This section describes the 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 Algebras.Signatures using (Signature; π; π₯) module Homomorphisms.Isomorphisms {π : Signature π π₯} where open import Homomorphisms.Noether{π = π} public open import MGS-Embeddings using (Nat; NatΞ ; NatΞ -is-embedding) public
Recall, f ~ g means f and g are extensionally (or pointwise) equal; i.e., β x, f x β‘ g x. We use this notion of equality of functions in the following definition of isomorphism.
_β _ : {π€ π¦ : Universe}(π¨ : Algebra π€ π)(π© : Algebra π¦ π) β π β π₯ β π€ β π¦ Μ π¨ β π© = Ξ£ f κ (hom π¨ π©) , Ξ£ g κ (hom π© π¨) , (β£ f β£ β β£ g β£ βΌ β£ πΎπΉ π© β£) Γ (β£ g β£ β β£ f β£ βΌ β£ πΎπΉ π¨ β£)
That is, two structures are isomorphic provided there are homomorphisms going back and forth between them which compose to the identity map.
β -refl : {π€ : Universe} {π¨ : Algebra π€ π} β π¨ β π¨ β -refl {π€}{π¨} = πΎπΉ π¨ , πΎπΉ π¨ , (Ξ» a β refl) , (Ξ» a β refl) β -sym : {π€ π¦ : Universe}{π¨ : Algebra π€ π}{π© : Algebra π¦ π} β π¨ β π© β π© β π¨ β -sym h = fst β₯ h β₯ , fst h , β₯ snd β₯ h β₯ β₯ , β£ snd β₯ h β₯ β£ module _ {π§ π¨ π© : Universe} where β -trans : {π¨ : Algebra π§ π}{π© : Algebra π¨ π}{πͺ : Algebra π© π} β π¨ β π© β π© β πͺ β π¨ β πͺ β -trans {π¨} {π©}{πͺ} ab bc = f , g , Ξ± , Ξ² where f1 : hom π¨ π© f1 = β£ ab β£ f2 : hom π© πͺ f2 = β£ bc β£ f : hom π¨ πͺ f = β-hom π¨ πͺ f1 f2 g1 : hom πͺ π© g1 = fst β₯ bc β₯ g2 : hom π© π¨ g2 = fst β₯ ab β₯ g : hom πͺ π¨ g = β-hom πͺ π¨ g1 g2 Ξ± : β£ f β£ β β£ g β£ βΌ β£ πΎπΉ πͺ β£ Ξ± x = (ap β£ f2 β£(β£ snd β₯ ab β₯ β£ (β£ g1 β£ x)))β(β£ snd β₯ bc β₯ β£) x Ξ² : β£ g β£ β β£ f β£ βΌ β£ πΎπΉ π¨ β£ Ξ² x = (ap β£ g2 β£(β₯ snd β₯ bc β₯ β₯ (β£ f1 β£ x)))β(β₯ snd β₯ ab β₯ β₯) x
Fortunately, the lift operation preserves isomorphism (i.e., itβs 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 Lift module _ {π€ π¦ : Universe} where Lift-β : {π¨ : Algebra π€ π} β π¨ β (Lift-alg π¨ π¦) Lift-β {π¨} = ππΎπ»π , (πβ΄πβ―π{π€}{π¦}{π¨}) , happly liftβΌlower , happly (lowerβΌlift{π¦}) Lift-hom : (π§ : Universe)(π¨ : Universe){π¨ : Algebra π€ π}(π© : Algebra π¦ π) β hom π¨ π© β hom (Lift-alg π¨ π§) (Lift-alg π© π¨) Lift-hom π§ π¨ {π¨} π© (f , fhom) = lift β f β lower , Ξ³ where lABh : is-homomorphism (Lift-alg π¨ π§) π© (f β lower) lABh = β-is-hom (Lift-alg π¨ π§) π© {lower}{f} (Ξ» _ _ β refl) fhom Ξ³ : is-homomorphism(Lift-alg π¨ π§)(Lift-alg π© π¨) (lift β (f β lower)) Ξ³ = β-is-hom (Lift-alg π¨ π§) (Lift-alg π© π¨){f β lower}{lift} lABh Ξ» _ _ β refl module _ {π€ π¦ : Universe} where Lift-alg-iso : {π¨ : Algebra π€ π}{π§ : Universe} {π© : Algebra π¦ π}{π¨ : Universe} ----------------------------------------- β π¨ β π© β (Lift-alg π¨ π§) β (Lift-alg π© π¨) Lift-alg-iso Aβ B = β -trans (β -trans (β -sym Lift-β ) Aβ B) Lift-β
The lift is also associative, up to isomorphism at least.
module _ {π π€ π¦ : Universe} where Lift-alg-assoc : {π¨ : Algebra π€ π} β Lift-alg π¨ (π¦ β π) β (Lift-alg (Lift-alg π¨ π¦) π) Lift-alg-assoc {π¨} = β -trans (β -trans Ξ³ Lift-β ) Lift-β where Ξ³ : Lift-alg π¨ (π¦ β π) β π¨ Ξ³ = β -sym Lift-β Lift-alg-associative : (π¨ : Algebra π€ π) β Lift-alg π¨ (π¦ β π) β (Lift-alg (Lift-alg π¨ π¦) π) Lift-alg-associative π¨ = Lift-alg-assoc {π¨}
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 _ {π π€ π¦ : Universe}{I : π Μ}{feππ€ : dfunext π π€}{feππ¦ : dfunext π π¦} where β¨ β : {π : I β Algebra π€ π}{β¬ : I β Algebra π¦ π} β Ξ i κ I , π i β β¬ i β β¨ π β β¨ β¬ β¨ β {π}{β¬} AB = Ξ³ where Ο : β£ β¨ π β£ β β£ β¨ β¬ β£ Ο a i = β£ fst (AB i) β£ (a i) Οhom : is-homomorphism (β¨ π) (β¨ β¬) Ο Οhom π a = feππ¦ (Ξ» i β β₯ fst (AB i) β₯ π (Ξ» x β a x i)) Ο : β£ β¨ β¬ β£ β β£ β¨ π β£ Ο b i = β£ fst β₯ AB i β₯ β£ (b i) Οhom : is-homomorphism (β¨ β¬) (β¨ π) Ο Οhom π π = feππ€ (Ξ» i β snd β£ snd (AB i) β£ π (Ξ» x β π x i)) Ο~Ο : Ο β Ο βΌ β£ πΎπΉ (β¨ β¬) β£ Ο~Ο π = feππ¦ Ξ» i β fst β₯ snd (AB i) β₯ (π i) Ο~Ο : Ο β Ο βΌ β£ πΎπΉ (β¨ π) β£ Ο~Ο a = feππ€ Ξ» i β snd β₯ snd (AB i) β₯ (a i) Ξ³ : β¨ π β β¨ β¬ Ξ³ = (Ο , Οhom) , ((Ο , Οhom) , Ο~Ο , Ο~Ο)
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, β i β π i β
β¬ (lift i) instead of Ξ i κ I , π i β
β¬ (lift i).)
module _ {π π© : Universe}{I : π Μ}{fizw : dfunext (π β π©) π¦}{fiu : dfunext π π€} where Lift-alg-β¨ β : {π : I β Algebra π€ π}{β¬ : (Lift{π©} I) β Algebra π¦ π} β (β i β π i β β¬ (lift i)) β Lift-alg (β¨ π) π© β β¨ β¬ Lift-alg-β¨ β {π}{β¬} AB = Ξ³ where Ο : β£ β¨ π β£ β β£ β¨ β¬ β£ Ο a i = β£ fst (AB (lower i)) β£ (a (lower i)) Οhom : is-homomorphism (β¨ π) (β¨ β¬) Ο Οhom π a = fizw (Ξ» i β (β₯ fst (AB (lower i)) β₯) π (Ξ» x β a x (lower i))) Ο : β£ β¨ β¬ β£ β β£ β¨ π β£ Ο b i = β£ fst β₯ AB i β₯ β£ (b (lift i)) Οhom : is-homomorphism (β¨ β¬) (β¨ π) Ο Οhom π π = fiu (Ξ» i β (snd β£ snd (AB i) β£) π (Ξ» x β π x (lift i))) Ο~Ο : Ο β Ο βΌ β£ πΎπΉ (β¨ β¬) β£ Ο~Ο π = fizw Ξ» i β fst β₯ snd (AB (lower i)) β₯ (π i) Ο~Ο : Ο β Ο βΌ β£ πΎπΉ (β¨ π) β£ Ο~Ο a = fiu Ξ» i β snd β₯ snd (AB i) β₯ (a i) Aβ B : β¨ π β β¨ β¬ Aβ B = (Ο , Οhom) , ((Ο , Οhom) , Ο~Ο , Ο~Ο) Ξ³ : Lift-alg (β¨ π) π© β β¨ β¬ Ξ³ = β -trans (β -sym Lift-β ) Aβ B
Finally, we prove some useful facts about embeddings that occasionally come in handy.
module _ {π π€ π¦ : Universe} where embedding-lift-nat : hfunext π π€ β hfunext π π¦ β {I : π Μ}{A : I β π€ Μ}{B : I β π¦ Μ} (h : Nat A B) β (β i β is-embedding (h i)) ------------------------------------------ β is-embedding(NatΞ h) embedding-lift-nat hfiu hfiw h hem = NatΞ -is-embedding hfiu hfiw h hem embedding-lift-nat' : hfunext π π€ β hfunext π π¦ β {I : π Μ}{π : I β Algebra π€ π}{β¬ : I β Algebra π¦ π} (h : Nat(fst β π)(fst β β¬)) β (β i β is-embedding (h i)) -------------------------------------------------------- β is-embedding(NatΞ h) embedding-lift-nat' hfiu hfiw h hem = NatΞ -is-embedding hfiu hfiw h hem embedding-lift : hfunext π π€ β hfunext π π¦ β {I : π Μ} β {π : I β Algebra π€ π}{β¬ : I β Algebra π¦ π} β (h : β i β β£ π i β£ β β£ β¬ i β£) β (β i β is-embedding (h i)) ---------------------------------------------------------- β is-embedding(Ξ» (x : β£ β¨ π β£) (i : I) β (h i)(x i)) embedding-lift hfiu hfiw {I}{π}{β¬} h hem = embedding-lift-nat' hfiu hfiw {I}{π}{β¬} h hem isoβembedding : {π€ π¦ : Universe}{π¨ : Algebra π€ π}{π© : Algebra π¦ π} β (Ο : π¨ β π©) β is-embedding (fst β£ Ο β£) isoβembedding Ο = equivs-are-embeddings (fst β£ Ο β£) (invertibles-are-equivs (fst β£ Ο β£) finv) where finv : invertible (fst β£ Ο β£) finv = β£ fst β₯ Ο β₯ β£ , (snd β₯ snd Ο β₯ , fst β₯ snd Ο β₯)
β Homomorphisms.Noether Homomorphisms.HomomorphicImages β