂.⨆ (Σ-map id (_⇔_.to (⊑⇔⊑ _ _) ∘_) s)) ↝⟨ Σ-cong (∀-cong ext λ _ → ∀-cong ext λ _ →
Eq.⇔↔≃ ext P₁.⊑-propositional P₂.⊑-propositional)
(λ _ → F.id) ⟩
(∃ λ (⊑≃⊑ : ∀ x y → (x P₁.⊑ y) ≃ (x P₂.⊑ y)) →
P₁.never ≡ P₂.never
×
(∀ x → P₁.now x ≡ P₂.now x)
×
∀ s → P₁.⨆ s ≡ P₂.⨆ (Σ-map id (_≃_.to (⊑≃⊑ _ _) ∘_) s)) ↝⟨ inverse $ Σ-cong
(∀-cong ext λ _ → ∀-cong ext λ _ →
≡≃≃ (_≃_.to
(Propositional-extensionality-is-univalence-for-propositions
ext)
prop-ext
P₁.⊑-propositional
P₂.⊑-propositional))
(λ _ → F.id) ⟩
(∃ λ (⊑≡⊑ : ∀ x y → (x P₁.⊑ y) ≡ (x P₂.⊑ y)) →
P₁.never ≡ P₂.never
×
(∀ x → P₁.now x ≡ P₂.now x)
×
∀ s → P₁.⨆ s ≡ P₂.⨆ (Σ-map id (≡⇒→ (⊑≡⊑ _ _) ∘_) s)) ↝⟨ equality-characterisation-Partiality-algebra-with₁ ⟩□
P₁ ≡ P₂ □
where
module P₁ = Partiality-algebra-with P₁
module P₂ = Partiality-algebra-with P₂
------------------------------------------------------------------------
-- Partiality algebra categories
-- Partiality algebras (with fixed levels and types) form
-- precategories.
precategory :
∀ {a} p q (A : Type a) → Precategory (a ⊔ lsuc (p ⊔ q)) (a ⊔ p ⊔ q)
Precategory.precategory (precategory p q A) =
Partiality-algebra p q A
, (λ P Q → Morphism P Q , Morphism-set)
, PA.id
, PA._∘_
, _↔_.to equality-characterisation-Morphism refl
, _↔_.to equality-characterisation-Morphism refl
, _↔_.to equality-characterisation-Morphism refl
-- A "standard notion of structure" built using
-- Partiality-algebra-with.
standard-notion-of-structure :
∀ {a} p q (A : Type a) →
Standard-notion-of-structure _ _ (precategory-Set p ext)
standard-notion-of-structure p q A = record
{ P = λ B → Partiality-algebra-with (proj₁ B) q A
; H = Is-morphism-with
; H-prop = λ { {p = P} {q = Q} _ →
Is-morphism-with-propositional P Q
}
; H-id = λ { {p = P} →
proj₂ $
_↔_.to Morphism↔Morphism-as-Σ
(PA.id {P = ⟨ P ⟩})
}
; H-∘ = λ { {p = P} {q = Q} {r = R}
f-morphism g-morphism →
proj₂ $
_↔_.to Morphism↔Morphism-as-Σ
(_↔_.from
(Morphism↔Morphism-as-Σ
{P₁ = ⟨ Q ⟩} {P₂ = ⟨ R ⟩})
(_ , g-morphism)
PA.∘
_↔_.from
(Morphism↔Morphism-as-Σ {P₁ = ⟨ P ⟩})
(_ , f-morphism))
}
; H-antisymmetric = λ P Q id-morphism-P→Q id-morphism-Q→P →
_↔_.to
(equality-characterisation-Partiality-algebra-with₂)
( (λ x y → record { to = proj₁ id-morphism-P→Q
; from = proj₁ id-morphism-Q→P
})
, proj₁ (proj₂ id-morphism-P→Q)
, proj₁ (proj₂ (proj₂ id-morphism-P→Q))
, proj₂ (proj₂ (proj₂ id-morphism-P→Q))
)
}
abstract
-- The precategory obtained from the standard notion of structure is
-- equal to the direct definition above.
precategories-equal :
∀ {a p q} {A : Type a} →
Standard-notion-of-structure.Str
(standard-notion-of-structure p q A)
≡
precategory p q A
precategories-equal {p = p} {q} {A} =
_↔_.to (equality-characterisation-Precategory ext univ univ)
( ≃Partiality-algebra
, (λ _ _ → Eq.↔⇒≃ $ inverse $ Morphism↔Morphism-as-Σ)
, (λ _ → refl)
, (λ _ _ _ _ _ → refl)
)
where
≃Partiality-algebra :
(∃ λ (T : Set p) → Partiality-algebra-with (proj₁ T) q A)
≃
Partiality-algebra p q A
≃Partiality-algebra =
(∃ λ (T : Set p) → Partiality-algebra-with (proj₁ T) q A) ↔⟨ inverse Σ-assoc ⟩
(∃ λ (T : Type p) →
Is-set T × Partiality-algebra-with T q A) ↔⟨ ∃-cong (λ _ → drop-⊤-left-× λ P → _⇔_.to contractible⇔↔⊤ $
propositional⇒inhabited⇒contractible
(H-level-propositional ext 2)
(Partiality-algebra-with.T-is-set P)) ⟩
(∃ λ (T : Type p) → Partiality-algebra-with T q A) ↝⟨ Eq.↔⇒≃ record
{ surjection = record
{ logical-equivalence = record
{ to = uncurry λ _ P → ⟨ P ⟩
; from = λ P → T P , partiality-algebra-with P
}
; right-inverse-of = λ _ → refl
}
; left-inverse-of = λ _ → refl
} ⟩□
Partiality-algebra p q A □
where
open Partiality-algebra
-- Thus the precategory is a category.
category :
∀ {a} p q (A : Type a) →
Category (a ⊔ lsuc (p ⊔ q)) (a ⊔ p ⊔ q)
Category.category (category p q A) =
precategory _ _ A
, subst (λ C → ∀ {P Q} → Eq.Is-equivalence
(Precategory.≡→≅ C {P} {Q}))
precategories-equal
(structure-identity-principle
ext
(category-Set _ ext (λ _ _ → univ))
(standard-notion-of-structure p q A))
private
precategory-category :
∀ {a p q} {A : Type a} →
Category.precategory (category p q A) ≡ precategory p q A
precategory-category = refl