This is the Base.Homomorphisms.Isomorphisms module of the Agda Universal Algebra Library. Here we formalize the informal notion of isomorphism between algebraic structures.
{-# OPTIONS --without-K --exact-split --safe #-} open import Overture using ( Signature ; 𝓞 ; 𝓥 ) module Base.Homomorphisms.Isomorphisms {𝑆 : Signature 𝓞 𝓥} where -- Imports from Agda and the Agda Standard Library ----------------------------------------------- open import Agda.Primitive using () renaming ( Set to Type ) open import Data.Product using ( _,_ ; Σ-syntax ; _×_ ) open import Function using ( _∘_ ) open import Level using ( Level ; _⊔_ ) open import Relation.Binary using ( Reflexive ; Sym ; Symmetric; Trans; Transitive ) open import Relation.Binary.PropositionalEquality as ≡ using ( _≡_ ; module ≡-Reasoning ) open import Axiom.Extensionality.Propositional using () renaming (Extensionality to funext ) -- Imports from the Agda Universal Algebra Library ----------------------------------------------- open import Overture using ( ∣_∣ ; ∥_∥ ; _≈_ ; _∙_ ; lower∼lift ; lift∼lower ) open import Base.Functions using ( IsInjective ) open import Base.Algebras {𝑆 = 𝑆} using ( Algebra ; Lift-Alg ; ⨅ ) open import Base.Homomorphisms.Basic {𝑆 = 𝑆} using ( hom ; 𝒾𝒹 ; 𝓁𝒾𝒻𝓉 ; 𝓁ℴ𝓌ℯ𝓇 ; is-homomorphism ) open import Base.Homomorphisms.Properties {𝑆 = 𝑆} using ( ∘-hom )
Recall, we use f ≈ g to denote the assertion that f and g are
extensionally (or point-wise) equal; i.e., ∀ x, f x ≡ g x. This notion
of equality of functions is used in the following definition of isomorphism
between two algebras, say, 𝑨 and 𝑩.
record _≅_ {α b : Level}(𝑨 : Algebra α)(𝑩 : Algebra b) : Type (𝓞 ⊔ 𝓥 ⊔ α ⊔ b) where constructor mkiso field to : hom 𝑨 𝑩 from : hom 𝑩 𝑨 to∼from : ∣ to ∣ ∘ ∣ from ∣ ≈ ∣ 𝒾𝒹 𝑩 ∣ from∼to : ∣ from ∣ ∘ ∣ to ∣ ≈ ∣ 𝒾𝒹 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.
We could define this using Sigma types, like this.
_≅_ : {α b : Level}(𝑨 : Algebra α)(𝑩 : Algebra b) → Type(𝓞 ⊔ 𝓥 ⊔ α ⊔ b)
𝑨 ≅ 𝑩 = Σ[ f ∈ (hom 𝑨 𝑩)] Σ[ g ∈ hom 𝑩 𝑨 ] ((∣ f ∣ ∘ ∣ g ∣ ≈ ∣ 𝒾𝒹 𝑩 ∣) × (∣ g ∣ ∘ ∣ f ∣ ≈ ∣ 𝒾𝒹 𝑨 ∣))
However, with four components, an equivalent record type is easier to work with.
private variable a b c ℓ : Level ≅-refl : Reflexive (_≅_ {a}) ≅-refl {α}{𝑨} = mkiso (𝒾𝒹 𝑨) (𝒾𝒹 𝑨) (λ _ → ≡.refl) λ _ → ≡.refl ≅-sym : Sym (_≅_ {a}) (_≅_ {b}) ≅-sym φ = mkiso (from φ) (to φ) (from∼to φ) (to∼from φ) ≅-trans : Trans (_≅_ {a})(_≅_ {b})(_≅_ {a}{ℓ}) ≅-trans {ℓ = ℓ}{𝑨}{𝑩}{𝑪} ab bc = mkiso f g τ ν where f : hom 𝑨 𝑪 f = ∘-hom 𝑨 𝑪 (to ab) (to bc) g : hom 𝑪 𝑨 g = ∘-hom 𝑪 𝑨 (from bc) (from ab) τ : ∣ f ∣ ∘ ∣ g ∣ ≈ ∣ 𝒾𝒹 𝑪 ∣ τ x = (≡.cong ∣ to bc ∣(to∼from ab (∣ from bc ∣ x)))∙(to∼from bc) x ν : ∣ g ∣ ∘ ∣ f ∣ ≈ ∣ 𝒾𝒹 𝑨 ∣ ν x = (≡.cong ∣ from ab ∣(from∼to bc (∣ to ab ∣ x)))∙(from∼to ab) x -- The "to" map of an isomorphism is injective. ≅toInjective : {a b : Level}{𝑨 : Algebra a}{𝑩 : Algebra b} (φ : 𝑨 ≅ 𝑩) → IsInjective ∣ to φ ∣ ≅toInjective (mkiso (f , _) (g , _) _ g∼f){a}{b} fafb = a ≡⟨ ≡.sym (g∼f a) ⟩ g (f a) ≡⟨ ≡.cong g fafb ⟩ g (f b) ≡⟨ g∼f b ⟩ b ∎ where open ≡-Reasoning -- The "from" map of an isomorphism is injective. ≅fromInjective : {a b : Level}{𝑨 : Algebra a}{𝑩 : Algebra b} (φ : 𝑨 ≅ 𝑩) → IsInjective ∣ from φ ∣ ≅fromInjective φ = ≅toInjective (≅-sym φ)
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 Agda’s universe hierarchy.
open Level Lift-≅ : {a b : Level}{𝑨 : Algebra a} → 𝑨 ≅ (Lift-Alg 𝑨 b) Lift-≅{b = b}{𝑨 = 𝑨} = record { to = 𝓁𝒾𝒻𝓉 𝑨 ; from = 𝓁ℴ𝓌ℯ𝓇 𝑨 ; to∼from = ≡.cong-app lift∼lower ; from∼to = ≡.cong-app (lower∼lift {b = b}) } Lift-Alg-iso : {a b : Level}{𝑨 : Algebra a}{𝓧 : Level} {𝑩 : Algebra b}{𝓨 : Level} → 𝑨 ≅ 𝑩 → (Lift-Alg 𝑨 𝓧) ≅ (Lift-Alg 𝑩 𝓨) Lift-Alg-iso A≅B = ≅-trans (≅-trans (≅-sym Lift-≅) A≅B) Lift-≅
The lift is also associative, up to isomorphism at least.
Lift-Alg-assoc : (ℓ₁ ℓ₂ : Level) {𝑨 : Algebra a} → Lift-Alg 𝑨 (ℓ₁ ⊔ ℓ₂) ≅ (Lift-Alg (Lift-Alg 𝑨 ℓ₁) ℓ₂) Lift-Alg-assoc ℓ₁ ℓ₂ {𝑨} = ≅-trans (≅-trans Goal Lift-≅) Lift-≅ where Goal : Lift-Alg 𝑨 (ℓ₁ ⊔ ℓ₂) ≅ 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 _ {a b ι : Level}{I : Type ι}{fiu : funext ι a}{fiw : funext ι b} where ⨅≅ : {𝒜 : I → Algebra a}{ℬ : I → Algebra b} → (∀ (i : I) → 𝒜 i ≅ ℬ i) → ⨅ 𝒜 ≅ ⨅ ℬ ⨅≅ {𝒜}{ℬ} AB = record { to = ϕ , ϕhom ; from = ψ , ψhom ; to∼from = ϕ∼ψ ; from∼to = ψ∼ϕ } where ϕ : ∣ ⨅ 𝒜 ∣ → ∣ ⨅ ℬ ∣ ϕ a i = ∣ to (AB i) ∣ (a i) ϕhom : is-homomorphism (⨅ 𝒜) (⨅ ℬ) ϕ ϕhom 𝑓 a = fiw (λ i → ∥ to (AB i) ∥ 𝑓 (λ x → a x i)) ψ : ∣ ⨅ ℬ ∣ → ∣ ⨅ 𝒜 ∣ ψ b i = ∣ from (AB i) ∣ (b i) ψhom : is-homomorphism (⨅ ℬ) (⨅ 𝒜) ψ ψhom 𝑓 𝒃 = fiu (λ i → ∥ from (AB i) ∥ 𝑓 (λ x → 𝒃 x i)) ϕ∼ψ : ϕ ∘ ψ ≈ ∣ 𝒾𝒹 (⨅ ℬ) ∣ ϕ∼ψ 𝒃 = fiw λ i → to∼from (AB i) (𝒃 i) ψ∼ϕ : ψ ∘ ϕ ≈ ∣ 𝒾𝒹 (⨅ 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 _ {a b γ ι : Level}{I : Type ι}{fizw : funext (ι ⊔ γ) b}{fiu : funext ι a} where Lift-Alg-⨅≅ : {𝒜 : I → Algebra a}{ℬ : (Lift γ I) → Algebra b} → (∀ i → 𝒜 i ≅ ℬ (lift i)) → Lift-Alg (⨅ 𝒜) γ ≅ ⨅ ℬ Lift-Alg-⨅≅ {𝒜}{ℬ} AB = Goal where ϕ : ∣ ⨅ 𝒜 ∣ → ∣ ⨅ ℬ ∣ ϕ a i = ∣ to (AB (lower i)) ∣ (a (lower i)) ϕhom : is-homomorphism (⨅ 𝒜) (⨅ ℬ) ϕ ϕhom 𝑓 a = fizw (λ i → (∥ to (AB (lower i)) ∥) 𝑓 (λ x → a x (lower i))) ψ : ∣ ⨅ ℬ ∣ → ∣ ⨅ 𝒜 ∣ ψ b i = ∣ from (AB i) ∣ (b (lift i)) ψhom : is-homomorphism (⨅ ℬ) (⨅ 𝒜) ψ ψhom 𝑓 𝒃 = fiu (λ i → ∥ from (AB i) ∥ 𝑓 (λ x → 𝒃 x (lift i))) ϕ∼ψ : ϕ ∘ ψ ≈ ∣ 𝒾𝒹 (⨅ ℬ) ∣ ϕ∼ψ 𝒃 = fizw λ i → to∼from (AB (lower i)) (𝒃 i) ψ∼ϕ : ψ ∘ ϕ ≈ ∣ 𝒾𝒹 (⨅ 𝒜) ∣ ψ∼ϕ a = fiu λ i → from∼to (AB i) (a i) A≅B : ⨅ 𝒜 ≅ ⨅ ℬ A≅B = record { to = ϕ , ϕhom ; from = ψ , ψhom ; to∼from = ϕ∼ψ ; from∼to = ψ∼ϕ } Goal : Lift-Alg (⨅ 𝒜) γ ≅ ⨅ ℬ Goal = ≅-trans (≅-sym Lift-≅) A≅B
← Base.Homomorphisms.Factor Base.Homomorphisms.HomomorphicImages →