�� The game plan Németi and Sain defined, for each composition � X = HS Σ , HS Σ + , etc., a class of cones, M � X ⊆ SubCat ( Cone ( C )) . For instance, M HSP consists of those cones such that B is epi- projective. ����������� B C Some Co-Birkhoff-Type Theorems – p.9/25
The game plan Németi and Sain defined, for each composition � X = HS Σ , HS Σ + , etc., a class of cones, M � X ⊆ SubCat ( Cone ( C )) . Next, we define, for each � X , an operator � SubCat ( Cocone ( C )) . X : SubCat ( C ) K � K � X V represents the M � X -theory of V . That is, X V = { c ∈ M � X | V ⊆ Inj ( c ) } . K � Some Co-Birkhoff-Type Theorems – p.9/25
The game plan Next, we define, for each � X , an operator � SubCat ( Cocone ( C )) . X : SubCat ( C ) K � K � X V represents the M � X -theory of V . That is, X V = { c ∈ M � X | V ⊆ Inj ( c ) } . K � Finally, we prove a whole slew of theorems of the form X V ) = � X V , Inj ( M � greatly impressing everybody. Some Co-Birkhoff-Type Theorems – p.9/25
The game plan Next, we define, for each � X , an operator � SubCat ( Cocone ( C )) . X : SubCat ( C ) K � K � X V represents the M � X -theory of V . That is, X V = { c ∈ M � X | V ⊆ Inj ( c ) } . K � Finally, we prove a whole slew of theorems of the form X V ) = � X V , Inj ( M � greatly impressing everybody. It’s been done. Some Co-Birkhoff-Type Theorems – p.9/25
The game plan Finally, we prove a whole slew of theorems of the form X V ) = � X V , Inj ( M � greatly impressing everybody. It’s been done. Plan B: Turn all the arrows around and see what you get. Hope someone is mildly interested. Some Co-Birkhoff-Type Theorems – p.9/25
Outline I. Some Birkhoff-type theorems II. Equations and injectivity III. Injectivity and cones IV. The abstract setting V. Projectivity and cocones VI. A cornucopia of closure operators VII. A slew of theorems VIII. Categories of coalgebras IX. Classes of automata X. Behavioral classes Some Co-Birkhoff-Type Theorems – p.10/25
Outline I. Some Birkhoff-type theorems II. Equations and injectivity III. Injectivity and cones IV. The abstract setting V. Projectivity and cocones VI. A cornucopia of closure operators VII. A slew of theorems VIII. Categories of coalgebras IX. Classes of automata X. Behavioral classes Some Co-Birkhoff-Type Theorems – p.10/25
The abstract setting We assume the following: • C has all coproducts. Some Co-Birkhoff-Type Theorems – p.11/25
The abstract setting We assume the following: • C has all coproducts. • C has a factorization system �H , S� . This assumption appeared earlier in our use of epis. Implic- itly, we were using the factorization system � Epi , Mono � in Set . Some Co-Birkhoff-Type Theorems – p.11/25
The abstract setting We assume the following: • C has all coproducts. • C has a factorization system �H , S� . • C is S -well-powered A category is S -well-powered if for each C ∈ C , the collection { j ∈ S | cod ( j ) = C } / ∼ = is a set. Some Co-Birkhoff-Type Theorems – p.11/25
� � � � The abstract setting We assume the following: • C has all coproducts. • C has a factorization system �H , S� . • C is S -well-powered • C has enough S -injectives. Recall an object C is S -injective if, for all A � � B in C , and all A � C , there is an extension B � C . ∀ A C ∃ B Some Co-Birkhoff-Type Theorems – p.11/25
The abstract setting We assume the following: • C has all coproducts. • C has a factorization system �H , S� . • C is S -well-powered • C has enough S -injectives. In Set , every non-empty set is Mono -injective. Some Co-Birkhoff-Type Theorems – p.11/25
The abstract setting We assume the following: • C has all coproducts. • C has a factorization system �H , S� . • C is S -well-powered • C has enough S -injectives. C has enough injectives if for every A in C , there is an S - injective C and a S -morphism A � � C . Some Co-Birkhoff-Type Theorems – p.11/25
� Projectivity and cocones A discrete cone is a pair c = � B, { f i : B � C i } i ∈ I � . B � � � ������ . . . � � � � C j C i Some Co-Birkhoff-Type Theorems – p.12/25
� Projectivity and cocones A discrete cocone is a pair c = � B, { f i : C i � B } i ∈ I � . B � ������ � . . . � � � � � C j C i Some Co-Birkhoff-Type Theorems – p.12/25
� � Projectivity and cocones A discrete cocone is a pair c = � B, { f i : C i � B } i ∈ I � . B � ������ � . . . � � � � � C j C i An object A is injective with respect to c if every B � A factors through some f i . � A ∀ B ∃ f i � ∃ C i Some Co-Birkhoff-Type Theorems – p.12/25
� � � � Projectivity and cocones A discrete cocone is a pair c = � B, { f i : C i � B } i ∈ I � . B � ������ � . . . � � � � � C j C i An object A is projective with respect to c if every A � B (co-)factors through some f i . ∀ B A ∃ f i ∃ C i Some Co-Birkhoff-Type Theorems – p.12/25
The cocone classes M � X Define • � ���� . . • M S cocones with injective vertex . � ���� • Some Co-Birkhoff-Type Theorems – p.13/25
� � The cocone classes M � X Define • � ���� . . • M S cocones with injective vertex . � ���� • • � ���� . • . cocones with S -morphisms M H . � ���� • Some Co-Birkhoff-Type Theorems – p.13/25
� � � The cocone classes M � X Define • � ���� . . • M S cocones with injective vertex . � ���� • • � ���� . • . cocones with S -morphisms M H . � ���� • • • M Σ cocones with one arrow Some Co-Birkhoff-Type Theorems – p.13/25
� � � � The cocone classes M � X Define • � ���� . . • M S cocones with injective vertex . � ���� • • � ���� . • . cocones with S -morphisms M H . � ���� • • • M Σ cocones with one arrow • • M Σ + cocones with 0 or 1 arrow • Some Co-Birkhoff-Type Theorems – p.13/25
� � � � The cocone classes M � X Define • � ���� . . • M S cocones with injective vertex . � ���� • • � ���� . • . cocones with S -morphisms M H . � ���� • • • M Σ cocones with one arrow • • M Σ + cocones with 0 or 1 arrow • For composites � X = X 1 . . . X n , X = M X 1 ∩ . . . ∩ M X n . M � Some Co-Birkhoff-Type Theorems – p.13/25
� � � � The cocone classes M � X Define • � ���� . . • M S cocones with injective vertex . � ���� • • � ���� . • . cocones with S -morphisms M H . � ���� • • • M Σ cocones with one arrow • • M Σ + cocones with 0 or 1 arrow • M � X can be considered the language of the theory at hand. Some Co-Birkhoff-Type Theorems – p.13/25
Outline I. Some Birkhoff-type theorems II. Equations and injectivity III. Injectivity and cones IV. The abstract setting V. Projectivity and cocones VI. A cornucopia of closure operators VII. A slew of theorems VIII. Categories of coalgebras IX. Classes of automata X. Behavioral classes Some Co-Birkhoff-Type Theorems – p.14/25
Outline I. Some Birkhoff-type theorems II. Equations and injectivity III. Injectivity and cones IV. The abstract setting V. Projectivity and cocones VI. A cornucopia of closure operators VII. A slew of theorems VIII. Categories of coalgebras IX. Classes of automata X. Behavioral classes Some Co-Birkhoff-Type Theorems – p.14/25
A cornucopia of closure operators We define the following operators � SubCat ( C ) . SubCat ( C ) Some Co-Birkhoff-Type Theorems – p.15/25
A cornucopia of closure operators We define the following operators � SubCat ( C ) . SubCat ( C ) H V = { B ∈ C | ∃ V ∋ C � � B } Note: The symbols H and S do double duty, as classes of arrows and also as closure operators. Some Co-Birkhoff-Type Theorems – p.15/25
A cornucopia of closure operators We define the following operators � SubCat ( C ) . SubCat ( C ) H V = { B ∈ C | ∃ V ∋ C � � B } S V = { B ∈ C | ∃ B � � C ∈ V } Some Co-Birkhoff-Type Theorems – p.15/25
A cornucopia of closure operators We define the following operators � SubCat ( C ) . SubCat ( C ) H V = { B ∈ C | ∃ V ∋ C � � B } S V = { B ∈ C | ∃ B � � C ∈ V } � Σ V = { B ∈ C | ∃{ A i } i ∈ I ⊆ V . B ∼ A i } = Some Co-Birkhoff-Type Theorems – p.15/25
A cornucopia of closure operators We define the following operators � SubCat ( C ) . SubCat ( C ) H V = { B ∈ C | ∃ V ∋ C � � B } S V = { B ∈ C | ∃ B � � C ∈ V } � Σ V = { B ∈ C | ∃{ A i } i ∈ I ⊆ V . B ∼ A i } = � Σ + V = { B ∈ C | ∃{ A i } i ∈ I ⊆ V . B ∼ A i , I � = ∅} = Some Co-Birkhoff-Type Theorems – p.15/25
A slew of theorems X be a composite of S , H , Σ and Σ + such that Let � • the operators occur in the order above; Some Co-Birkhoff-Type Theorems – p.16/25
A slew of theorems X be a composite of S , H , Σ and Σ + such that Let � • the operators occur in the order above; • H occurs in � X . Some Co-Birkhoff-Type Theorems – p.16/25
A slew of theorems X be a composite of S , H , Σ and Σ + such that Let � • the operators occur in the order above; • H occurs in � X . I.e., let � X be one of H , H Σ , H Σ + , SH , SH Σ , SH Σ + Some Co-Birkhoff-Type Theorems – p.16/25
A slew of theorems X be a composite of S , H , Σ and Σ + such that Let � • the operators occur in the order above; • H occurs in � X . I.e., let � X be one of H , H Σ , H Σ + , SH , SH Σ , SH Σ + X V ) = � X V Proj ( K � X V = { c ∈ M V | V ⊆ Proj ( c ) } Here, K � Some Co-Birkhoff-Type Theorems – p.16/25
A slew of theorems X be a composite of S , H , Σ and Σ + such that Let � • the operators occur in the order above; • H occurs in � X . I.e., let � X be one of H , H Σ , H Σ + , SH , SH Σ , SH Σ + X V ) = � X V Proj ( K � Compare: Mod Th V = HSP V (Birkhoff) Some Co-Birkhoff-Type Theorems – p.16/25
Outline I. Some Birkhoff-type theorems II. Equations and injectivity III. Injectivity and cones IV. The abstract setting V. Projectivity and cocones VI. A cornucopia of closure operators VII. A slew of theorems VIII. Categories of coalgebras IX. Classes of automata X. Behavioral classes Some Co-Birkhoff-Type Theorems – p.17/25
Outline I. Some Birkhoff-type theorems II. Equations and injectivity III. Injectivity and cones IV. The abstract setting V. Projectivity and cocones VI. A cornucopia of closure operators VII. A slew of theorems VIII. Categories of coalgebras IX. Classes of automata X. Behavioral classes Some Co-Birkhoff-Type Theorems – p.17/25
Categories of coalgebras Let C satisfy our previous requirements and Γ: C � C be given. Let U : C Γ � C be the forgetful functor. • U creates coproducts, so C Γ has them. Some Co-Birkhoff-Type Theorems – p.18/25
Categories of coalgebras Let C satisfy our previous requirements and Γ: C � C be given. Let U : C Γ � C be the forgetful functor. • U creates coproducts, so C Γ has them. • If Γ preserves S -morphisms, then � U − 1 H , U − 1 S� form a factorization system for C Γ . Some Co-Birkhoff-Type Theorems – p.18/25
Categories of coalgebras Let C satisfy our previous requirements and Γ: C � C be given. Let U : C Γ � C be the forgetful functor. • U creates coproducts, so C Γ has them. • If Γ preserves S -morphisms, then � U − 1 H , U − 1 S� form a factorization system for C Γ . • C Γ is U − 1 S -well-powered. Some Co-Birkhoff-Type Theorems – p.18/25
Categories of coalgebras Let C satisfy our previous requirements and Γ: C � C be given. Let U : C Γ � C be the forgetful functor. • U creates coproducts, so C Γ has them. • If Γ preserves S -morphisms, then � U − 1 H , U − 1 S� form a factorization system for C Γ . • C Γ is U − 1 S -well-powered. • If U ⊣ H , then C Γ has enough (cofree) injectives. Some Co-Birkhoff-Type Theorems – p.18/25
Categories of coalgebras Let C satisfy our previous requirements and Γ: C � C be given. Let U : C Γ � C be the forgetful functor. • U creates coproducts, so C Γ has them. • If Γ preserves S -morphisms, then � U − 1 H , U − 1 S� form a factorization system for C Γ . • C Γ is U − 1 S -well-powered. • If U ⊣ H , then C Γ has enough (cofree) injectives. Thus, if Γ preserves S -morphisms and C Γ has cofree coal- gebras, then C Γ satisfies our abstract setting. Some Co-Birkhoff-Type Theorems – p.18/25
Categories of coalgebras Let C satisfy our previous requirements and Γ: C � C be given. Let U : C Γ � C be the forgetful functor. • U creates coproducts, so C Γ has them. • If Γ preserves S -morphisms, then � U − 1 H , U − 1 S� form a factorization system for C Γ . • C Γ is U − 1 S -well-powered. • If U ⊣ H , then C Γ has enough (cofree) injectives. Moreover, we may restrict our attention to cocones with cofree vertices, in the case that � X contains S . Some Co-Birkhoff-Type Theorems – p.18/25
Deterministic automata and languages Fix an alphabet I . Let Γ: Set � Set be the functor X �→ 2 × X I . Some Co-Birkhoff-Type Theorems – p.19/25
Deterministic automata and languages Fix an alphabet I . Let Γ: Set � Set be the functor X �→ 2 × X I . A Γ -coalgebra � A, α � is an automaton accepting input from I and outputting either 0 or 1 , where out α ( a ) = π 1 ◦ α ( a ) trans α ( a ) = π 2 ◦ α ( a ) Some Co-Birkhoff-Type Theorems – p.19/25
Deterministic automata and languages Let σ ∈ I <ω and define eval α : A × I <ω � A by eval α ( a, ()) = a, eval α ( a, σ ∗ i ) = trans α ( eval α ( a, σ ))( i ) . eval α ( a, σ ) is the final state of the calculation beginning in a with input σ . Some Co-Birkhoff-Type Theorems – p.19/25
Deterministic automata and languages Define � P ( I <ω ) acc α : A by acc α ( a ) = { σ ∈ I <ω | out α ◦ eval α ( a, σ ) = 1 } . acc α ( a ) is the set of all words accepted by state a . Some Co-Birkhoff-Type Theorems – p.19/25
Some classes of automata Fix a language L ⊆ I <ω . V {� A, α � | . . . } V closed under ∀ a ∈ A . acc ( a ) = L SH Σ Some Co-Birkhoff-Type Theorems – p.20/25
Some classes of automata Fix a language L ⊆ I <ω . V {� A, α � | . . . } V closed under ∀ a ∈ A . acc ( a ) = L SH Σ A � = ∅ ⇒ ∃ a ∈ A . acc ( a ) = L H Σ Some Co-Birkhoff-Type Theorems – p.20/25
Some classes of automata Fix a language L ⊆ I <ω . V {� A, α � | . . . } V closed under ∀ a ∈ A . acc ( a ) = L SH Σ A � = ∅ ⇒ ∃ a ∈ A . acc ( a ) = L H Σ H Σ + ∃ a ∈ A . acc ( a ) = L Some Co-Birkhoff-Type Theorems – p.20/25
Some classes of automata Fix a language L ⊆ I <ω . V {� A, α � | . . . } V closed under ∀ a ∈ A . acc ( a ) = L SH Σ A � = ∅ ⇒ ∃ a ∈ A . acc ( a ) = L H Σ H Σ + ∃ a ∈ A . acc ( a ) = L ∃ ! a ∈ A . acc ( a ) = L H Some Co-Birkhoff-Type Theorems – p.20/25
Some classes of automata Fix a language L ⊆ I <ω . V {� A, α � | . . . } V closed under ∀ a ∈ A . acc ( a ) = L SH Σ A � = ∅ ⇒ ∃ a ∈ A . acc ( a ) = L H Σ H Σ + ∃ a ∈ A . acc ( a ) = L ∃ ! a ∈ A . acc ( a ) = L H ∗ � a ∃ ! a ∈ A . acc ( a ) = L and ∀ b ∈ A . b SH Some Co-Birkhoff-Type Theorems – p.20/25
Some classes of automata Fix a language L ⊆ I <ω . V {� A, α � | . . . } V closed under ∀ a ∈ A . acc ( a ) = L SH Σ A � = ∅ ⇒ ∃ a ∈ A . acc ( a ) = L H Σ H Σ + ∃ a ∈ A . acc ( a ) = L ∃ ! a ∈ A . acc ( a ) = L H ∗ � a ∃ ! a ∈ A . acc ( a ) = L and ∀ b ∈ A . b SH In fact, there’s a “hidden” closure operator here. Some Co-Birkhoff-Type Theorems – p.20/25
Some classes of automata Fix a language L ⊆ I <ω . V {� A, α � | . . . } V closed under H − SH Σ ∀ a ∈ A . acc ( a ) = L H − H Σ A � = ∅ ⇒ ∃ a ∈ A . acc ( a ) = L H − H Σ + ∃ a ∈ A . acc ( a ) = L ∃ ! a ∈ A . acc ( a ) = L H ∗ � a ∃ ! a ∈ A . acc ( a ) = L and ∀ b ∈ A . b SH The H − operator closes a class of coalgebras under domains of H -morphisms. Some Co-Birkhoff-Type Theorems – p.20/25
Outline I. Some Birkhoff-type theorems II. Equations and injectivity III. Injectivity and cones IV. The abstract setting V. Projectivity and cocones VI. A cornucopia of closure operators VII. A slew of theorems VIII. Categories of coalgebras IX. Classes of automata X. Behavioral classes Some Co-Birkhoff-Type Theorems – p.21/25
Outline I. Some Birkhoff-type theorems II. Equations and injectivity III. Injectivity and cones IV. The abstract setting V. Projectivity and cocones VI. A cornucopia of closure operators VII. A slew of theorems VIII. Categories of coalgebras IX. Classes of automata X. Behavioral classes Some Co-Birkhoff-Type Theorems – p.21/25
Behavioral classes Consider the following operators. H − V = { B ∈ C | ∃ B � � A ∈ V } Some Co-Birkhoff-Type Theorems – p.22/25
� � Behavioral classes Consider the following operators. H − V = { B ∈ C | ∃ B � � A ∈ V } � � A ∈ V } B V = { B ∈ C | ∃ relation B R Here, a relation is an S -morphism R � � B × A (we assume that C has finite products). Some Co-Birkhoff-Type Theorems – p.22/25
� � � � Behavioral classes Consider the following operators. H − V = { B ∈ C | ∃ B � � A ∈ V } � � A ∈ V } B V = { B ∈ C | ∃ relation B R � � A ∈ V } Q V = { B ∈ C | ∃ B C Some Co-Birkhoff-Type Theorems – p.22/25
� � � � Behavioral classes H − V = { B ∈ C | ∃ B � � A ∈ V } � � A ∈ V } B V = { B ∈ C | ∃ relation B R � � A ∈ V } Q V = { B ∈ C | ∃ B C H − H V = BB V = QQ V . Some Co-Birkhoff-Type Theorems – p.22/25
� � � � Behavioral classes H − V = { B ∈ C | ∃ B � � A ∈ V } � � A ∈ V } B V = { B ∈ C | ∃ relation B R � � A ∈ V } Q V = { B ∈ C | ∃ B C H − H V = BB V = QQ V . If, in E , epis are stable under pullback, then also H − H V = B V = Q V . Some Co-Birkhoff-Type Theorems – p.22/25
� � � � The cocone classes M � X Recall • � ���� . . • M S cocones with injective vertex . � ���� • • � ���� . • . cocones with S -morphisms M H . � ���� • • • M Σ cocones with one arrow • • M Σ + cocones with 0 or 1 arrow • Some Co-Birkhoff-Type Theorems – p.23/25
� � � � � � The cocone classes M � X • � ���� . • . M S cocones with injective vertex . � ���� • • � ���� . • . cocones with S -morphisms M H . � ���� • • • M Σ cocones with one arrow • • M Σ + cocones with 0 or 1 arrow • • � ���� . cocones with vertex ≤ 1 • . M H − 1 . � ���� • Some Co-Birkhoff-Type Theorems – p.23/25
� � � � � � The cocone classes M � X • � ���� . • . M S cocones with injective vertex . � ���� • • � ���� . • . cocones with S -morphisms M H . � ���� • • • M Σ cocones with one arrow • • M Σ + cocones with 0 or 1 arrow • • � ���� . cocones with vertex ≤ 1 • . M H − 1 . � ���� • As before, for composites � X = X 1 . . . X n , X = M X 1 ∩ . . . ∩ M X n . M � Some Co-Birkhoff-Type Theorems – p.23/25
� � � � � � The cocone classes M � X • � ���� . • . M S cocones with injective vertex . � ���� • • � ���� . • . cocones with S -morphisms M H . � ���� • • • M Σ cocones with one arrow • • M Σ + cocones with 0 or 1 arrow • • � ���� . cocones with vertex ≤ 1 • . M H − 1 . � ���� • X | V ⊆ Proj ( � X V = { c ∈ M � X ) } . Also as before, K � Some Co-Birkhoff-Type Theorems – p.23/25
An augmented slew X be a composite of H − , S , H , Σ and Σ + such that Let � • the operators occur in the order above; • H occurs in � X . X V ) = � X V Proj ( K � Some Co-Birkhoff-Type Theorems – p.24/25
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