Universal Algebra in HoTT Andreas Lynge and Bas Spitters Aarhus - PowerPoint PPT Presentation

Universal Algebra in HoTT Andreas Lynge and Bas Spitters Aarhus University August 13, 2019

Introduction ‚ Universal algebra is a general study of algebraic structures. The results in universal algebra apply to all “algebras”, e.g. groups, rings, modules. ‚ We have formalized a part of universal algebra in the HoTT library for Coq, including the three isomorphism theorems. ‚ Based on the math-classes library. ‚ Type theoretic universal algebra often relies on setoids. ‚ We avoid setoids in the HoTT library, quotient sets are HITs.

Group Example (Group) A group is an h-set G : Set with ‚ unit e : G ‚ multiplication ¨ : G Ñ G Ñ G ‚ inversion p´q ´ 1 : G Ñ G ‚ satisfying certain equations, e.g. x ¨ x ´ 1 “ e for all x : G .

Group acting on a set Example (Group) A group is an h-set G : Set with ‚ unit e : G ‚ multiplication ¨ : G Ñ G Ñ G ‚ inversion p´q ´ 1 : G Ñ G ‚ satisfying certain equations, e.g. x ¨ x ´ 1 “ e for all x : G . Example (Group acting on a set) A group acting on a set is a group G and an h-set S with ‚ action α : G Ñ S Ñ S ‚ α p x ¨ y q “ α p x q ˝ α p y q ‚ α p e q “ id S

Signature Definition (Signature) A signature σ : Signature consists of ‚ Sort p σ q : U ‚ Symbol p σ q : U ‚ for each u : Symbol p σ q , σ u : Sort p σ q ˆ List p Sort p σ qq .

Algebra Definition (Signature) A signature σ : Signature consists of ‚ Sort p σ q : U ‚ Symbol p σ q : U ‚ for each u : Symbol p σ q , σ u : Sort p σ q ˆ List p Sort p σ qq . Definition (Algebra) An algebra A : Algebra p σ q for σ : Signature consists of ‚ for each s : Sort p σ q , A s : Set ‚ for each u : Symbol p σ q , u A : A s 1 Ñ A s 2 Ñ ¨ ¨ ¨ Ñ A s n , where p s 1 , r s 2 , . . . , s n sq : ” σ u .

Example (Group acting on a set) A group G acting on a set S , ‚ unit e : G ‚ multiplication ¨ : G Ñ G Ñ G ‚ inversion p´q ´ 1 : G Ñ G ‚ action α : G Ñ S Ñ S , is an algebra A : Algebra p σ q for σ : Signature with ‚ Sort p σ q ” t g , s u ‚ Symbol p σ q ” t u , m , i , a u ‚ σ u ” p g , rsq , σ m ” p g , r g , g sq , σ i ” p g , r g sq , σ a ” p g , r s , s sq . Carriers A g : ” G and A s : ” S , and operations ‚ u A : A g is unit ‚ m A : A g Ñ A g Ñ A g is multiplication i A : A g Ñ A g is inversion ‚ a A : A g Ñ A s Ñ A s is the action. ‚

Let A , B , C : Algebra p σ q .

Homomorphism Let A , B , C : Algebra p σ q . Definition (Homomorphism) A homomorphism f : A Ñ B consists of ‚ f s : A s Ñ B s for all s : Sort p σ q ‚ f s t p u A p x 1 , . . . , x n qq “ u B p f s 1 p x 1 q , . . . , f s n p x n qq , for all u : Symbol p σ q .

Isomorphism Let A , B , C : Algebra p σ q . Definition (Homomorphism) A homomorphism f : A Ñ B consists of ‚ f s : A s Ñ B s for all s : Sort p σ q ‚ f s t p u A p x 1 , . . . , x n qq “ u B p f s 1 p x 1 q , . . . , f s n p x n qq , for all u : Symbol p σ q . Definition (Isomorphism) An isomorphism is a homomorphism f : A Ñ B where f s : A s Ñ B s is an equivalence for all s : Sort p σ q .

Isomorphic Let A , B , C : Algebra p σ q . Definition (Homomorphism) A homomorphism f : A Ñ B consists of ‚ f s : A s Ñ B s for all s : Sort p σ q ‚ f s t p u A p x 1 , . . . , x n qq “ u B p f s 1 p x 1 q , . . . , f s n p x n qq , for all u : Symbol p σ q . Definition (Isomorphism) An isomorphism is a homomorphism f : A Ñ B where f s : A s Ñ B s is an equivalence for all s : Sort p σ q . Definition (Isomorphic) Write A – B for there is an isomorphism A Ñ B .

Isomorphic implies equal Theorem (Isomrophic implies equal) If A – B then A “ B. ‚ Coquand and Danielsson, Isomorphism is equality.

Lemma Theorem (Isomrophic implies equal) If A – B then A “ B. ‚ Coquand and Danielsson, Isomorphism is equality. Lemma Suppose ‚ X , Y : Sort p σ q Ñ Set ‚ α : X s 1 Ñ ¨ ¨ ¨ Ñ X s n Ñ X t and β : Y s 1 Ñ ¨ ¨ ¨ Ñ Y s n Ñ Y t ‚ f : ś s X s » Y s ‚ f t p α p x 1 , . . . , x n qq “ β p f s 1 p x 1 q , . . . , f s n p x n qq . Then ś s X s “ Y s hkkikkj transport p λ Z . Z s 1 Ѩ¨¨Ñ Z sn Ñ Z t q p funext p ua ˝ f qq p α q “ β loooooooomoooooooon X “ Y

Precategory of algebras Lemma (Precategory of algebras) There is a precategory σ - Alg of Algebra p σ q and homomorphisms, ‚ p 1 A q s ” λ x . x, s : Sort p σ q ‚ p gf q s ” g s ˝ f s , f : A Ñ B, g : B Ñ C

Equal is equivalent to isomorphic Lemma (Precategory of algebras) There is a precategory σ - Alg of Algebra p σ q and homomorphisms, ‚ p 1 A q s ” λ x . x, s : Sort p σ q ‚ p gf q s ” g s ˝ f s , f : A Ñ B, g : B Ñ C Theorem (Equal is equivalent to isomorphic) The function p A “ B q Ñ p A – B q is an equivalence.

Univalent category of algebras Lemma (Precategory of algebras) There is a precategory σ - Alg of Algebra p σ q and homomorphisms, ‚ p 1 A q s ” λ x . x, s : Sort p σ q ‚ p gf q s ” g s ˝ f s , f : A Ñ B, g : B Ñ C Theorem (Equal is equivalent to isomorphic) The function p A “ B q Ñ p A – B q is an equivalence. Theorem (Univalent category of algebras) The precategory σ - Alg is a univalent category. ‚ HoTT book, http://homotopytypetheory.org/book . ‚ Arhens and Lumsdaine, Displayed Categories.

Congruence Definition (Congruence) A congruence on A is a family of mere equivalence relations Θ : ś s p A s Ñ A s Ñ Prop q where Θ s 1 p x 1 , y 1 q ˆ ¨ ¨ ¨ ˆ Θ s n p x n , y n q implies u A p x 1 , . . . , x n q , u A p y 1 , . . . , y n q ` ˘ Θ s t for all u : Symbol p σ q .

Quotient algebra Definition (Congruence) A congruence on A is a family of mere equivalence relations Θ : ś s p A s Ñ A s Ñ Prop q where Θ s 1 p x 1 , y 1 q ˆ ¨ ¨ ¨ ˆ Θ s n p x n , y n q implies u A p x 1 , . . . , x n q , u A p y 1 , . . . , y n q ` ˘ Θ s t for all u : Symbol p σ q . Definition (Quotient algebra) Let Θ : ś s p A s Ñ A s Ñ Prop q be a congruence. The quotient algebra A { Θ consists of ‚ p A { Θ q s : ” A s { Θ s , the set-quotient ‚ operations u A { Θ ` ˘ ` u A p x 1 , . . . , x n q ˘ q 1 p x 1 q , . . . , q n p x n q “ q t , where q i : A s i Ñ A s i { Θ s i are the set-quotient constructors.

Suppose Θ : ś s p A s Ñ A s Ñ Prop q is a congruence.

Quotient homomorphism Suppose Θ : ś s p A s Ñ A s Ñ Prop q is a congruence. Lemma (Quotient homomorphism) There is a homomorphism ρ : A Ñ A { Θ , pointwise A s Ñ A s { Θ s .

Quotient universal property Suppose Θ : ś s p A s Ñ A s Ñ Prop q is a congruence. Lemma (Quotient homomorphism) There is a homomorphism ρ : A Ñ A { Θ , pointwise A s Ñ A s { Θ s . Lemma (Quotient universal property) Precomposition with ρ : A Ñ A { Θ induces an equivalence p A { Θ Ñ B q » ř f : A Ñ B resp p f q , where resp p f q : ” ś ś ` ˘ Θ s p x , y q Ñ f s p x q “ f s p y q . s :Sort p σ q x , y : A s ρ Let f : A Ñ B such that resp p f q . Then there A { Θ A is a unique p : A { Θ Ñ B satisfying f “ pq . p f Coequalizers in σ - Alg are quotient algebras. B

Product algebra Product algebra Let F : I Ñ Algebra p σ q . The product algebra Ś i F p i q has carriers p Ś i F p i qq s ” ś i p F p i qq s There are projection homomorphisms π j : Ś i F p i q Ñ F p j q . Products in σ - Alg are product algebras.

Subalgebra Product algebra Let F : I Ñ Algebra p σ q . The product algebra Ś i F p i q has carriers p Ś i F p i qq s ” ś i p F p i qq s There are projection homomorphisms π j : Ś i F p i q Ñ F p j q . Products in σ - Alg are product algebras. Subalgebra Let P : ś s p A s Ñ Prop q such that, for any u : Symbol p σ q , P n ` 1 p u A p x 1 , . . . , x n qq , P s 1 p x 1 q ˆ ¨ ¨ ¨ ˆ P s n p x n q implies where p s 1 , r s 2 , . . . , s n ` 1 sq ” σ u . Then there is a subalgebra A & P with carriers p A & P q s ” ř x : A s P s p x q There exists an inclusion homomorphism p A & P q Ñ A . Equalizers in σ - Alg are subalgebras.

First isomorphism theorem Theorem (First isomorphism/identification theorem) Let f : A Ñ B be a homomorphism. ‚ ker p f qp s , x , y q : ” ` ˘ f s p x q “ f s p y q is a congruence. ‚ inim p f qp s , y q : ” � ř x p f s p x q “ y q � is closed under operations, so it induces a subalgebra B & inim p f q of B. ‚ There exists an isomorphism A { ker p f q Ñ B & inim p f q .

First identification theorem Theorem (First isomorphism/identification theorem) Let f : A Ñ B be a homomorphism. ‚ ker p f qp s , x , y q : ” ` ˘ f s p x q “ f s p y q is a congruence. ‚ inim p f qp s , y q : ” � ř x p f s p x q “ y q � is closed under operations, so it induces a subalgebra B & inim p f q of B. ‚ There exists an isomorphism A { ker p f q Ñ B & inim p f q . ‚ Therefore A { ker p f q “ B & inim p f q .

The category of algebras is regular Theorem (First isomorphism/identification theorem) Let f : A Ñ B be a homomorphism. ‚ ker p f qp s , x , y q : ” ` ˘ f s p x q “ f s p y q is a congruence. ‚ inim p f qp s , y q : ” � ř x p f s p x q “ y q � is closed under operations, so it induces a subalgebra B & inim p f q of B. ‚ There exists an isomorphism A { ker p f q Ñ B & inim p f q . ‚ Therefore A { ker p f q “ B & inim p f q . Category σ - Alg is regular, ‚ f : A Ñ B image factorizes A Ñ B & inim p f q ã Ñ B ‚ images are pullback stable. ‚ σ - Alg is complete