 Isomorphisms

This is the Structures.Isos module of the Agda Universal Algebra Library.

{-# OPTIONS --without-K --exact-split --safe #-}

module Structures.Isos where

-- Imports from Agda and the Agda Standard Library ---------------------
open import Agda.Primitive using ( _⊔_ ; lsuc ) 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.Base using ( _∘_ )
open import Level using ( Level ; Lift )
open import Relation.Binary.PropositionalEquality using ( cong ; refl ; cong-app ; module ≡-Reasoning )

-- Imports from the Agda Universal Algebra Library ---------------------------------------------
open import Overture.Preliminaries using ( ∣_∣ ; _≈_ ; ∥_∥ ; _∙_ ; lower∼lift ; lift∼lower )
open import Structures.Basic       using ( signature ; structure ; Lift-Strucˡ ; Lift-Strucʳ
; Lift-Struc ; sigl ; siglˡ ; siglʳ )
open import Structures.Homs        using ( hom ; 𝒾𝒹 ; ∘-hom ; 𝓁𝒾𝒻𝓉 ; 𝓁ℴ𝓌ℯ𝓇 ; 𝓁𝒾𝒻𝓉ˡ
; 𝓁ℴ𝓌ℯ𝓇ˡ ; 𝓁𝒾𝒻𝓉ʳ ; 𝓁ℴ𝓌ℯ𝓇ʳ ; is-hom )
open import Structures.Products    using (  ; ℓp ;  ; class-product )

private variable
𝓞₀ 𝓥₀ 𝓞₁ 𝓥₁ α ρᵃ β ρᵇ γ ρᶜ ρ  ι : Level
𝐹 : signature 𝓞₀ 𝓥₀
𝑅 : signature 𝓞₁ 𝓥₁

Definition of Isomorphism

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.

Isomorphism is an equivalence relation

module _ {𝑨 : structure 𝐹 𝑅 {α}{ρᵃ}} where

≅-refl : 𝑨  𝑨
≅-refl = mkiso 𝒾𝒹 𝒾𝒹  _  refl)  _  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

Lift is an algebraic invariant

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 [Overture.Lifts][].

open Level

module _ {𝑨 : structure 𝐹 𝑅{α}{ρᵃ}} where

Lift-≅ˡ : 𝑨  (Lift-Strucˡ  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  𝑨)
Lift-≅ʳ  = record { to = 𝓁𝒾𝒻𝓉ʳ
; from = 𝓁ℴ𝓌ℯ𝓇ʳ
; to∼from = cong-app refl
; from∼to = cong-app refl
}

Lift-≅ : 𝑨  (Lift-Struc  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' : Level)  𝑨  𝑩  Lift-Strucˡ  𝑨  Lift-Strucˡ ℓ' 𝑩
Lift-Strucˡ-iso  ℓ' A≅B = ≅-trans ( ≅-trans (≅-sym Lift-≅ˡ) A≅B ) Lift-≅ˡ

Lift-Struc-iso : ( ρ ℓ' ρ' : Level)  𝑨  𝑩  Lift-Struc  ρ 𝑨  Lift-Struc ℓ' ρ' 𝑩
Lift-Struc-iso  ρ ℓ' ρ' A≅B = ≅-trans ( ≅-trans (≅-sym Lift-≅) A≅B ) Lift-≅

Lift associativity

The lift is also associative, up to isomorphism at least.

module _ {𝑨 : structure 𝐹 𝑅 {α}{ρᵃ} } where

Lift-Struc-assocˡ : { ℓ' : Level}  Lift-Strucˡ (  ℓ') 𝑨  (Lift-Strucˡ  (Lift-Strucˡ ℓ' 𝑨))
Lift-Struc-assocˡ {}{ℓ'} = ≅-trans (≅-trans Goal Lift-≅ˡ) Lift-≅ˡ
where
Goal : Lift-Strucˡ (  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' ρ ρ' : Level}
Lift-Struc (  ℓ') (ρ  ρ') 𝑨  (Lift-Struc  ρ (Lift-Struc ℓ' ρ' 𝑨))
Lift-Struc-assoc {}{ℓ'}{ρ}{ρ'} = ≅-trans (≅-trans Goal Lift-≅ ) Lift-≅
where
Goal : Lift-Struc (  ℓ') (ρ  ρ') 𝑨  𝑨
Goal = ≅-sym Lift-≅

Products preserve isomorphisms

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}}{ : I  structure 𝐹 𝑅{β}{ρᵇ}} where

open structure
open ≡-Reasoning
⨅≅ : funext ι α  funext ι β  (∀ (i : I)  𝒜 i   i)   𝒜

⨅≅ fiu fiw AB = record { to = ϕ , ϕhom ; from = ψ , ψhom ; to∼from = ϕ~ψ ; from∼to = ψ~ϕ }
where
ϕ : carrier ( 𝒜)  carrier ( )
ϕ a i =  to (AB i)  (a i)

ϕhom : is-hom ( 𝒜) ( ) ϕ
ϕhom =  r a x 𝔦  fst  to (AB 𝔦)  r  z  a z 𝔦) (x 𝔦)) ,
λ f a  fiw  i  snd  to (AB i)  f  z  a z i))
ψ : carrier ( )  carrier ( 𝒜)
ψ b i =  from (AB i)  (b i)

ψhom : is-hom ( ) ( 𝒜) ψ
ψhom =  r a x 𝔦  fst  from (AB 𝔦)  r  z  a z 𝔦) (x 𝔦)) ,
f a  fiu  i  snd  from (AB i)  f  z  a z i)))

ϕ~ψ : ϕ  ψ   𝒾𝒹 {𝑨 =  }
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}}{ : (Lift γ I)  structure 𝐹 𝑅 {β}{ρᵇ}} where

open structure

Lift-Struc-⨅≅ : funext (ι  γ) β  funext ι α
(∀ i  𝒜 i   (lift i))  Lift-Strucˡ γ ( 𝒜)

Lift-Struc-⨅≅ fizw fiu AB = Goal
where
ϕ : carrier ( 𝒜)   carrier ( )
ϕ a i =  to (AB (lower i))  (a (lower i))

ϕhom : is-hom ( 𝒜) ( ) ϕ
ϕhom =  r a x i  fst  to (AB (lower i))  r  x₁  a x₁ (lower i)) (x (lower i)))
, λ f a  fizw  i  snd  to (AB (lower i))  f  x  a x (lower i)))

ψ : carrier ( )  carrier ( 𝒜)
ψ b i =  from (AB i)  (b (lift i))

ψhom : is-hom ( ) ( 𝒜) ψ
ψhom =  r a x i  fst  from (AB i)  r  x₁  a x₁ (lift i)) (x (lift i)))
, λ f a  fiu  i  snd  from (AB i)  f  x  a x (lift i)))

ϕ~ψ : ϕ  ψ   𝒾𝒹 {𝑨 = ( )}
ϕ~ψ b = fizw  i  to∼from (AB (lower i)) (b i))

ψ~ϕ : ψ  ϕ   𝒾𝒹 {𝑨 = ( 𝒜)}
ψ~ϕ a = fiu  i  from∼to (AB i) (a i))

A≅B :  𝒜
A≅B = mkiso (ϕ , ϕhom) (ψ , ψhom) ϕ~ψ ψ~ϕ

Goal : Lift-Strucˡ γ ( 𝒜)
Goal = ≅-trans (≅-sym Lift-≅ˡ) A≅B