The syntactic Frobenius theory Signature Σ, each σ ∈ Σ equipped with arity and coarity, σ : n → m . Freely generated (syntactic) Cartesian bicategory CB Σ has objects N and morphisms . . . . S 1 . . � � � � � . . . Mor( CB Σ ) ::= ǫ . . . � � � � S 1 S 2 � . . . . . � � � � � . . S 2 . . � � � � . . . . � � � � σ . . � � � � modulo the laws of Cartesian bicategories. A model M : CB Σ → Rel consists of • a set V = M (1)
The syntactic Frobenius theory Signature Σ, each σ ∈ Σ equipped with arity and coarity, σ : n → m . Freely generated (syntactic) Cartesian bicategory CB Σ has objects N and morphisms . . . . S 1 . . � � � � � . . . Mor( CB Σ ) ::= ǫ . . . � � � � S 1 S 2 � . . . . . � � � � � . . S 2 . . � � � � . . . . � � � � σ . . � � � � modulo the laws of Cartesian bicategories. A model M : CB Σ → Rel consists of • a set V = M (1) • relations M ( σ ) ⊆ V n × V m for σ ∈ Σ, σ : n → m
Example Frobenius theories Example • ≤ R
Example Frobenius theories Example • ensures that R is reflexive ≤ R
Example Frobenius theories Example • ensures that R is reflexive ≤ R • ≤ R R R
Example Frobenius theories Example • ensures that R is reflexive ≤ R • ensures that R is transitive ≤ R R R
Example Frobenius theories Example • ensures that R is reflexive ≤ R • ensures that R is transitive ≤ R R R • R and ≤ ≤ R R R
Example Frobenius theories Example • ensures that R is reflexive ≤ R • ensures that R is transitive ≤ R R R • R and ≤ ≤ R R R ensure that R is a function.
Example Frobenius theories Example • ensures that R is reflexive ≤ R • ensures that R is transitive ≤ R R R • R and ≤ ≤ R R R ensure that R is a function. • ≤
Example Frobenius theories Example • ensures that R is reflexive ≤ R • ensures that R is transitive ≤ R R R • R and ≤ ≤ R R R ensure that R is a function. • ensures that the underlying set is nonempty. ≤
Presentations Definition Fix a signature Σ and let E be a set of (well-typed) inequalities of morphisms in CB Σ .
Presentations Definition Fix a signature Σ and let E be a set of (well-typed) inequalities of morphisms in CB Σ . The Frobenius theory CB Σ /E has the morphisms of CB Σ taken modulo E .
Presentations Definition Fix a signature Σ and let E be a set of (well-typed) inequalities of morphisms in CB Σ . The Frobenius theory CB Σ /E has the morphisms of CB Σ taken modulo E . Lemma A model of CB Σ /E is the same thing as a model of CB Σ satisfying E .
Presentations Definition Fix a signature Σ and let E be a set of (well-typed) inequalities of morphisms in CB Σ . The Frobenius theory CB Σ /E has the morphisms of CB Σ taken modulo E . Lemma A model of CB Σ /E is the same thing as a model of CB Σ satisfying E . Lemma Every Frobenius theory is of the shape CB Σ /E for some Σ , E .
Contents 1 Cartesian bicategories 2 Frobenius theories 3 Completeness
Σ-structures • A model M : CB Σ → Rel consists of • a set V = M (1) • relations M ( σ ) ⊆ V n × V m for σ ∈ Σ, σ : n → m
Σ-structures • A model M : CB Σ → Rel consists of • a set V = M (1) • relations M ( σ ) ⊆ V n × V m for σ ∈ Σ, σ : n → m • In model theory, these are called Σ-structures.
Σ-structures • A model M : CB Σ → Rel consists of • a set V = M (1) • relations M ( σ ) ⊆ V n × V m for σ ∈ Σ, σ : n → m • In model theory, these are called Σ-structures. • A morphism between Σ-structures is a function between the underlying sets respecting the relations.
Σ-structures • A model M : CB Σ → Rel consists of • a set V = M (1) • relations M ( σ ) ⊆ V n × V m for σ ∈ Σ, σ : n → m • In model theory, these are called Σ-structures. • A morphism between Σ-structures is a function between the underlying sets respecting the relations. • One can view a set as a Σ-structure – with empty relations
Σ-structures • A model M : CB Σ → Rel consists of • a set V = M (1) • relations M ( σ ) ⊆ V n × V m for σ ∈ Σ, σ : n → m • In model theory, these are called Σ-structures. • A morphism between Σ-structures is a function between the underlying sets respecting the relations. • One can view a set as a Σ-structure – with empty relations • For S a Σ-structure, a morphism n → S is an n -tuple in S
Universal models Example We can translate a morphism R : n → m in CB Σ to a finite ι R ω R model U R with n − → U R ← − − m (called universal model).
Universal models Example We can translate a morphism R : n → m in CB Σ to a finite ι R ω R model U R with n − → U R ← − − m (called universal model). σ σ τ
Universal models Example We can translate a morphism R : n → m in CB Σ to a finite ι R ω R model U R with n − → U R ← − − m (called universal model). z σ y σ x τ
Universal models Example We can translate a morphism R : n → m in CB Σ to a finite ι R ω R model U R with n − → U R ← − − m (called universal model). z σ y σ x τ U R (1) = { x, y, z }
Universal models Example We can translate a morphism R : n → m in CB Σ to a finite ι R ω R model U R with n − → U R ← − − m (called universal model). z σ y σ x τ U R (1) = { x, y, z } U R ( σ ) = { ( x, y ) , ( y, z ) } , U R ( τ ) = { x }
Universal models Example We can translate a morphism R : n → m in CB Σ to a finite ι R ω R model U R with n − → U R ← − − m (called universal model). z σ y σ x τ U R (1) = { x, y, z } U R ( σ ) = { ( x, y ) , ( y, z ) } , U R ( τ ) = { x } ι R = x, ω R = y
Connection with Completeness Theorem (SYCO 1) ι R ω R The assignment of R : n → m to n − → U R ← − − m is a bijection
Connection with Completeness Theorem (SYCO 1) ι R ω R The assignment of R : n → m to n − → U R ← − − m is a bijection between morphisms in CB Σ and discrete cospans of finite Σ -structures.
Connection with Completeness Theorem (SYCO 1) ι R ω R The assignment of R : n → m to n − → U R ← − − m is a bijection between morphisms in CB Σ and discrete cospans of finite Σ -structures. S ≤ R if and only if there is α : U R → U S such that U R ι R ω R n m α ι S ω S U S
Connection with Completeness Theorem (SYCO 1) ι R ω R The assignment of R : n → m to n − → U R ← − − m is a bijection between morphisms in CB Σ and discrete cospans of finite Σ -structures. S ≤ R if and only if there is α : U R → U S such that U R ι R ω R n m α ι S ω S U S Connects semantics to syntax.
The ( · ) E construction Idea: Saturate a Σ-structure with respect to the axioms E .
The ( · ) E construction Idea: Saturate a Σ-structure with respect to the axioms E . Theorem There is a functor ( · ) E : Mod CB Σ → Mod CB Σ with a natural transformation ζ S : S → S E with the following property:
The ( · ) E construction Idea: Saturate a Σ-structure with respect to the axioms E . Theorem There is a functor ( · ) E : Mod CB Σ → Mod CB Σ with a natural transformation ζ S : S → S E with the following property: If A ≤ B is an axiom in E , and ( x, y ) ∈ S ( A ) then ( ζ ( x ) , ζ ( y )) ∈ S E ( B ) .
The ( · ) E construction Idea: Saturate a Σ-structure with respect to the axioms E . Theorem There is a functor ( · ) E : Mod CB Σ → Mod CB Σ with a natural transformation ζ S : S → S E with the following property: If A ≤ B is an axiom in E , and ( x, y ) ∈ S ( A ) then ( ζ ( x ) , ζ ( y )) ∈ S E ( B ) . Definition An algebra for the pointed endofunctor ( · ) E is a Σ-structure S with a morphism α : S E → S such that α ◦ ζ S = id S
The ( · ) E construction Idea: Saturate a Σ-structure with respect to the axioms E . Theorem There is a functor ( · ) E : Mod CB Σ → Mod CB Σ with a natural transformation ζ S : S → S E with the following property: If A ≤ B is an axiom in E , and ( x, y ) ∈ S ( A ) then ( ζ ( x ) , ζ ( y )) ∈ S E ( B ) . Definition An algebra for the pointed endofunctor ( · ) E is a Σ-structure S with a morphism α : S E → S such that α ◦ ζ S = id S Lemma ( · ) E -algebras are models for CB Σ /E .
Example � � Take Σ = ∅ , E = ≤
Example � � Take Σ = ∅ , E = , Mod CB Σ /E is the category of ≤ non-empty sets.
Example � � Take Σ = ∅ , E = , Mod CB Σ /E is the category of ≤ non-empty sets. • S E = S + 1
Example � � Take Σ = ∅ , E = , Mod CB Σ /E is the category of ≤ non-empty sets. • S E = S + 1 • ( · ) E –algebras are pointed sets
Example � � Take Σ = ∅ , E = , Mod CB Σ /E is the category of ≤ non-empty sets. • S E = S + 1 • ( · ) E –algebras are pointed sets The category of ( · ) E –algebras is better behaved than Mod CB Σ /E .
The free algebra ( · ) E -Alg Mod CB Σ U Mod CB Σ /E
The free algebra ( · ) E -Alg Mod CB Σ ⊥ U Mod CB Σ /E
The free algebra ( · ) Eω ( · ) E -Alg Mod CB Σ ⊥ U Mod CB Σ /E
The free algebra ( · ) Eω ( · ) E -Alg Mod CB Σ ⊥ U Mod CB Σ /E S S E S E 2 · · · S E ω
The free algebra ( · ) Eω ( · ) E -Alg Mod CB Σ ⊥ U Mod CB Σ /E S S E S E 2 · · · S E ω
From semantics to syntax Theorem S ≤ R in CB Σ if and only if there is α : U R → U S such that U R ι R ω R n m α ι S ω S U S
From semantics to syntax Theorem S ≤ R in CB Σ /E if and only if there is α : ( U R ) E ω → ( U S ) E ω such that ( U R ) E ω n m α ( U S ) E ω
From semantics to syntax Proof sketch. ( U R ) E ω n m α ( U S ) E ω
From semantics to syntax Proof sketch. U R n m α ( U S ) E ω
From semantics to syntax Proof sketch. U R n m α ( U S ) E ω • U R is compact
From semantics to syntax Proof sketch. U R n m α ( U S ) E k • U R is compact
From semantics to syntax Proof sketch. U R n m α ( U S ) E k • U R is compact • n → ( U S ) E i ← m correspond to string diagrams S i
From semantics to syntax Proof sketch. U R n m α ( U S ) E k • U R is compact • n → ( U S ) E i ← m correspond to string diagrams S i • S k ≤ R in CB Σ , hence in CB Σ /E
From semantics to syntax Proof sketch. U R n m α ( U S ) E k • U R is compact • n → ( U S ) E i ← m correspond to string diagrams S i • S k ≤ R in CB Σ , hence in CB Σ /E • S i +1 is obtained by blindly applying all axioms to S i
From semantics to syntax Proof sketch. U R n m α ( U S ) E k • U R is compact • n → ( U S ) E i ← m correspond to string diagrams S i • S k ≤ R in CB Σ , hence in CB Σ /E • S i +1 is obtained by blindly applying all axioms to S i • S = S 0 ≤ S 1 ≤ S 2 ≤ · · · ≤ S k ≤ R in CB Σ /E
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