This is the Base.Structures.Isos module of the Agda Universal Algebra Library.
{-# OPTIONS --without-K --exact-split --safe #-} module Base.Structures.Isos where -- Imports from Agda and the Agda Standard Library --------------------- open import Agda.Primitive using () renaming ( Set to Type ) open import Axiom.Extensionality.Propositional using () renaming (Extensionality to funext) open import Data.Product using ( _,_ ; Σ-syntax ; _×_ ) renaming ( proj₁ to fst ; proj₂ to snd ) open import Function using ( _∘_ ) open import Level using ( _⊔_ ; Level ; Lift ) open import Relation.Binary.PropositionalEquality as ≡ using ( module ≡-Reasoning ; cong-app ) -- Imports from the Agda Universal Algebra Library --------------------------------------------- open import Overture using ( ∣_∣ ; _≈_ ; ∥_∥ ; _∙_ ; lower∼lift ; lift∼lower ) open import Base.Structures.Basic using ( signature ; structure ; Lift-Strucˡ ) using ( Lift-Strucʳ ; Lift-Struc ; sigl ) using ( siglˡ ; siglʳ ) open import Base.Structures.Homs using ( hom ; 𝒾𝒹 ; ∘-hom ; 𝓁𝒾𝒻𝓉 ; 𝓁ℴ𝓌ℯ𝓇 ; 𝓁𝒾𝒻𝓉ˡ ) using ( 𝓁ℴ𝓌ℯ𝓇ˡ ; 𝓁𝒾𝒻𝓉ʳ ; 𝓁ℴ𝓌ℯ𝓇ʳ ; is-hom ) open import Base.Structures.Products using ( ⨅ ; ℓp ; ℑ ; class-product ) private variable 𝓞₀ 𝓥₀ 𝓞₁ 𝓥₁ α ρᵃ β ρᵇ γ ρᶜ ρ ℓ ι : Level 𝐹 : signature 𝓞₀ 𝓥₀ 𝑅 : signature 𝓞₁ 𝓥₁
Recall, f ≈ g means f and g are extensionally (or pointwise) equal; i.e., ∀ x, f x ≡ g x.
We use this notion of equality of functions in the following definition of isomorphism.
record _≅_ (𝑨 : structure 𝐹 𝑅 {α}{ρᵃ}) (𝑩 : structure 𝐹 𝑅 {β}{ρᵇ}) : Type (sigl 𝐹 ⊔ sigl 𝑅 ⊔ α ⊔ ρᵃ ⊔ β ⊔ ρᵇ) where constructor mkiso field to : hom 𝑨 𝑩 from : hom 𝑩 𝑨 to∼from : ∣ to ∣ ∘ ∣ from ∣ ≈ ∣ 𝒾𝒹 {𝑨 = 𝑩} ∣ from∼to : ∣ from ∣ ∘ ∣ to ∣ ≈ ∣ 𝒾𝒹 {𝑨 = 𝑨} ∣ open _≅_ public
That is, two structures are isomorphic provided there are homomorphisms going back and forth between them which compose to the identity map.
module _ {𝑨 : structure 𝐹 𝑅 {α}{ρᵃ}} where ≅-refl : 𝑨 ≅ 𝑨 ≅-refl = mkiso 𝒾𝒹 𝒾𝒹 (λ _ → ≡.refl) (λ _ → ≡.refl) module _ {𝑩 : structure 𝐹 𝑅 {β}{ρᵇ}} where ≅-sym : 𝑨 ≅ 𝑩 → 𝑩 ≅ 𝑨 ≅-sym φ = mkiso (from φ) (to φ) (from∼to φ) (to∼from φ) module _ {𝑪 : structure 𝐹 𝑅 {γ}{ρᶜ}} where ≅-trans : 𝑨 ≅ 𝑩 → 𝑩 ≅ 𝑪 → 𝑨 ≅ 𝑪 ≅-trans φab φbc = mkiso f g τ ν where f : hom 𝑨 𝑪 f = ∘-hom {𝑨 = 𝑨}{𝑩}{𝑪} (to φab) (to φbc) g : hom 𝑪 𝑨 g = ∘-hom {𝑨 = 𝑪}{𝑩}{𝑨} (from φbc) (from φab) τ : ∣ f ∣ ∘ ∣ g ∣ ≈ ∣ 𝒾𝒹 {𝑨 = 𝑪} ∣ τ x = ( ≡.cong ∣ to φbc ∣ (to∼from φab (∣ from φbc ∣ x)) ) ∙ (to∼from φbc) x ν : ∣ g ∣ ∘ ∣ f ∣ ≈ ∣ 𝒾𝒹 {𝑨 = 𝑨} ∣ ν x = ( ≡.cong ∣ from φab ∣ (from∼to φbc (∣ to φab ∣ x)) ) ∙ (from∼to φab) x
Fortunately, the lift operation preserves isomorphism (i.e., it’s an algebraic invariant). As our focus is universal algebra, this is important and is what makes the lift operation a workable solution to the technical problems that arise from the noncumulativity of the universe hierarchy discussed in the [Base.Overture][] module.
open Level module _ {𝑨 : structure 𝐹 𝑅{α}{ρᵃ}} where Lift-≅ˡ : 𝑨 ≅ (Lift-Strucˡ ℓ 𝑨) Lift-≅ˡ = record { to = 𝓁𝒾𝒻𝓉ˡ ; from = 𝓁ℴ𝓌ℯ𝓇ˡ {𝑨 = 𝑨} ; to∼from = cong-app lift∼lower ; from∼to = cong-app (lower∼lift{α}{ρᵃ}) } Lift-≅ʳ : 𝑨 ≅ (Lift-Strucʳ ℓ 𝑨) Lift-≅ʳ = record { to = 𝓁𝒾𝒻𝓉ʳ ; from = 𝓁ℴ𝓌ℯ𝓇ʳ ; to∼from = cong-app ≡.refl ; from∼to = cong-app ≡.refl } Lift-≅ : 𝑨 ≅ (Lift-Struc ℓ ρ 𝑨) Lift-≅ = record { to = 𝓁𝒾𝒻𝓉 ; from = 𝓁ℴ𝓌ℯ𝓇 {𝑨 = 𝑨} ; to∼from = cong-app lift∼lower ; from∼to = cong-app (lower∼lift{α}{ρᵃ}) } module _ {𝑨 : structure 𝐹 𝑅{α}{ρᵃ}} {𝑩 : structure 𝐹 𝑅{β}{ρᵇ}} where Lift-Strucˡ-iso : (ℓ ℓ' : Level) → 𝑨 ≅ 𝑩 → Lift-Strucˡ ℓ 𝑨 ≅ Lift-Strucˡ ℓ' 𝑩 Lift-Strucˡ-iso ℓ ℓ' A≅B = ≅-trans ( ≅-trans (≅-sym Lift-≅ˡ) A≅B ) Lift-≅ˡ Lift-Struc-iso : (ℓ ρ ℓ' ρ' : Level) → 𝑨 ≅ 𝑩 → Lift-Struc ℓ ρ 𝑨 ≅ Lift-Struc ℓ' ρ' 𝑩 Lift-Struc-iso ℓ ρ ℓ' ρ' A≅B = ≅-trans ( ≅-trans (≅-sym Lift-≅) A≅B ) Lift-≅
The lift is also associative, up to isomorphism at least.
module _ {𝑨 : structure 𝐹 𝑅 {α}{ρᵃ} } where Lift-Struc-assocˡ : {ℓ ℓ' : Level} → Lift-Strucˡ (ℓ ⊔ ℓ') 𝑨 ≅ (Lift-Strucˡ ℓ (Lift-Strucˡ ℓ' 𝑨)) Lift-Struc-assocˡ {ℓ}{ℓ'} = ≅-trans (≅-trans Goal Lift-≅ˡ) Lift-≅ˡ where Goal : Lift-Strucˡ (ℓ ⊔ ℓ') 𝑨 ≅ 𝑨 Goal = ≅-sym Lift-≅ˡ Lift-Struc-assocʳ : {ρ ρ' : Level} → Lift-Strucʳ (ρ ⊔ ρ') 𝑨 ≅ (Lift-Strucʳ ρ (Lift-Strucʳ ρ' 𝑨)) Lift-Struc-assocʳ {ρ}{ρ'} = ≅-trans (≅-trans Goal Lift-≅ʳ) Lift-≅ʳ where Goal : Lift-Strucʳ (ρ ⊔ ρ') 𝑨 ≅ 𝑨 Goal = ≅-sym Lift-≅ʳ Lift-Struc-assoc : {ℓ ℓ' ρ ρ' : Level} → Lift-Struc (ℓ ⊔ ℓ') (ρ ⊔ ρ') 𝑨 ≅ (Lift-Struc ℓ ρ (Lift-Struc ℓ' ρ' 𝑨)) Lift-Struc-assoc {ℓ}{ℓ'}{ρ}{ρ'} = ≅-trans (≅-trans Goal Lift-≅ ) Lift-≅ where Goal : Lift-Struc (ℓ ⊔ ℓ') (ρ ⊔ ρ') 𝑨 ≅ 𝑨 Goal = ≅-sym Lift-≅
Products of isomorphic families of algebras are themselves isomorphic. The proof looks a bit technical, but it is as straightforward as it ought to be.
module _ {I : Type ι} {𝒜 : I → structure 𝐹 𝑅{α}{ρᵃ}} {ℬ : I → structure 𝐹 𝑅{β}{ρᵇ}} where open structure open ≡-Reasoning ⨅≅ : funext ι α → funext ι β → (∀ (i : I) → 𝒜 i ≅ ℬ i) → ⨅ 𝒜 ≅ ⨅ ℬ ⨅≅ fiu fiw AB = record { to = ϕ , ϕhom ; from = ψ , ψhom ; to∼from = ϕ~ψ ; from∼to = ψ~ϕ } where ϕ : carrier (⨅ 𝒜) → carrier (⨅ ℬ) ϕ a i = ∣ to (AB i) ∣ (a i) ϕhom : is-hom (⨅ 𝒜) (⨅ ℬ) ϕ ϕhom = ( λ r a x 𝔦 → fst ∥ to (AB 𝔦) ∥ r (λ z → a z 𝔦) (x 𝔦)) , λ f a → fiw (λ i → snd ∥ to (AB i) ∥ f λ z → a z i) ψ : carrier (⨅ ℬ) → carrier (⨅ 𝒜) ψ b i = ∣ from (AB i) ∣ (b i) ψhom : is-hom (⨅ ℬ) (⨅ 𝒜) ψ ψhom = ( λ r a x 𝔦 → fst ∥ from (AB 𝔦) ∥ r (λ z → a z 𝔦) (x 𝔦)) , λ f a → fiu (λ i → snd ∥ from (AB i) ∥ f λ z → a z i) ϕ~ψ : ϕ ∘ ψ ≈ ∣ 𝒾𝒹 {𝑨 = ⨅ ℬ} ∣ ϕ~ψ 𝒃 = fiw λ i → (to∼from (AB i)) (𝒃 i) ψ~ϕ : ψ ∘ ϕ ≈ ∣ 𝒾𝒹 {𝑨 = ⨅ 𝒜} ∣ ψ~ϕ a = fiu λ i → (from∼to (AB i)) (a i)
A nearly identical proof goes through for isomorphisms of lifted products (though,
just for fun, we use the universal quantifier syntax here to express the dependent
function type in the statement of the lemma, instead of the Pi notation we used in
the statement of the previous lemma; that is, ∀ i → 𝒜 i ≅ ℬ (lift i) instead of
Π i ꞉ I , 𝒜 i ≅ ℬ (lift i).)
module _ {I : Type ι} {𝒜 : I → structure 𝐹 𝑅 {α}{ρᵃ}} {ℬ : (Lift γ I) → structure 𝐹 𝑅 {β}{ρᵇ}} where open structure Lift-Struc-⨅≅ : funext (ι ⊔ γ) β → funext ι α → (∀ i → 𝒜 i ≅ ℬ (lift i)) → Lift-Strucˡ γ (⨅ 𝒜) ≅ ⨅ ℬ Lift-Struc-⨅≅ fizw fiu AB = Goal where ϕ : carrier (⨅ 𝒜) → carrier (⨅ ℬ) ϕ a i = ∣ to (AB (lower i)) ∣ (a (lower i)) ϕhom : is-hom (⨅ 𝒜) (⨅ ℬ) ϕ ϕhom = ( λ r a x i → fst ∥ to (AB (lower i)) ∥ r (λ x₁ → a x₁ (lower i)) (x (lower i))) , λ f a → fizw (λ i → snd ∥ to (AB (lower i)) ∥ f λ x → a x (lower i)) ψ : carrier (⨅ ℬ) → carrier (⨅ 𝒜) ψ b i = ∣ from (AB i) ∣ (b (lift i)) ψhom : is-hom (⨅ ℬ) (⨅ 𝒜) ψ ψhom = ( λ r a x i → fst ∥ from (AB i) ∥ r (λ x₁ → a x₁ (lift i)) (x (lift i))) , λ f a → fiu (λ i → snd ∥ from (AB i) ∥ f λ x → a x (lift i)) ϕ~ψ : ϕ ∘ ψ ≈ ∣ 𝒾𝒹 {𝑨 = (⨅ ℬ)} ∣ ϕ~ψ b = fizw λ i → to∼from (AB (lower i)) (b i) ψ~ϕ : ψ ∘ ϕ ≈ ∣ 𝒾𝒹 {𝑨 = (⨅ 𝒜)} ∣ ψ~ϕ a = fiu λ i → from∼to (AB i) (a i) A≅B : ⨅ 𝒜 ≅ ⨅ ℬ A≅B = mkiso (ϕ , ϕhom) (ψ , ψhom) ϕ~ψ ψ~ϕ Goal : Lift-Strucˡ γ (⨅ 𝒜) ≅ ⨅ ℬ Goal = ≅-trans (≅-sym Lift-≅ˡ) A≅B